Spherical and Spheroidal Harmonics: Examples and Computations
Thesis
Presented in Partial Fulfillment of the Requirements for the Degree Master of Science in the Graduate School of The Ohio State University
By
Lin Zhao, B.S., M.S.
Graduate Program in Mathematics
The Ohio State University
2017
Thesis Committee:
Ghaith A. Hiary, Advisor Roman Holowinsky © Copyright by
Lin Zhao
2017 Abstract
This thesis introduces the theory of spherical and spheroidal harmonics, with em-
phasis on algorithms, computations, and explicit examples. In particular, we collect
in one place several viewpoints on spherical and spheroidal harmonics, and clarify how
the various viewpoints are related and how to transition between them. The ultimate
goal is to be able to apply the spheroidal harmonics in calculating the spacing dis-
tributions of eigenvalues of the Gaussian unitary ensemble (GUE) random matrices.
These eigenvalue statistics are conjectured to model corresponding statistics for zeros
of the Riemann zeta function. New results are presented for comparing statistics of
zeros and eigenvalues.
ii Acknowledgments
First and foremost, I am deeply indebted to my advisor, Prof. Ghaith A. Hiary
for his guidance over my master’s study in Department of Mathematics. This thesis would not have been possible without his consistent support, great patience, and
continual encouragement. I feel very fortunate to be introduced to this interesting
topic and it has been a wonderful journey delving into it. I had the opportunity
to see broad connections to various aspects of mathematics, as well as interesting
applications in many other disciplines. The many discussions with Prof. Hiary have
been tremendously helpful. I benefited a lot from his extremely organized and rigorous way of thinking, as well as his meticulousness. I will also be amused by his jokes about
mathematicians and physicists from time to time.
I am also very thankful to Prof. Roman Holowinsky for serving on my committees
and for trying to accommodate my schedule at his best. I owe many thanks to Mr.
Roman Nitze, who has been remarkably patient and kind to help me on every step
of applying the dual degree. I am also very grateful for all the professors who taught
me from the math department. Their passion and dedication to the classes greatly
inspired me to think deeply, thoroughly, and creatively. I am privileged to be able to
learn from them.
Finally, I would like to thank my family for their love, care, encouragement, and
support over the years. This thesis is dedicated to them.
iii Vita
2010 ...... B.S., Control Science and Engineering, Harbin Institute of Technology, China. 2012 ...... M.S., Control Science and Engineering, Harbin Institute of Technology, China. 2012–present ...... Graduate Research Associate, Electrical and Computer Engineering, The Ohio State University 2016-present ...... M.S. Student, Mathematics, The Ohio State University
Publications
Research Publications
L. Zhao, W. Zhang, J. Hu, A. Abate, and C.J. Tomlin. “On the optimal solu- tions of the infinite-horizon linear sensor scheduling problem”, IEEE Transactions on Automatic Control, 59(10):2825-2830, March 2014.
L. Zhao and W. Zhang. A unified stochastic hybrid system approach to aggregate modeling of responsive loads, IEEE Transaction on Automatic Control, 2018, to be published.
L. Zhao, W. Zhang, H. Hao, and K. Kalsi. A geometric approach to aggregate flexibility modeling of thermostatically controlled loads, IEEE Transactions on Power Systems, 32(6):4721-4731, November 2017.
L. Zhao, H. Hao, and W. Zhang. Extracting flexibility of heterogeneous deferrable loads via polytopic projection approximation, The 55th IEEE Conference on Decision and Control, Las Vegas, USA, 2016.
iv L. Zhao, and W. Zhang. A geometric approach to virtual battery modeling of thermostatically controlled loads, The American Control Conference, Boston, MA, 2016.
L. Zhao and W. Zhang. A unified stochastic hybrid system approach to aggregated load modeling for demand response, The 54th IEEE Conf. on Decision and Control, Osaka, Japan, 2015.
S. Li, W. Zhang, L. Zhao, J. Lian and K. Kalsi. On social optima of non-cooperative mean field games, The 55th IEEE Conference on Decision and Control, Las Vegas, USA, 2016.
Fields of Study
Major Field: Mathematics
v Table of Contents
Page
Abstract ...... ii
Acknowledgments ...... iii
Vita ...... iv
List of Tables ...... viii
List of Figures ...... ix
1. Introduction ...... 1
2. Harmonic Analysis on the Sphere Sn−1 in Euclidean Coordinates .... 5
2.1 Laplacian and Invariant Integrals on Spheres ...... 7 2.2 Eigenfunctions of the Spherical Laplacian ...... 10 2.3 Decomposition of Homogeneous Polynomials ...... 12 2.4 L2 Spectral Decomposition on Spheres ...... 15
3. Spherical Harmonics Basis in Euclidean Coordinates ...... 18
3.1 Kelvin Transform ...... 18
3.2 Projection of Homogeneous Polynomials to Hd spaces ...... 23 3.3 Examples ...... 25
4. Harmonic Analysis on the sphere Sn−1 in Spherical Coordinates ..... 29
4.1 Spherical Harmonics (n = 3) ...... 29 4.1.1 Legendre Equation and Polynomials ...... 31 4.1.2 Rodrigues’ Formula ...... 34
vi 4.1.3 General Solution of Spherical Laplace Equation ...... 35 4.2 Hyperspherical Harmonics (n > 3) ...... 41 4.2.1 Gegenbauer polynomials ...... 41 4.2.2 Hyperspherical Harmonics ...... 43
5. Harmonics Analysis on the Spheroid ...... 45
5.1 Prolate Spheroidal Coordinates ...... 45 5.2 Spheroidal Harmonics ...... 48 5.3 Series Expansion Representation of Spheroidal Wave Functions .. 50
6. Application Example ...... 58
6.1 Linear Prolate Function ...... 59 6.2 GUE Eigenvalue Spacing Distribution ...... 61 6.3 Simulation Results ...... 63
Bibliography ...... 69
Appendices 75
A. Harmonic Analysis on the Flat Space Rm ...... 75
B. Harmonic Analysis on the Torus Rm/Zm ...... 80
C. Recurrence Formulas ...... 82
D. Mathematica Code ...... 85
vii List of Tables
Table Page
3.1 Basis of Hd for different dimensions ...... 26
viii List of Figures
Figure Page
2.1 Spherical coordinate systems (left) and Mathematica illustration with ∈ 8 ϕ [0, 5 π] (right) ...... 6
4.1 Legendre Polynomials ...... 34
5 5 3 4.2 Mathematica density plot for ReY10(θ, ϕ) (left) and ReY10(θ, ϕ)+ReY5 (θ, ϕ) (right) ...... 40
4.3 Gegenbauer polynomials ...... 43
5.1 Prolate spheroidal coordinate systems (left) and Mathematica illustra- ∈ 8 tion with ϕ [0, 5 π] (right) ...... 47
∈ 8 5.2 Constant coordinate curves of equation (5.3) for ϕ [0, 5 π] with dif- ferent values of ξ = cosh µ, where µ ∈ {0.4, 0.6, 0.9, 1.2, 1.5}...... 47
5.3 Examples of prolate spheroidal wave functions: angular functions (left) and radial functions (right) ...... 55
¯ 5.4 Renormalized angular functions Sml(5, η) ...... 56
5.5 R10,10(5, ξ) obtained from Mathematica built-in function (left) and Graham’s modified function (right) ...... 56
5.6 Mathematica density plot for R3,4(3, ξ)S3,4(3, η) cos 3ϕ on the left and R3,4(3, ξ)S3,4(3, η) cos 3ϕ + R2,4(3, ξ)S2,4(3, η) cos 2ϕ on the right ... 57
6.1 Probability density of the nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). 64
ix 6.2 Probability density of the next-nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right)...... 65
6.3 Probability density of the 3rd nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right)...... 67
6.4 Probability density of the 4th nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right)...... 67
6.5 Difference of the probability densities between the empirical dataand
GUE prediction. Left: k = 0, N0 = 4 with different number of Rie- mann zeta zeros; Right: k = 1 with N0 = 4 of the right figure in Fig. 6.2...... 68
6.6 Difference of the probability densities between the empirical dataand
GUE prediction. Left: k = 2 with N0 = 5, 6; Right: k = 3 with N0 = 5, 6, 7...... 68
x Chapter 1: Introduction
Harmonic analysis on the sphere is perhaps less familiar than the well-studied
examples of harmonic analysis on the flat space Rn and the torus Rm/Zm. This is
partly because Sn−1 does not have a natural group structure, in contrast to Rn or
Rm/Zm which are abelian additive groups. Still, the sphere Sn−1, like the torus, is a
compact space with a natural group action on it. Specifically, it is acted on transitively
by the rotation group SO(n), which simplifies some details of the analysis and building
of theory.
One feature of spherical harmonics is that they are computationally more involved
than harmonics on the circle. Therefore, we first take a somewhat abstract approach
in Chapter 2, which will hopefully facilitate the conceptual understanding. Various
results are established under the familiar Euclidean coordinates. The spherical har-
monics arise as the eigenfunctions of the spherical Laplacian ∆S. It is shown that the
only eigenvalues of ∆S are −d(n + d − 2) (Theorem 1), and the only eigenfunctions
of ∆S are those homogeneous harmonic polynomials restricted to the sphere (Theo- rem 3). The highlight of this chapter is Theorem 2, which proves the decomposition
of the space L2(Sn−1) of square integrable functions on Sn−1 into an infinite direct
n−1 sum of the spaces Hd(S ) of homogeneous harmonic polynomials of degree d.
1 In Chapter 3, we describe an algorithm for generating a basis for the spherical
harmonics in Euclidean coordinates (Theorem 6). The algorithms use the Kelvin
Transform and is quite efficient in practice. Specifically, the algorithm is basedon the projection of homogeneous polynomials onto the subspaces of homogeneous har- monic polynomials of a given degree. One feature of this algorithm is that it is based on the differentiation operation instead of the integration, and therefore, it canbeex- ecuted very efficiently. However, in general, the generated basis are not automatically orthogonal with respect to the usual L2 inner product on Sn−1 under the spherical
area measure. We provide examples of the generated basis using the Kelvin Transform
and the corresponding orthonormalized basis using the Gram–Schmidt procedure.
In Chapter 4, we switch from Euclidean to spherical coordinates, which leads to
“closed-form”, though somewhat complicated, expressions for spherical harmonics in
terms of the associated Legendre functions (on 3-dimensional spheres) and Gegen-
bauer polynomials (on n > 3 dimensional spheres). The associated Legendre func-
tions are usually defined succinctly through the Legendre polynomials, which gives
the Zonal function solutions to the eigenfunctions of the Laplacian equation. We
prove many useful properties of these special functions with the goal of building up
a foundation when we later discuss the spheroidal harmonics. The generalization of
the spherical harmonics of S2 to higher dimensions is sometimes referred to as the
hyperspherical harmonics. Many theorems for the spherical harmonics have counter-
parts in the hyperspherical case, and are in fact specialization of it. For example,
the associated Legendre functions that appear in the S2 case are a special case of
the Gegenbauer polynomials. We briefly overview several facts of the hyperspherical
2 harmonics at the end of this chapter, but we don’t provide complete proofs except in the spherical case.
In Chapter 5, we switch the space itself from spheres to 3-dimensional space under the prolate spheroidal coordinates where the analysis is being done. The spheroidal harmonics arise as the solutions of the Helmholtz equation ∆f + κ2f = 0 for some constant κ (referred to as the wave number). Under the spheroidal coordinates, it can be separated into three components consisting of the radial function, angular func- tion (the first two are called spheroidal wave functions), and the azimuthal (wave) function. We derive the differential equations satisfied by the spheroidal wave func- tions and show that the spherical harmonics can be thought of as a limiting case of the spheroidal harmonics. This also naturally leads to the discussion of the series expansions for radial functions and angular functions using the associated Legendre functions and spherical Bessel functions, respectively. This is intended to provide insights of the computational aspect of the spheroidal wave functions.
We conclude in Chapter 6 with an example computation motivated by connections between the Riemann zeta function and the Gaussian unitary ensemble (GUE), which uses spheroidal wave functions to calculate (up to the 4-th nearest neighbor) spacing distributions of the eigenvalues of GUE matrices. Moreover, we compare the results to numerical data for spacings of zeros of the Riemann zeta function. Our results demonstrate good agreement between the behavior of spacings of zeta zeros and GUE predictions. These might provide further supportive evidence for the GUE conjecture for the Riemann zeta function.
Throughout, we provide explicit examples whenever appropriate. Most of our examples and computations were done using the Mathematica software. The sample
3 Mathematica code for the computation in Chapter 6 is included in Appendix D. We also include in appendices a brief overview of harmonic analysis on Rn and Rm/Zm, which might be useful when carrying out comparisons between the various settings. In addition, some useful recurrence formulas for the Legendre polynomials and associated
Legendre functions are proved in Appendix C.
4 Chapter 2: Harmonic Analysis on the Sphere Sn−1 in Euclidean Coordinates
n−1 { ∈ Rn| ∥ ∥ } The harmonic analysis on the sphere S = x x 2 = 1 is more compli- cated than on the flat space Rn or torus Rn/Zn, by the fact that it is not a group
in a natural way. Let O(n) be the orthogonal group defined by O(n) = {U ∈
n×n T R |U U = In}, where In denotes the n × n identity matrix. Then the special
orthogonal group SO(n) is the subgroup of O(n) consisting of matrices with de-
terminant 1. The sphere is compact and is acted on transitively by the SO(n). The
left figure in Fig. 2.1 illustrates the group action of SO(n) on Sn−1 for n = 3, where the points (0, 0, 1) rotates to (sin θ cos ϕ, sin θ sin ϕ, cos θ) under the group action of
multiplication by the matrix
cos θ 0 − sin θ cos ϕ sin ϕ 0 k = 0 1 0 − sin ϕ cos ϕ 0 ∈ SO(3). sin θ 0 cos θ 0 0 1
For n = 3, the spherical coordinates are simply given by
x = r sin θ cos ϕ,
y = r sin θ sin ϕ,
z = r cos θ.
5 𝑧𝑧
𝜃𝜃
𝑦𝑦 𝜙𝜙
𝑥𝑥
Figure 2.1: Spherical coordinate systems (left) and Mathematica illustration with ∈ 8 ϕ [0, 5 π] (right)
The right figure in Fig. 2.1 illustrates some surfaces defined by the above coordinate
system when one of the coordinates is fixed. For example, if the radius (in the above
figure r = 1) is fixed, then the spherical coordinate system is defined on a unitsphere.
π 3π If the altitude θ is fixed (in the above figure, θ = 4 and 4 ), then the coordinate ∈ 8 system is defined on cones. We plot these surfaces with ϕ [0, 5 π] to give a better view of the inside. In addition, if the azimuth angle ϕ is fixed, then the coordinate
system is defined on a half plane (not shown in the figure).
Note that the sphere Sn−1 can be identified with the quotient space SO(n −
1)\SO(n). This can be shown by the orbit-stabilizer formula (see, for example,
Dummit and Foote’s textbook [16, Page 114]). Fix a point in Sn−1, for example,
en = (0,..., 0, 1). The stabilizer of en is SO(n − 1) × {1}. Then from the proof of the orbit-stabilizer formula, we know that there is a bijection between the cosets of
6 SO(n − 1) × {1} in SO(n) and the elements of Sn−1 (which is the equivalence class
n−1 of en under SO(n) since SO(n) acts upon S transitively.
When n = 2, the space S1 is just a circle, hence, the harmonic analysis on S1 is just the Fourier series analysis on R/Z. More generally, for n ≥ 3, the harmonic analysis is about the representation of a function using spherical harmonics. The spherical harmonics are the eigenfunctions of the Laplacian operator defined on the surface spheres (the latter is also referred to as the spherical Laplacian). Given the spherical coordinates, we can derive the spherical Laplacian by change of variables from the Euclidean Laplacian (A.1) (see for example Terras’s book [62, page 109-
111], Courant and Hilbert [14, page 224]). More generally, it can be derived from the Laplacian-Beltrami operator [47, Chap. 3] [45, Chap. 2.2] [5, Chap. 3], which
is the Laplacian defined on an arbitrary Riemannian manifold. See, for example,
Shubin [55, page 164], Gallier and Quaintance’s lecture notes [26, Chap.26].
In the following, we will review the abstract construction of the spherical Laplacian
and harmonics as in the ambient Euclidean space. We show that the spherical har-
monics are exactly the homogeneous harmonics polynomials restricted to the sphere.
The material presented here can also be found in great details in Garrett’s note [27],
as well as [26, 47].
2.1 Laplacian and Invariant Integrals on Spheres
We will define the spherical Laplacian and the integral (or measure) on spheres.
Both of them are rotationally invariant. The rotation is defined through the action of
SO(n). Specifically, the action of k ∈ SO(n) on a function f : Sn−1 7→ C is defined
7 by
(k · f)(x) = f(xk), where we use the notation of right matrix multiplication on row vectors. The usual
Euclidean Laplacian (A.1) is SO(n)-invariant (commute with the group action of
SO(n)): for a differentiable functions g on Rn, we have
∆(k · g)(x) = (k · (∆g))(x), ∀k ∈ SO(n).
This can be easily shown by direct calculation. Then for a function f on Sn−1, we can
create a function F on Rn − 0 by F (x) = f(x/|x|), and define the spherical Laplacian
∆S applied to f to be the restriction of ∆F to the sphere Sn−1, that is,
S ∆ f = (∆F ) |Sn−1 . ( ) − x n 1 S 1 | − − For example, let f(x) = x1 be defined on S . Then ∆ f = ∆ |x| Sn 1 = (1 n)x1 on Sn−1. Since ∆ has only real coefficients, from the above definition, we immediately
have ∆Sf = ∆Sf¯.
The SO(n)-invariance of the spherical Laplacian follows by
S ∆ (k · f) = (∆(k · F ))|Sn−1 = (k · (∆F ))|Sn−1 = k · (∆F )|Sn−1 , where note that the group action of SO(n) commutes with the map from f to F
(the first equality), and also commutes with the restriction to the sphere (thelast
equality). The second equality is due to the SO(n)-invariance of ∆. Hence, the ∆S
is SO(n)-invariant.
Now we define a measure on the sphere that is SO(n)-invariant. This can be
shown by utilizing a smooth non-negative function γ on [0, ∞) with the property
8 ∫ | |2 n−1 that Rn γ( x )dx = 1. Specifically, for a continuous function f on S , we define
the integration ∫ ∫ x | |2 f = γ( x )f(| |)dx, (2.1) Sn−1 Rn x and then for k ∈ SO(n) we have ∫ ∫ ∫ xk x · | |2 | −1|2 k f = γ( x )f(| |)dx = γ( xk )f(| |)dx = Sn−1 Rn xk Rn ∫ x ∫ x | |2 γ( x )f(| |)dx = f, Rn x Sn−1 where note the fact that |xk−1| = |x| and the Jacobian determinant |k−1| = 1.
The measure on the sphere is used to define the function space of square-integrable
functions on the sphere, { ∫ } L2(Sn−1) = f : Sn−1 7→ C| |f|2< ∞ . (2.2) Sn−1 with the Hermitian inner product ∫ ⟨f, g⟩ = fg,¯ (2.3) Sn−1 where g¯ is the complex conjugate of g : Sn−1 7→ C.
Furthermore, it can be shown that ∆S is self-adjoint and semi-negative definite, i.e., ∫ ∫ (∆Sf)φ = f(∆Sφ), (self-adjointness) (2.4) Sn−1 ∫ Sn−1 (∆Sf)f¯ ≤ 0, (semi-negative definite) (2.5) Sn−1
see Garrett’s note [27] for the proof.
Then the following results can be obtained immediately by a standard linear
algebra argument.
9 Lemma 1. The eigenvalues of ∆S are non-positive real numbers. Eigenfunctions f, g associated with distinct eigenvalues are orthogonal with respect to the Hermitian ∫ ⟨ ⟩ inner product f, g = Sn−1 fg¯.
Proof. It is clear that the self-adjointness (2.4) implies that all the eigenvalues of ∆S must be real. Suppose f : Sn−1 7→ C is a eigenfunction of ∆S such that ∆Sf = λf, ∫ ∫ S ¯ | |2 ≤ then from (2.5) we have Sn−1 (∆ f)f = Sn−1 λ f 0. Hence, all eigenvalues of ∆S are non-positive. Suppose f and φ are two eigenfunctions of ∆S associated with ∫ ∫ ̸ S eigenvalues λ = µ. By the property (2.4), it follows that λ Sn−1 fφ = Sn−1 (∆ f)φ = ∫ ∫ ∫ S Sn−1 f(∆ φ) = µ Sn−1 fφ, which implies that Sn−1 fφ = 0, that is f and φ are orthogonal with respective to the inner product ⟨, ⟩.
2.2 Eigenfunctions of the Spherical Laplacian
Let Ck(X) denote the space of k-th continuously differentiable functions defined
on X. A function G ∈ C2(Rn) is called harmonic if ∆G = 0, that is, if it is annihilated
by the Euclidean Laplacian ∆ and it is called positive-homogeneous of degree µ ≥ 0,
if for all t > 0 and x ∈ Rn, G(tx) = tµG(x).
The following results can be proved by direct calculations and use Euler’s identity on homogeneous functions 1, see for example [27]. A more concise treatment can be
found in Shubin [55, page 168-169] and Morimoto [45, page 21].
Proposition 1. For G positive-homogeneous of degree µ and harmonic, the restriction
n−1 S G|Sn−1 of G to the sphere S is an eigenfunction for ∆ , i.e.,
S ∆ (G|Sn−1 ) = −µ(µ + n − 2)(G|Sn−1 ), (2.6)
1 T ∂G see [28, page 51]. It states that for positive-homogeneous G of degree µ, x ∂x = µG, where the superscript T denotes the transpose.
10 and the corresponding eigenvalue is −µ(µ + n − 2).
The most tractable positive-homogeneous functions are homogeneous polynomials.
Denote by Pd ⊂ C[x1, ··· , xn] the homogeneous polynomials of (total) degree d, d ∈ Z≥0. With the multi-index notation, the homogeneous polynomials have the form: ∑ a cax , ca ∈ C. |a|=d ( ) n + d − 1 This is a C-vector space of finite dimension . Let H be the subset of n − 1 d 2 harmonic polynomials in Pd, for example, given n = 3, |x| − x1x3 − x1x2 − x2x3 is ∑ H | |2 3 2 an element of 2, where x = i=1 xi . Furthermore, we can show that the spectrum (i.e., the eigenvalues) of the spherical
Laplacian only takes integer values −d(d + n − 2), for d ∈ Z≥0. The following proof is adapted from Shubin [55, page 169].
Theorem 1. The eigenvalues of the operator ∆S are λ = −d(d + n − 2), where d ∈ Z≥0.
Proof. Since all eigenvalues of ∆S are non-positive (see Lemma 1), we may assume
that µ ≥ 0 or µ ≤ 2−n. Note that the quadratic function λ(µ) = −µ(µ+n−2) takes
all values λ ≤ 0 for µ ≥ 0 and each of them exactly once. It is therefore clear that
the eigenvalues λ ≤ 0 are in a one-to-one correspondence with those µ ≥ 0, for which
there exists a non-trivial function G(ω) such that rµG(ω) is a harmonic function on
Rn\ x 0, where ω = |x| (see Proposition 1). But then, by the removable singularity theorem, the function rµG(ω) is harmonic everywhere on Rn and consequently, is a harmonic polynomial by the Liouville theorem [11, Theorem 2] [30, page 531]. In
particular, µ must be an integer.
11 Moreover, we can show that the restriction to the sphere of the homogeneous
harmonic polynomials are the only eigenfunctions of ∆S (Theorem 3). To show this,
S we shall first explore the structure of Pd with respect to ∆ , which will give an
algebraic direct-sum decomposition of Pd into the Hj spaces.
2.3 Decomposition of Homogeneous Polynomials
To decompose Pd into the direct sum of the Hj spaces, we adopt an approach which makes use of a particularly defined complex Hermitian form Morimoto [45, page 22].
A different approach motivated from the solution of the Dirichlet problem fortheball is documented in Axler et al. [8].
Define the following complex Hermitian form (·): C[x1, ··· , xn]×C[x1, ··· , xn] 7→
C by: ¯ (P,Q) = (Q(D)P (x))|x=0, (2.7)
where Q(D) means to replace xi by ∂/∂xi in a polynomial. It is easy to see from the
definition that:
(∆f, g) = (f, r2g), (2.8) ∑ 2 n 2 2 where r = i=1 xi . That is, multiplication by r is adjoint to application of ∆ with respect to (, ).
The following result can be shown by direct computation.
Lemma 2. The Hermitian form (2.7) is positive-definite, that is (P,P ) = 0, if and only if P = 0.
Proof. Observe that (P,Q) = 0 for homogeneous polynomials P , Q unless they are
of the same degree. Considering the homogeneous polynomials Pd, then the distinct
12 monomials of degree d constitute an orthogonal basis with respect to the Hermitian form, since (using the multi-index notation Da and xe) { ̸ a e 0 if any ai = ei, ((D )x )|x=0 = a1! ··· an! if ai = ei, ∀i.
From this, we can see that the Hermitian form is positive definite on Pd.
Note that a positive-definitive Hermitian form is an inner product. Moreover,
the complex vector space C[x1, ··· , xn] together with the positive-definite Hermitian
form (2.7) is called a (infinite dimensional) pre-Hilbert space. It is not a Hilbert space because it is not a complete normed vector space under the norm induced by the ∑ n t x Hermitian form (e.g. the partial sums t=1 x /t! of the Taylor expansion of e is a
Cauchy sequence, but does not have a limit in C[x1, ··· , xn]).
The following results can be proved very conveniently using the properties of the
Hermitian form. It gives a direct-sum decomposition of the finite-dimensional vector
space Pd. For convenience, let ⌊h⌋ denote the floor function, i.e., the largest integer
less or equal to h.
Lemma 3. The map ∆ : Pd 7→ Pd−2 is surjective. Furthermore, we have a direct
sum decomposition
2 4 2⌊d/2⌋ Pd = Hd ⊕ r Hd−2 ⊕ r Hd−2 ⊕ · · · ⊕ r H⌊d/2⌋, (2.9) where the summands of the direct sum are pairwise orthogonal with respect to the
Hermitian form (2.7).
Proof. Suppose the map ∆ is not surjective, then by the standard results of linear
algebra, there exists nonzero Q ∈ Pd−2 which is orthogonal to the image of ∆, i.e.,
13 2 2 (∆P,Q) = 0 for all P ∈ Pd. Since r Q ∈ Pd, let P = r Q and we have by (2.8) that
(∆P,Q) = (P,P ) = 0, which contradicts with the positiveness of the Hermitian form.
2 For the second assertion, we first claim that Pd = Hd ⊕ r Pd−2. Since ∆ is surjective and ker∆ = Hd, we have dim(Pd) = dim(Hd) + dim(Pd−2) (see for ex-
ample [16, Cor. 8, page 413]). Hence, we only need the orthogonality between Hd
2 and r Pd−2, which is obvious in view of the identity (2.8). The decomposition (2.9)
follows immediately by induction.
Note that the decomposition (2.9) implies that each f ∈ Pd can be written
uniquely as the summation of finite homogeneous harmonic polynomials, which fol-
lows from the property of the direct sum [16, page 353]. It can also be proved easily
by showing that the difference of any two decompositions is 0 using the Hermitian
form. ( ) ( ) n + d − 1 n + d − 3 Corollary 1. The dimension of H is − for d > 1. d n − 1 n − 1
Proof. From the proof of Lemma 3, we see that dim(H ) = dim(P ) − dim(P − ), ( ) d d d 2 n + d − 1 while dim(P ) = . d n − 1
Since any polynomial in C[x1, ··· , xn] is a sum of homogeneous polynomials, we
immediately have
n−1 Corollary 2. The restriction to S of every polynomial in C[x1, ··· , xn] is a sum of restrictions to Sn−1 of homogeneous harmonic polynomials.
The above decomposition of Pd leads to an orthogonal decomposition of the
infinite-dimensional vector space L2(Sn−1).
14 2.4 L2 Spectral Decomposition on Spheres
n−1 n−1 Let Hd(S ) denote the space of restrictions of polynomials in Hd to S . They are the spherical harmonics of degree d. Let C(Sn−1) be the space of continuous
n−1 n−1 function on S . We have shown in the previous section that Hd(S ) are all the eigenfunctions of ∆S, and that the space of all the polynomials can be decomposed
2 n−1 into the direct sum of Hd’s. In this section, we further show that L (S ) can be
n−1 decomposed into an infinite direct sum of Hd(S )’s.
Recall from the standard Hilbert space theory [37, page 159] that the complex
+ Hilbert space H is said to be a direct-sum of the subspaces Hm, m ∈ Z and we write ⊕∞ H = m=0Hm , if
1. The subspaces Hm are closed for every m.
2. The subspaces Hm are pairwise orthogonal, i.e., if h ∈ Hm and g ∈ Hm′ , then
⟨f, g⟩ = 0 whenever m ≠ m′.
3. Every element f ∈ H has a representation of the form
f = h0 + h1 + ··· + hm + ··· ,
+ where hm ∈ Hm (m ∈ Z ) and the sum converges in the norm of H.
∪∞ H n−1 Proposition 2. The set of all finite linear combinations of elements in d=0 d(S ) is
1. dense in C(Sn−1) with respect to the L∞ norm;
2. dense in L2(Sn−1) with respect to the L2 norm.
Proof. Since Sn−1 is compact, by the Stone-Weierstrass approximation theorem (see
for example Lang [38, Chap. III] Corollary 1.3 for the case of real valued functions
15 and Theorem 1.4 the complex valued functions), if f is continuous on Sn−1, then it
n−1 can be approximated uniformly by polynomials Pj restricted to S . Then we obtain the conclusion of Part 1 in view of Corollary 2. For Part II, note that C(Sn−1) is dense in L2(Sn−1) in the L2 sense ( see for example Taylor [61, page 44]. Since the
sphere is compact which has finite total measure, density in the L∞ norm implies
density in L2 norm, and therefore Part II follows.
Now we can prove the following important result.
2 n−1 ⊕∞ H n−1 Theorem 2. L (S ) = d=0 d(S ).
Proof. By the maximum principle [8, page 7, Corollary 1.9], we see that a harmonic polynomial is uniquely defined by its restriction to Sn−1. Therefore, by Corollary. 1,
n−1 2 n−1 Hd(S ) is finite dimensional, and hence, it is a closed subspace of L (S ). More-
n−1 over, the subspaces Hd(S ) for different d are pairwise orthogonal since they are
the eigenspaces associated with distinct eigenvalues of ∆S (see Lemma 1). Together with Proposition 2, we reach the conclusion.
Theorem 3. The eigenspace of ∆S associated with the eigenvalue −d(d + n − 2) is
n−1 exactly Hd(S ).
Proof. Let f ∈ C2(Sn−1) be an eigenfunction of ∆S associated with the eigenvalue
−d(d + n − 2). Since f also belongs to L2(Sn−1). By Theorem 2, there is a decompo- ∑ 2 n−1 n−1 sition of f = hm in the L (S ) sense, where hm ∈ Hm(S ) for all m. By the orthogonality of the eigenspaces of ∆S associated with different eigenvalues, we must have f = hd almost everywhere. Since f is continuous, f = hd everywhere.
Combining the above theorem with Theorem 1, we see that the only eigenfunctions
of ∆S are those homogeneous harmonic polynomials restricted to the sphere.
16 Example 1. For the circle S1, the eigenspace associated with d = 0 is the complex number C with unit modulus, which is of one dimension, and for d > 0 the (−d2)- eigenspace is 2-dimensional, with basis (x iy)d (restricted to S1, or equivalently e2πiθd as we already seen in previous section). While for n > 2, the dimension of eigenspaces are unbounded as d goes to infinity by Corollary 1.
17 Chapter 3: Spherical Harmonics Basis in Euclidean Coordinates
In this chapter, we describe a method of generating the basis of the spherical
harmonics as reported in Axler et al. [8, page 92]. To begin with, we will introduce
the so-called Kelvin transform.
3.1 Kelvin Transform
Consider the space Rn ∪ {∞}, which is the one-point compactification of Rn.
Define the map x 7→ x∗, where x ̸ ∞ |x|2 if x = 0, ∗ x = 0 if x = ∞ ∞ if x = 0
This is called the inversion of Rn ∪ {∞} relative to the unit sphere. Graphically, for
any point x∈ / {0, ∞}, its image x∗ lies on the ray from the origin determined by x,
and with length |x∗| = 1/|x|. It is easy to verify that this map is continuous, and
satisfies (x∗)∗ = x. For any set E ⊂ Rn ∪ {∞}, define the set E∗ = {x∗ : x ∈ E}.
Clearly, we have S∗ = S for the sphere. Now given a function u defined on a set
E ⊂ Rn\{0}, we define the function K[u] on E∗ by
K[u](x) = |x|2−nu(x∗),
18 the function K[u] is called the Kelvin transform of u. Clearly, the transform K is linear in u. It can be easily verified that K [K[u]] = u for all functions u as above.
An important property of the Kelvin transform is that it preserves the harmonic functions. Following the proof given in [8, page 62], we first show the following two lemmas.
Lemma 4. If f on Rn is positive-homogeneous of degree d, then
∆(|x|2−n−2df) = |x|2−n−2d∆f.
Proof. Using the product rule of the Laplacian ∆(uv) = v∆u + 2∇u · ∇v + u∆v, it
can be easily shown that
∆(|x|tf) = |x|t∆f + 2t|x|t−2x · ∇f + t(t + n − 2)|x|t−2f.
By Euler’s identity on homogeneous functions, we have x · ∇f = d · f. Hence, the above equation becomes,
∆(|x|tf) = |x|t∆f + t(2d + t + n − 2)|x|t−2f.
Now taking t = 2 − n − 2d, we will have the desired result.
Observing that if u on Rn is positive-homogeneous of degree d, then
K[u] = |x|2−n−2du. (3.1)
Moreover, taking the Laplacian of both sides of u(tx) = tdu(x), we have respectively
∑n 2 T 2 2 ∆u(tx) = t ei (Dxu)(tx)ei = t (∆u)(tx), i=1 ∆tdu(x) = td∆u(x) = td(∆u)(x),
19 which implies (∆u)(tx) = td−2(∆u)(x), that is, ∆u is of homogeneous degree d − 2.
n The convention here is that ei is the canonical basis of the Euclidean space R , and
2 Dxu takes the Hessian matrix of u. We use the above two formulas in the proof of the following lemma. This lemma shows that the Kelvin transform comes close to
commuting with the Laplacian.
Lemma 5. If u is a C2 function on an open subset of Rn\{0}, then
∆(K[u]) = K[|x|4∆u]. (3.2)
Proof. We have
∆(K[u]) = ∆(|x|2−n−2du) = |x|2−n−2d∆u = K[|x|4∆u], where the first and the third equalities follow from(3.1). In particular, notice that
|x|4∆u is of homogeneous degree d + 2. The second equality is by Lemma 4.
Clearly, the relation (3.2) holds for all the homogeneous polynomials. By the
linearity of the K transform, we see that (3.2) holds all the polynomials. Since
polynomials are dense in the C2 norm, we see that it also holds for the C2 functions.
Theorem 4. Let E be a subset of Rn\{0}. Then u is harmonic on E if and only if
K[u] is harmonic on E∗.
Proof. This is implied by Lemma 5.
Now we will show that the Kelvin transform of certain polynomials gives a basis
of the the spherical harmonics. The following two lemmas will be useful.
Recall the definition that for Q a polynomial, Q(D) means to replace xi by ∂/∂xi.
20 2−n Lemma 6. Suppose n > 2 and p ∈ Pd, then K[p(D)|x| ] ∈ Hd.
2−n Proof. We first prove K[p(D)|x| ] ∈ Pd, that is, they are homogeneous polynomials.
By the linearity of the Kelvin transform, we only need to prove it for the monomials.
Clearly, for d = 0, K[p(D)|x|2−n] is just a constant, for which the relation holds
trivially. Assuming that the relation holds for some fixed d, we will show that it also
holds for d + 1.
Let a be a multi-index with |a| = d. By the induction hypothesis, there exists
a 2−n u ∈ Hd such that K[D |x| ] = u. Take the Kelvin transform of both sides, yielding
Da|x|2−n = |x|2−n−2du.
Taking the derivative of both sides the above equation with respective to xj for a
fixed j, we have
a 2−n −n−2d 2−n−2d DjD |x| = (2 − n − 2d)xj|x| u + |x| Dju
2−n−2(d+1) 2 = |x| [(2 − n − 2d)xju + |x| Dju]
= |x|2−n−2(d+1)v, (3.3)
where v ∈ Pd+1. Taking the Kelvin transform again of the above equation, we have
a 2−n K[DjD |x| ] = v,
a 2−n which implies that K[DjD |x| ] ∈ Pd+1. Since j is arbitrary, the relation holds for
differentiation with respect to an arbitrary multi-index oforder m + 1. Therefore, by
2−n induction, we proved K[p(D)|x| ] ∈ Pd, for p ∈ Pd.
Now we prove that K[p(D)|x|2−n] is also harmonic. Note that |x|2−n is harmonic
for n > 2. In addition, it is easy to verify that every partial derivative of any har-
monic function is harmonic (note that harmonic function is smooth, see for example
21 Evans [21, page 28]). Hence, p(D)|x|2−n is harmonic. Since Kelvin transform preserves
the harmonic property (Theorem 4), we conclude that K[p(D)|x|2−n] is harmonic.
From the proof of Lemma 3, we see p ∈ Pd can be uniquely decomposed into
2 p = hd +|x| q, where hd ∈ Hd and q ∈ Pd−2. Therefore p determines two homogeneous
2−n harmonic polynomials K[p(D)|x| ] and hd. The following lemma gives their relation
by refining the proof of the last lemma. For n > 2, we define the constant
∏d−1 cd = (2 − n − 2i), d > 0, i=0 with the convention that cd = 1 for d = 0. It will be useful later.
Lemma 7. Suppose n > 2 and p ∈ Pd, then
2−n 2 K[p(D)|x| ] = cd(p − |x| q),
for some q ∈ Pd−2 with the convention that q = 0 for d < 2.
Proof. The proof is a refinement of the last lemma. In view of the linearity ofthe
Kelvin transform, we shall prove for monomials only. The desired result obviously
holds for d = 0 and d = 1. By induction, we assume that the result hold for for some
fixed d, and then showing that it also holds for d + 1.
Let a be a multi-index with |a| = m. By our induction hypothesis, there exists q ∈ Pd−2 such that
a 2−n a 2 K[D |x| ] = cd(x − |x| q).
a 2 Setting u = cd(x − |x| q). Following the same procedure as in (3.3), we take the
Kelvin transform of both sides of the above equation, then differentiate them with
22 respect to xj, and we have
| |2 a 2−n 2−n−2(d+1) a 2 x Dju DjD |x| = |x| cd[(2 − n − 2d)xj(x − |x| q) + ] cd 2−n−2(d+1) a 2 = |x| cd+1(xjx − |x| v),
where v ∈ Pd−1. Taking the Kelvin transform again, we have
a 2−n a 2 K[DjD |x| ] = cd+1(xjx − |x| v).
a Since xjx represent an arbitrary monomial of order d + 1, we proved the desired
relation by induction.
3.2 Projection of Homogeneous Polynomials to Hd spaces
We are in the position to prove the following formulas of calculating the projec-
tion of a homogeneous polynomial onto the Hd spaces under the inner product of either (2.7) or (2.3).
Theorem 5. Suppose n > 2 and p ∈ Pd. Then
2−n 1. The orthogonal projection of p onto Hd is K[p(D)|x| ]/cd under the inner
product (2.7).
n−1 2−n 2. The orthogonal projection of p|S onto Hd(S ) is (p(D)|x| ) |Sn−1 /cd under the L2-inner product (2.3).
Proof. By Lemma 7, we can write
K[p(D)|x|2−n] p = + |x|2q, (3.4) cd
for some q ∈ Pd−2. Lemma 6 shows that the first term on the right side of(3.4) is in
Hd. Therefore, by Lemma 3 we see (3.4) is the unique decomposition of p under the
inner product (2.7).
23 For Part 2, restrict both sides of (3.4) to Sn−1 and we have
(p(D)|x|2−n) p|Sn−1 = |Sn−1 + q|Sn−1 , cd where we used the property that the Kelvin transform of a sphere is itself. Note that
n−1 the first term on the right side of the above equation belongs to Hd(S ), while
n−1 n−1 n−1 q|Sn−1 ∈ Hd−2(S ) ⊕ Hd−2(S ) ⊕ · · · ⊕ H[d/2](S ), which are orthogonal to
each other under (2.3) because they are eigenspaces of ∆S associated with distinct
n−1 eigenvalues (see Lemma 1). Hence, the projection of p|Sn−1 onto Hd(S ) is exactly
2−n (p(D)|x| ) |Sn−1 /cd.
a 2−n The above theorem implies that {K[D |x| ]: |a| = d} spans Hd and that
a 2−n n−1 {(D |x| )|Sn−1 : |a| = d} spans Hd(S ). The main result of this subsection is the
following theorem, which finds an explicit basis for each of these spanning sets.
Theorem 6. If n > 2 then the set
a 2−n {K[D |x| ]: |a| = d and a1 ≤ 1}, (3.5)
is a vector space basis of Hd, and the set
a 2−n {(D |x| )|Sn−1 : |a| = d and a1 ≤ 1}, (3.6)
n−1 is a vector space basis of Hd(S ).
Proof. Let B denote the set (3.5) for short. To prove the results of the theorem, we will show 1) K[Da|x|2−n] is in the span of B for the multi-index a with |a| = d, and
2) the cardinality of B is at most the dimension of Hd.
For the first part, we will prove by induction on a1, the first component of the
a 2−n multi-index a of degree d. Clearly, if a1 is 0 or 1, then K[D |x| ] is in B by definition.
24 b 2−n Now consider the case a1 > 1. By induction, we can assume that K[D |x| ] is in the span of B for all multi-indices b of degree d whose first components are less than
2−n a1. Using the fact that ∆|x| ≡ 0, we have
− − − a| |2 n a1 2 a2 ··· an 2| |2 n K[D x ] = K[D1 D2 Dn (D1 x )] ∑n − − − a1 2 a2 ··· an 2| |2 n = K[D1 D2 Dn ( Di x )] i=2 ∑n − − − a1 2 a2 ··· an 2| |2 n = K[D1 D2 Dn (Di x )]. i=2
Note that by the hypothesis of our induction, each of the summands in the last line is
in the span of B, and therefore K[Da|x|2−n] is in the span of B for every multi-index
a of degree d. Hence, we conclude that B spans Hd.
Furthermore, if Part 2) holds, then we can readily conclude that B is a vector
space basis of Hd. Note that for the set B, the multi-indices a are of length d and
cannot be of the form (b + 2, b , ··· , b ) with |b| = m − 2. Therefore the cardinality ( 1 ) 2 ( n ) n + d − 1 n + d − 3 of B is at most − . We know from Lemma 1 that this n − 1 n − 1
difference is exactly the dimension of Hd. Hence, Part 2) follows.
The second assertion follows by the restriction of (3.5) to Sn−1, since the elements
of Hd are uniquely determined by its restriction to the sphere.
For the case of n = 2, analogous formulas exists to generate an orthogonal basis
for the spherical harmonics (see [8, page 107]).
3.3 Examples
Table 3.1 gives several examples of the basis functions obtained with the methods of Theorem 6 (as well as [8, page 107] for the 2 dimensional case). They are calculated
using the Mathematica software package HFT.m [7, 8]. Clearly, the basis is just the
25 Table 3.1: Basis of Hd for different dimensions d = 2 d = 3 | |2 − 3 | |2 − 2 3 x x2 4x2, n = 2 x 2x2, x1x2 | |2 − 2 x x1 4x1x2 | |2 − 3 3 x x2 5x2, | |2 − 3 3 x x3 5x3, | |2 − 2 | |2 − 2 x 3x2, x x3 5x2x3, | |2 − 2 | |2 − 2 n = 3 x 3x3, x x2 5x2x3, | |2 − 2 x1x2, x2x3, x1x3 x x1 5x1x2, | |2 − 2 x x1 5x1x3, x1x2x3 | |2 − 2 x 4x2, | |2 − 2 x 4x3, | |2 − 2 n = 4 x 4x4 (left out) x1x2, x2x3, x3x4, x1x3, x2x4, x1x4
{ }n constant for the degree d = 0, and is the coordinate set xi i=1 for d = 1 for different dimensions. In addition, for n = 2, note that the basis calculated by [8, page 107] is
d 2πiθd n−1 different from (x iy) for Hd and e for Hd(S ).
Note that the above obtained basis functions do not constitute an orthogonal set
2 n−1 of basis functions of L (S ). In fact, within the space Hd, the eigenfunctions are
generally not orthogonal to each other. We will show this using the (normalized)
n−1 surface area measure Ωn on rS . Note that the surface integral with respect to
Ωn can be calculated under the spherical coordinate (ϕ, θ1, ··· , θn−2, r) with ϕ ∈
[0, 2π], θi ∈ [0, π] i = 0, 1, ··· , n − 2, r > 0. They are related to the Euclidean
coordinates by equation (3.7) at the top of the next page. Denote this mapping
by x = ψ(ϕ, θ1, ··· , θn−2, r) and let Jn be the determinant of the Jacobian matrix
26 x1 = r sin θn−2 ··· sin θ2 sin θ1 cos ϕ,
x2 = r sin θn−2 ··· sin θ2 sin θ1 sin ϕ,
x3 = r sin θn−2 ··· sin θ2 cos θ1, . . (3.7)
xn−2 = r sin θn−2 sin θn−3 cos θn−4,
xn−1 = r sin θn−2 cos θn−3,
xn = r cos θn−2.
∂x/∂(ϕ, θ1, ··· , θn−2, r). It can be calculated that
n−1 n−2 2 Jn = r sin θn−2 ··· sin θ2 sin θ1.
Then by the change of variable formula, we can obtain the integral of f on the sphere
n−1 rS with respect to Ωn as: ∫ ∫
f(x)dΩn = f(ψ(ϕ, θ1, ··· , θn−2, r)) n−1 rS 0≤θi≤π,0≤ϕ≤2π
× Jn(ϕ, θ1, ··· , θn−2, r)dθ1 ··· dθn−2dϕ.
In particular, for n = 3, d = 2, we can calculate ∫ 8π | |2 − 2 | |2 − 2 − ( x 3x2)( x 3x3)dΩ2 = , S2 5 which shows that the two basis of H2 are not orthogonal.
Furthermore, we can use the Gram–Schmidt orthogonalization process to obtain
the an orthogonal set of the basis of Hd. They can be further normalized by dividing
2 n−1 their L (S ) norm. For n = 3, d = 2, we have dim(H2)=5, which is the same as
27 2 calculated by Corollary 1. Then we can compute an orthonormal basis of H2(S ) as √ √ 1 5 ( ) 1 15 ( ) − 2 − 2 2 − v1 = 1 3x2 , v2 = x2 + 2x3 1 , 4 π √ √4 π 1 15 1 15 v3 = x1x2, v4 = x2x3, 2√ π 2 π 1 15 v = x x . 5 2 π 1 3
28 Chapter 4: Harmonic Analysis on the sphere Sn−1 in Spherical Coordinates
In the last chapter, we defined the spherical Laplacian in the ambient Euclidean
coordinates. We identified that the restriction of the homogeneous harmonic polyno-
mials to the spheres are exactly the eigenfunctions of the Laplacian, i.e., the spherical
harmonics. In this chapter, we will characterize the spherical harmonics directly un-
der the spherical coordinates. When n = 3, an orthonormal set of spherical harmonics
can be expressed in terms of the so-called associated Legendre functions (see for
example Hobson [33] and many other references [5,45,47]. More generally for arbitrary
dimension n > 2, it can be represented in terms of the Gegenbauer polynomials
(or ultraspherical polynomials, see for example Szegő [60, Chap. 4.7, page 80]).
4.1 Spherical Harmonics (n = 3)
We first present the detailed derivation of the solutions of the Laplacian equation
for the 3 dimensional case where n = 3. By the change of coordinates, we can obtained
the Laplacian operator under the spherical coordinates as:
∂2 2 ∂ 1 ∂2 1 ∂2 cot θ ∂ ∆ = + + + + , ∂r2 r ∂r r2 sin2 θ ∂2ϕ r2 ∂2θ r2 ∂θ
29 where r > 0, θ ∈ (0, π), and ϕ ∈ (0, 2π) (see Fig. 2.1). Restricted on the unit sphere, we have 1 ∂2 ∂2 cos θ ∂ ∆S = + + . sin2 θ ∂2ϕ ∂2θ sin θ ∂θ Now we will find the solutions to the Laplacian equation ∆f = 0 under the
spherical coordinates. We shall see later (in Theorem 11) that if in addition f is
S positive homogenous of degree d > 0, then ∆ f|S2 = −d(d + 1)f|S2 , that is, f|S2 is a
spherical harmonic.
Using separation of variables, we denote f(r, θ, ϕ) = ρ(r)Θ(θ)Φ(ϕ) and obtain
ρ′′(r) ρ′(r) Φ′′(ϕ) Θ′′(θ) cot θΘ′(θ) r2 + 2r + + + = 0. ρ(r) ρ(r) sin2 θΦ(ϕ) Θ(θ) Θ(θ)
Separating the variable r from θ and ϕ, we have
r2ρ′′(r) + 2rρ′(r) − λρ = 0, (4.1) Φ′′(ϕ) Θ′′(θ) cot θΘ′(θ) + + = −λ, sin2 θΦ(ϕ) Θ(θ) Θ(θ) where λ is the separation constant. We further separate θ from ϕ to obtain
Φ′′(ϕ) + αΦ(ϕ) = 0, (4.2) cos θ α Θ′′(θ) + Θ′(θ) + (λ − )Θ(θ) = 0, (4.3) sin θ sin2 θ where α is the separation constant.
The ρ-equation is the Cauchy-Euler equation, which has two fundamental
c1 c2 2 solutions ρ1(r) = r and ρ1(r) = r , where c1, c2 are the roots of s + s − λ = 0.
Assuming Φ(ϕ) is 2π-periodic, we observe that α = m2 for non-negative integers m
and the eigenfunctions are Φ(ϕ) = cos(mϕ) and Φ(ϕ) = sin(mϕ) (or eimϕ). Moreover,
if α = 0, then Φ(ϕ) = cos(0) = 1, then the eigenfunction f is independent of ϕ and
30 in this case the solution Θ(θ) is usually called the Zonal spherical harmonics. We
now consider the case of α = 0 first. To solve for Θ(θ), we make a change of variables t = cos θ for θ ∈ [0, π], and thus t ∈ [−1, 1]. Let w(t) = Θ(θ) = Θ(arccos t), thus we have
√ Θ′(θ) = − sin θw′(cos θ) = − 1 − t2w′(t),
Θ′′(θ) = sin2 θw′′(cos θ) − cos θw′(cos θ) = (1 − t2)w′′(t) − tw′(t).
Using the above expressions in (4.3), we have
(1 − t2)w′′(t) − 2tw′(t) + λw(t) = 0. (4.4)
Equation (4.4) is known as the Legendre equation. It is also called singular
Sturm-Liouville equation, since (1 − t2) equals 0 when t = 1. In the following, we will construct the solutions of the Legendre equation for t ∈ [−1, 1]. For more general cases like t > 1 or a complex number t, the readers can refer to, for example,
Hobson [33] and Lebedev [40].
4.1.1 Legendre Equation and Polynomials
We can find the power series solutions expanded around t = 0 with radius of ∑ ∞ i convergence at least 1. Therefore, we substitute w(t) = i=0 cit into (4.4), and obtain ∑∞ ∑∞ ∑∞ 2 i−2 i−1 i (1 − t ) i(i − 1)cit − 2t icit + λ cit = 0, i=2 i=1 i=0 which can be further written as
∑∞ ∑∞ ∑∞ ∑∞ i−2 i i i i(i − 1)cit − i(i − 1)cit − 2icit + λ cit = 0. i=2 i=2 i=1 i=0
31 The first term can be re-indexed and grouped with the rest of the terms toobtain
∑∞ ∑∞ i 2 i (i + 2)(i + 1)ci+2t − (i + i − λ)cit = 0. i=0 i=0
Therefore, we must have the coefficients of ti equal 0, which yields
(i + 1)i − λ c = c , i = 0, 1, 2,... (4.5) i+2 (i + 2)(i + 1) i
The above iteration formula implies that there are two independent solutions
of (4.4), that is, we have ( ) ∞ − ∑ ∏i 1 t2i w(t) =c [(2k + 1)(2k) − λ] 0 (2i)! i=0 k=0( ) ∞ ∑ ∏i t2i+1 + c [(2k)(2k − 1) − λ] 1 (2i + 1)! i=0 k=1
:=w1(t) + w2(t), (4.6)
for arbitrary c0 and c1, where the definitions of w1 and w2 are obvious. By the ratio
test, it can be easily shown that w1 and w2 is bounded (the series are convergent) on
the interval (−1, 1). Moreover, for t = 1, the convergence of the serires defining w1
and w2 also depends on the value of λ. For λ ≠ d(d + 1), both the serires of w1 and
w2 are infinite, and from the ratio test we see that both of them are divergent. For
λ = d(d + 1), we have from (4.5) that
(d − i)(d + i + 1) c = − c , i = 0, 1, 2,... (4.7) i+2 (i + 2)(i + 1) i
and we can easily verify that
• if d = 2p (even), then w1 has finite summands which is a polynomial of degree
2p, while w2 has infinite summands which is unbounded.
32 • if d = 2p + 1 (odd), then w1 has infinite summands which is unbounded, while
w2 has finite summands which is a polynomial of degree 2p + 1.
The above analysis proved that
Theorem 7. The Legendre equation (4.4) has a bounded solution on [−1, 1] if and only if λ = d(d + 1) with non-negative integer d.
Therefore, in the following, we will only consider the polynomial solution of the
Legendre equation with λ = d(d + 1) and non-negative integer d. In this case, we can
rewrite the solution of the Legendre equation as in terms of only cd, which is the last
non-zero power series coefficient. Using (4.7), we can obtain the formula of cd−2k in
terms of cd: ∏ − k 2k 1 − (−1) j=0 (d j) c − = ∏ c . d 2k 2k k−1 − − d j=0 (2d 1 2j) Hence, the polynomial solution is ∏ ⌊d/2⌋ 2k−1 ∑ (−1)k (d − j) w(t) = ∏ j=0 c td−2k, 2k k−1 − − d k=0 j=0 (2d 1 2j) where cd is an arbitrary constant. The d-th Legendre polynomials Pd(t) is the
above w(t) with coefficient (2d)! c = , d 2d(d!)2
the choice of which is to have Pd(1) = 1. After some simplification, we can further
express the d-th Legendre polynomials as
⌊d/2⌋ 1 ∑ (−1)k(2d − 2k)! P (t) = td−2k. (4.8) d 2d k!(d − k)!(d − 2k)! k=0
The first few Legendre polynomials are listed in the below. They are alsoillus- trated in Fig. 4.1. Note that Pd(t) is an odd (or even) function if d is odd (or even).
33 Figure 4.1: Legendre Polynomials
P0(t) = 1,( ) P1(t) = t, ( ) P (t) = 1 3t2 − 1 , P (t) = 1 5t3 − 3t , 2 2 ( ) 3 2 ( ) P (t) = 1 35t4 − 30t2 + 3 , P (t) = 1 63t5 − 70t3 + 15t , 4 8 ( ) 5 8 ( ) 1 6 − 4 2 − 1 7 − 5 3 − P6(t) = 16 231t 315t + 105t 5 ,P7(t) = 16 429t 693t + 315t 35t ,
It is worth mentioning that the Legendre polynomials can also be obtained from applying the Gram–Schmidt orthonormalization to the functions 1, t, t2,... on the interval [−1, 1] with the usual L2 inner product in the Euclidean space (see Reed and
Simon [54, page 47], Courant and Hilbert [14, page 82-85]).
4.1.2 Rodrigues’ Formula
The following theorem gives a more compact form of the Legendre polynomials.
It was discovered by the French mathematician Benjamin Olinde Rodrigues in 1816.
Theorem 8. (Rodrigues’ Formula) The lth Legendre polynomial Pl is generated by
1 dl [ ] P (t) = (t2 − 1)l . (4.9) l 2ll! dtl
Proof. Let y = (t2 − 1)l. We first show by induction that the kth derivative y(k) of y satisfies
d2y(k) dy(k) (1 − t2) + 2(l − k − 1)t + (2l − k)(k + 1)y(k) = 0. (4.10) dt2 dt 34 First, it is clear that y = y(0) and y′ = 2tl(t2−1)l−1. Therefore, we see (1−t2)y′+2lty =
0. Further differentiate this equation, yielding
(1 − t2)y′′ + 2(l − 1)ty′ + 2ly = 0, which shows (4.10) holds for k = 0. Suppose (4.10) holds for k − 1, which is
dy(k) (1 − t2) + 2(l − k)ty(k) + (2l − k + 1)(k)y(k−1) = 0. dt
Differentiating the above equation again shows4.10 ( ) holds for k.
Now let k = l in (4.10), we have y(l) satisfies the Legendre equation with λ =
l(l + 1). Hence, from the previous analysis, we know that y(l) must be a constant
l l 2 multiple of Pl. From (4.8) we see that the coefficient of t in Pl is (2l)!/(2 (l!) ), while
the coefficient of xl in y(l) is that of
dl(t2l) (2l)! = (2l)(2l − 1) ··· (2l − l + 1)tl = tl. dtl l!
(l) l Comparing the two coefficients, we see that Pl = y /(2 l!), which is (4.9).
More insights of the discovery of the Rodrigues’ formula can be found in Hob-
son [33].
4.1.3 General Solution of Spherical Laplace Equation
In the above, we considered the special case α = 0 in solving the Laplacian
equation in the spherical coordinates. The solutions are independent of ϕ, which can
be represented in terms of Legendre polynomials as Pd(cos θ). The functions Pd(cos θ)
are also called Zonal spherical harmonics. More generally, when α = m2 with integer m, we know Φ(ϕ) = eimϕ. In addition, for the Θ-equation (4.3), changing the
35 variable by t = cos θ and denote w(t) = Θ(θ), we can have
m2 (1 − t2)w′′(t) − 2tw′(t) + (λ − )w(t) = 0. (4.11) 1 − t2
Define the associated Legendre functions for t ∈ [−1, 1],
dmP (t) P m(t) = (1 − t2)m/2 l , (4.12) l dtm
for degree l = 0, 1, 2, ··· , and order m = 0, 1, 2, ··· l, where Pl(t) is the Legendre
polynomial of degree l. We now show that:
Theorem 9. The associated Legendre function gives a solution to the Sturm–Liouville equation (4.11) with λ = l(l + 1).
Proof. In the proof of the Rodrigues’ formula in Theorem 8, we see that if y = (t2 −1)l, then the k-th derivative y(k) of y satisfies the differential equation (4.10) for arbitrary
m non-negative integer k. In the case of Pl , we have k = l + m. Therefore, denote
m m v(t) = d Pl(t)/dt , and we will have from (4.10) that,
d2v dv (1 − t2) − 2(m + 1)t + (l − m)(l + m + 1)v = 0. dt2 dt
m − 2 m/2 Now using the above equality, it is easy to verify that Pl (t) = (1 t ) v(t) satisfies the Sturm–Liouville equation4.11 ( ) with λ = l(l + 1).
It is worth mentioning that some authors (e.g., Arfken et al. [4, Chapter 15.4, page
743] and Press et al. [53, page 246]) include the so-called Condon-Shortley phase term
(−1)m in the above definition (4.12). The Mathematica function LegendreP[l, m, t] also adopted such a convention.
0 Clearly, Pl (t) = Pl(t). Moreover, since the Pl(t) is a polynomial of degree l, then
m Pl (t) = 0 if m > l.
36 Note that the equation (4.11) is invariant under the sign change of m. Indeed,
by writing (d/dt)−1(d/dt) = identity, the definition (4.12) allows m to take negative values −1, −2, · · · − l. Similar to the proof of Theorem 9, it can be shown that
−m −m Pl (t) is also a solution to (4.12). In fact, as the following lemma shows Pl (t) is
m proportional to Pl (t).
Lemma 8. For 0 ≤ m ≤ l, we have (l − m)! P −m(t) = (−1)m P m(t). (4.13) l (l + m)! l ( ) ∑ l Proof. Using the Leibniz formula (uv)l = l u(l−s)v(s) on the (l + m)-th s=0 s differentiation of (t2 − 1)l = (t + 1)l(t − 1)l, we have dl+m (t + 1)l(t − 1)l dtl+m ∑l+m (l + m)! dl+m−s ds = (t + 1)l (t − 1)l. (l + m − s)!s! dtl+m−s dts s=0 Note that the nonzero summands in the above equation are when s ≤ l and
l + m − s ≤ l. We also have the constraint of 0 ≤ l + m − s in order to apply the
Leibniz formula. Thus, if m ≥ 0, then the sum is over s = m to l; if m < 0, then the
sum is over 0 to l + m = l − |m|. Further using the relation dk(t 1)i i! = (t 1)i−k, dtk (i − k)! then it follows that { ∑ − − dl+m 2 − l 2 l (l+m)! (t+1)s m (t−1)l s dtl+m (t 1) = (l!) s=m (l+m−s)!s! (s−m)! (l−s)! ∑ − dl−m 2 − l 2 l−m (l−m)! (t+1)j+m (t−1)l j dtl−m (t 1) = (l!) j=0 (l−m−j)!j! (j+m)! (l−j)! Change the variable j = s − m and we have dl−m ∑l (l − m)! (t + 1)s (t − 1)l+m−s (t2 − 1)l = (l!)2 dtl−m (l − s)!(s − m)! s! (l + m − s)! s=m (l − m)! dl+m = (t2 − 1)m (t2 − 1)l. (l + m)! dtl+m
37 Using the above relation in the definition (4.12) and the Rodrigues’ formula (4.9), we can immediately obtain (4.13).
The associated Legendre functions can be viewed as a collection of different com-
plete orthogonal sets for L2[−1, 1].
{ ···} { m | | | | | | Theorem 10. For a fixed m in 0, 1, 2, , the set Pl (t) l = m , m+1 , m+ 2|, ···} is a complete orthogonality set for L2[−1, 1].
Proof. The orthogonality directly comes from the Sturm–Liouville theory. It can
m m ̸ ′ also be derived from Lemma 1. Since both Pl (cos θ) and Pl′ (cos θ), l = l are eigenfunctions of the spherical Laplacian, we must have from Lemma 1 that they are
orthogonal to each other in L2(S2). Their inner product in L2(S2) can be evaluated
as ∫ ∫ 2π π m m Pl (cos θ)Pl′ (cos θ) sin θdθdϕ = 0. 0 0 ∫ 1 m m After the change of variable by t = cos θ, we immediately have 2π −1 Pl (t)Pl′ (t)dt = 0. Hence, we proved the orthogonality in L2[−1, 1].
m m − 2 m/2 m Let us denote Pl (t) = Pm+k(t) = (1 t ) fk (t). By the Rodrigues’ formula
m and the definition of Pm+k(t), we have
1 d2m+k [ ] f m = (t2 − 1)l . k 2ll! dt2m+k
{ m For the completeness, observe that it suffices to show the completeness of fk , k = 0, 1, 2, ···} in L2([−1, 1], (1−t2)m). The latter notation is the weighted L2-space with inner product given by ∫ 1 (f, g) = f(t)g(t)(1 − t2)mdx. −1
38 Clearly, the basis {1, t, t2, ···} is complete in L2([−1, 1], (1 − t2)m) (by a similar
m argument to the proof of Proposition 2). Therefore, it suffices to show that fk is orthogonal to tj when j = 0, 1, ··· , k − 1. It follows that ∫ ∫ [ ] 1 1 1 d2m+k f mtj(1 − t2)mdx = (t2 − 1)l tj(1 − t2)mdx k 2ll! dt2m+k −1 −∫1 1 1 d2m+k−1 [ ] d [ ] − 2 − l j − 2 m = l 2m+k−1 (t 1) t (1 t ) dx, (4.14) 2 l! −1 dt dt where we used the integration by part and notice that boundary terms are all 0. We
can repeat this process until we have
d2m+j+1 [ ] tj(1 − t2)m = 0, dt2m+j+1
··· − m for j = 0, 1, , k 1. Hence, the integral (4.14) is 0, which shows that fk is orthogonal to tj when j = 0, 1, ··· , k − 1.
2 − m In addition, it can be shown that the square root of the L ([ 1, 1]) norm of Pl (t) is ∫ 1 2 (l + m)! [P m(t)]2 dt = . (4.15) l − −1 2l + 1 (l m)! Its proof can be referred to Arfken et al. [4, page 746-747].
The above analysis immediately lead to (see also Terras [62, page 114])
Theorem 11. (1) A complete orthogonal set of eigenfunctions of ∆S is
m imϕ m ··· | | ≤ Yl (θ, ϕ) = e Pl (cos θ), for l = 0, 1, 2, , m l. (4.16)
m We call σYl a (surface) spherical harmonic of degree l and order m, where σ is a normalizing constant.
l m (2) Let f(x, y, z) = r Yl (θ, ϕ), using the change of coordinates formula (3.7). m S m − m Then Yl is a spherical harmonic satisfying ∆ Yl = l(l + 1)Yl if and only if f(x, y, z) is a homogeneous harmonic polynomial of degree l.
39 Since we already proved that the spherical harmonics are dense in L2(S2), and (4.16)
is a complete orthogonal set of the spherical harmonics,it follows that (4.16) is also √ 2 2 (2l+1)(l−m)! a complete orthogonal set for L (S ). If we further take σ = 4π(l+m)! , then { m ··· | | ≤ } 2 2 σYl , for l = 0, 1, 2, , m l is an orthonormal set of basis for L (S ). In addition, note that by Corollary 1 the dimension of H for n = 3 dimensional ( ) ( ) l n + l − 1 n + l − 3 sphere is − = 2l + 1, which is the same as what we have n − 1 n − 1 from (4.16).
Out[4]=
5 Figure 4.2: Mathematica density plot for ReY10(θ, ϕ) (left) and 5 3 ReY10(θ, ϕ)+ReY5 (θ, ϕ) (right)
The following table of first few degrees of spherical harmonics are generated di-
rectly using Mathematica function SphericalHarmonicY[l, m, θ, ϕ], where the normal- √ √ (2l+1) (l−m)! izing constant σ = 4π (l+m)! is applied.
40 √ √ Y 0(θ, ϕ) = 1 3 cos θ Y 0(θ, ϕ) = 1 1 1 2 π√ 0 2 √π Y 1(θ, ϕ) = ∓ 1 3 sin θe iϕ Y 0(θ, ϕ) = 1 7 (5 cos3 θ − 3 cos θ) 1 √2 2π 3 4 π√ Y 0(θ, ϕ) = 1 5 (3 cos2 θ − 1) Y 1(θ, ϕ) = ∓ 1 21 sin θ(5 cos2 θ − 1)eiϕ 2 4 π√ 3 √8 π 1 ∓ 1 15 iϕ 2 1 105 2 2iϕ Y2 (θ, ϕ) = 2 2π sin θ cos θe Y3 (θ, ϕ) = 4 2π sin θ cos θe √ √ 2 1 15 2 2iϕ 3 ∓ 1 35 3 3iϕ Y2 (θ, ϕ) = 4 2π sin θe Y3 (θ, ϕ) = 8 π sin θe Following the code provided in [62, page 116], we make the density plot on the
m sphere according to the real part of the Yl in Fig. 4.2.
4.2 Hyperspherical Harmonics (n > 3)
There is a natural generalization of the 3D spherical harmonics to the high di-
mensional case. Specifically for n > 3, these spherical harmonics are usually called
hyperspherical harmonics. Most of the familiar theorems for the 3D spherical
harmonics have the nD generalization (see for example, Avery [6]). They are widely
used in the study of quantum theory [6]. We shall briefly recite the theorem of the
hyperspherical harmonics here and give some examples. More details can be referred
to Dai and Xu [15, Chap. 1].
4.2.1 Gegenbauer polynomials
As the spherical harmonics are defined through the Legendre polynomials, the
hyperspherical harmonic is defined through the so-called Gegenbauer polynomials
(also called ultraspherical polynomials, see, for example, Szegő [60, Chap. 4.7]).
The Gegenbauer polynomials is defined as the coefficients of the series expansion of
the generating function (see for example Müller [47, Chap. 9]): ∑∞ 1 l m = h C (t), m ∈ R≥ , (1 − 2th + h2)m l 0 l=0 m where Cl is called the Gegenbauer polynomials of degree l and order m. A more explicit expression [60, Chap. 4.7, page 80, eq.(4.7.6)] using hypergeometric functions
41 can be given as ( ) (2m) 1 1 − t Cm(t) = l F −l, l + 2m, m + ; , (4.17) l l! 2 1 2 2 where (m)l is the rising factorial (Pochhammer’s function) defined as
∏l−1 (m)l := m(m + 1) ··· (m + l − 1) = (m + i), i=0 ( ) l + m − 1 and therefore (m)l = ; the hypergeometric function F is then defined l! l 2 1 by [60, page 63] ∞ ∑ (a) (b) ti F (a, b, c; t) := 1 + i i , 2 1 (c) i! i=1 i which is convergent for |t| < 1 and note that the series terminates if either a or b is a
non-positive integer. Other different representations can be found in Olver [52, page
442], Abramowitz and Stegun [1, 80], as well as Erdelyi et al. [19, page 175].
m ··· The Cl , l = 0, 1, 2, satisfy the Gegenbauer differential equation
(1 − t2)w′′(t) − (2m + 1)tw′(t) + n(n + 2m)w = 0,
and moreover, they are orthogonal on the interval (−1, 1) with the weighting function
(1 − t2)m−1/2.
The first few Gegenbauer polynomials are listed below. They are generated bythe
Mathematica function GegenbauerC[l, m, t]. Some plots of the Gegenbauer functions
are shown in Fig. 4.3.
m m − 2 C0 (t) = 1,C2 (t) = m + 2m(1 + m)t , m m − 4 3 C1 (t) = 2mt, C3 (t) = 2m(1 + m)t + 3 m(1 + m)(2 + m)t .
42 Figure 4.3: Gegenbauer polynomials
4.2.2 Hyperspherical Harmonics
The hyperspherical harmonics are given in the following theorem, which is adapted
from Erdelyi et al. [20, page 239-240]. Similar theorems can also be found in [6, page
40] and Theorem 1.5.1 of [15, Page 18].
Theorem 12. (1) For n > 2, a complete orthogonal set of eigenfunctions of ∆S is
n∏−3 1 − 1 im − ϕ m mk+1+ p k n 2 − − k+1 2 2 − − ··· Yl(m; θ, ϕ) = e (sin θn k 2) Cmk−mk+1 (cos θn k 2), for l = 0, 1, 2, , k=0 (4.18) where m = (m0, m1, ··· , mn−2) are all integers satisfying l = m0 ≥ m1 ≥ · · · ≥ mn−2 ≥ 0, and θ = (θ1, ··· θn−2) are coordinates defined in (3.7). We call σYl a
(surface) hyperspherical harmonic of degree l on Sn−2, where σ is a normalizing constant.
l (2) Let f(x) = r Yl(m; θ, ϕ), using the change of coordinates formula (3.7). Then
S Yl is a hyperspherical harmonic satisfying ∆ Yl = −l(l + n − 2)Yl if and only if f(x) is a homogeneous harmonic polynomial of degree l.
It can be easily verified that the above theorem gives the same orthonormal basis
as those in 4.16 when n = 3. One way to see this is to notice the direct relation
43 between the associated Legendre functions and the Gegenbauer polynomials (see for
example [15, Page 421, Appendix B.4]):
m 1 m 2 m d 2 m m+ P (t) = (1 − t ) 2 P (t) = (2m − 1)!!(1 − t ) 2 C 2 (t), l dtm l l−m ∏ − m − (2m)! where the double factorial is evaluated as (2m 1)!! = i=1(2i 1) = 2mm! . Alter-
m natively, we can verify this from the hypergeometric function representations of Pl
m m and Cl . For Pl , we have (see, for example, Beals and Wong [10, page 296], Laursen and Mita [39]): ( ) ( ) m − m 1 (l + m)! 2 m 1 t P (t) = − (1 − t ) 2 F m − l, m + l + 1, m + 1; . l 2 (l − m)!m! 2 1 2
Using the above formula and equation (4.17), we immediately see that Theorem 11
is a special case of Theorem 12.
For the choice of the normalizing constant σ that makes (4.18) an orthonormal set
of basis of L2(Sn−1), please refer to [15, Page 18]. Applications of 4D hyperspherical harmonics (n = 4) to the medical image representation can be found in [34].
44 Chapter 5: Harmonics Analysis on the Spheroid
Spheroid is obtained by rotating an ellipsoid about either its semi-major axis or
semi minor axis. For the purpose of this thesis, we will only consider the former case, which gives the so-called prolate spheroidal coordinates.
5.1 Prolate Spheroidal Coordinates
The prolate spheroidal coordinate system is illustrated in Fig. 5.1. It can be
formed by rotating a two-dimensional elliptic coordinate system, which consists of el-
lipses and hyperbolas, about the semi-major axis of the ellipse. The prolate spheroidal
coordinate coordinates (ξ, η, ϕ) with 1 ≤ ξ ≤ ∞, −1 ≤ η ≤ 1, and 0 ≤ ϕ ≤ 2π are
related to the Cartesian coordinates (x, y, z) by the following transformation: √ 2 − − 2 x = a√(ξ 1)(1 η ) cos ϕ, y = a (ξ2 − 1)(1 − η2) sin ϕ, (5.1) z = aξη, where a is the focal distance as shown in Fig. 5.1. Equivalently, it can also be denoted
by, x = a sinh µ sin ν cos ϕ, y = a sinh µ sin ν sin ϕ, (5.2) z = a cosh µ cos ν, where the coordinate system (5.2) is related to (5.1) by ξ=cosh µ, µ ∈ [0, ∞), η =
cos ν, ν ∈ [0, π), and recall that the hyperbolic functions satisfy cosh2 µ − sinh2 µ = 1.
45 Spheroidal coordinate system is a kind of curvilinear coordinate systems in the 3
dimensional Euclidean space (see Hobson [33, Chapter X] and Falloon [23]). Indeed,
let ξ be constant, we see (5.1) defines an ellipsoid of revolution:
x2 + y2 z2 + = 1. (5.3) a2(ξ2 − 1) a2ξ2
The case of ξ = 1 is degenerated where the coordinate is defined on the line segment
x = 0, y = 0, −a ≤ z ≤ a. From the above equation, we can see that the variable
ξ scales the semi-major axis and semi minor axis. Hence, it measures how prolate
the spheroid is. It can be seen that the spheroid is elongated along the z-axis as
ξ decreases, while more like a ball when ξ increases. Fig. 5.2 illustrates this effect,
∈ 8 where to have a clearer comparison we only plot the ellipsoid for ϕ [0, 5 π]. Now let η ∈ (−1, 1) be constant, and we will have from (5.1) hyperboloid of two
sheets: x2 + y2 z2 − = −1. a2(1 − η2) a2η2
The cases of η = 1 are degenerated where the coordinate system are defined on the
two line segments starting from the two foci (0, 0, a) to infinity along the z-axis,
respectively.
Finally, let ϕ be constant, we have a half plane through the z-axis ϕ = arctan(y/x).
As analogous to spherical coordinate systems, we illustrate the ellipsoid and the hy-
∈ 8 perboloids in the right of Fig. 5.1, where only the part of ϕ [0, 5 π] are plotted to give a better view of the structure.
46 푧
휂 =1 휂 =const
퐹 휉 =const 휙 = 휋/2 푎
휙 =0 0 푦 휙 푎
푥 퐹
휂 =1 휙 =const
Figure 5.1: Prolate spheroidal coordinate systems (left) and Mathematica illustration ∈ 8 with ϕ [0, 5 π] (right)
∈ 8 Figure 5.2: Constant coordinate curves of equation (5.3) for ϕ [0, 5 π] with different values of ξ = cosh µ, where µ ∈ {0.4, 0.6, 0.9, 1.2, 1.5}.
47 5.2 Spheroidal Harmonics
The spheroidal harmonics are the solutions of the Helmholtz equation in
the spheroidal coordinates. The Helmholtz equation is given by
∆f + κ2f = 0, where κ is referred to as the wave number, which is the spatial frequency of a wave (i.e.,
how many wave cycles within unit spatial distance). Similar to the transformation of
the Laplacian equation in the 3D spherical coordinates, we can have the Laplacian in
the spheroidal coordinates (5.1) as { ( ) ( )} 1 ∂ ∂ ∂ ∂ 1 1 ∂2 ∆ = (ξ2 − 1) + (1 − η2) + , a2(ξ2 − η2) ∂ξ ∂ξ ∂η ∂η a2 (ξ2 − 1)(1 − η2) ∂ϕ2 where the detailed derivation can be found in Flammer [24, page 10]. It follows that
the Helmholtz equation in spheroidal coordinates is ( ) ( ) ∂ ∂f(ξ, η, ϕ) ∂ ∂f(ξ, η, ϕ) (ξ2 − 1) + (1 − η2) ∂ξ ∂ξ ∂η ∂η c2(ξ2 − η2) ∂2f(ξ, η, ϕ) + = c2(η2 − ξ2)f(ξ, η, ϕ), (5.4) (ξ2 − 1)(1 − η2) ∂ϕ2 where we defined the spheroidicity parameter c = κa. Therefore, the wave number
can also be thought of relatively as a scaling factor of the unit length. Moreoever, if we
restrict the solution f(ξ, η, ϕ) of (5.4) to a spheroid, that is, if we fix ξ to be a constant
ξ0, then we see that f(ξ0, η, ϕ) is not an eigenfunction of the Laplacian under the
2 2 − 2 spheroidal coordinates, because the right side of (5.4), which is c (η ξ0 )f(ξ0, η, ϕ), is not a constant multiple of f. This is in contrast to the case of spherical harmonics, where the restriction of the solutions of the (Euclidean) Laplacian equation to the
sphere gives you the eigenfunctions of the spherical Laplacian. However, still f(x, y, z)
48 using the change of coordinate (5.1) is an eigenfunction of the Euclidean Laplacian with eigenvalue −κ2.
By the separation of variables, we can assume f(ξ, η, ϕ) = R(ξ)S(η)Φ(ϕ), where
R(ξ) is called the radial function, S(η) is called the angular function, and Φ(ϕ) is the azimuthal function.
Then we can have the following 3 ordinary differential equations (ODE): ( ) ∂ ∂R(ξ) α (ξ2 − 1) − (λ − c2ξ2 − )R(ξ) = 0, (5.5) 2 − ∂ξ ( ∂ξ ) ξ 1 ∂ ∂S(η) α (1 − η2) + (λ − c2η2 − )S(η) = 0, (5.6) ∂η ∂η 1 − η2 ∂2Φ(ϕ) + αΦ(ϕ) = 0, (5.7) ∂ϕ2 where λ and α are the separation constants. The solution to the last equation can be √ easily obtained as Φ(ϕ) = ei αϕ. Again, as same as the case of spherical harmonics,
for physical problems we assume Φ(ϕ) is a periodic function such that Φ(0) = Φ(2π), √ and therefore, α = m must be an integer.
There is no closed form solution of the spheroidal differential equations (5.5)-
(5.6). They are usually represented by infinite series expansions of various other
functions, such as the associated Legendre functions and the spherical Bessel func-
tions. The derivation of the corresponding coefficients are discussed in details in
Flammer [24]. The computation of the spheroidal wave functions (i.e., the an- gular functions and the radial functions) are generally very difficult [2]. This is the
major reason why the spheroidal harmonics are not as popular as spherical harmon-
ics in practice. However, the spheroidal harmonics have many important and unique
applications in atomic, band-limited functions, acoustic scattering problems, etc. A very recent review about prolate spheroidal harmonics can be found in [65]. The
49 handbook edited by Olver et al. [51, Chapter 30] gives a concise summary of the
important results of spheroidal harmonics. In addition, the various notations for
spheroidal harmonics used in the literature are tabled and compared in Abramowitz
and Stegun [1, Chapter 21.11, page 758] and Flammer [24, page 14-15].
Lastly, we note that all three equations (5.5)-(5.7) are Sturm–Liouville equations.
Therefore, spheroidal harmonics with different λ and m are orthogonal to each other
under the L2 inner product defined by ∫ ∫ ∫ 2π 1 ∞ ⟨ ⟩ f, g sphrd := f(ξ, η, ϕ)¯g(ξ, η, ϕ)dξdηdϕ, 0 −1 1 where the subscript sphrd is to remind us of the space under the spheroidal coordi-
nates.
In contrast, in the spherical harmonics case, only two (see eq. 4.2-4.3) of the
three ODEs separated from the Laplacian equation under the spherical coordinates
are Sturm–Liouville equations (the other one is the Cauchy-Euler equation). Hence,
the spherical harmonics are only defined on the sphere surface (orthogonal under the
L2(Sn−1) inner product).
In the following, we will briefly review the representations of spheroidal wave
functions in order to gain more understanding of them.
5.3 Series Expansion Representation of Spheroidal Wave Func- tions
Equations (5.5) and (5.6) are essentially the same (see Morse and Feshbach [46,
page 643]). Indeed, if we set ξ = z, and R = (ξ2 − 1)m/2ψ(z) in (5.5) or in (5.6) we
set z = η, and S = (1 − η2)m/2ψ(z), then the equation for ψ from both equations will
50 reduce to
(z2 − 1)ψ′′ + 2(m + 1)zψ′ + (λ + c2z)ψ = 0.
Therefore, we can see that the both radial function and the angular function are the
solution of the above equation, but on different domain. Moreover, it can be shown
that the numbers λ for which (5.5) and (5.6) have non-trivial finite solutions at the
singular points ξ = 1 and η = 1, respectively, are certain discrete real positive values
(see for example, Morse and Feshbach [46, page 1503], Chu and Stratton [13], and
Flammer [24]). We shall denote them by λml(c), l = m, m + 1, m + 2, ··· . They are
the Sturm–Liouville eigenvalues of the spheroidal differential equations. In addition,
note that in the case of m = 0, R and S will be the same up to a real scale factor
(due to different normalization scheme).
When c = 0, equation (5.6) becomes (4.11) which is satisfied by the associated
m Legendre functions Pl , and λml(c) reduces to l(l+1). Recall that c = κa and a is the focal distance. It is not hard to see that when c → 0, the prolate spheroidal coordinate
system reduces to the spherical coordinate system (see for example, Falloon [23, page
15]). Hence, the angular functions Sml(c, η) of degree l and order m, corresponding to
m λml(c), are usually expanded in terms of the associated Legendre functions Pm+k(η) when c is small (usually less than 10),
∑∞ ′ ml m Sml(c, η) = dk (c)Pm+k(η), (5.8) k=0,1 where the prime ( ’ ) symbol means that the summation is over only even values of k when l − m is even, and over only odd values of k when l − m is odd. Hence, Sml(c, η)
m ml has the same even or odd property with respect to η as Pl (η). The coefficients dk (c) are sometimes referred to as the D constants.
51 Plug in the expansion (5.8) into (5.6), and use the recurrence formula (see (C.6)
of Appendix C) for associated Legendre functions, yielding the following 3-term re-
cursion relation in terms of the D constants: [ (2m + k + 2)(2m + k + 1)c2 dml (c) + (m + k)(m + k + 1) − λ (c) (2m + 2k + 3)(2m + 2k + 5) k+2 ml ] 2(m + k)(m + k + 1) − 2m2 − 1 k(k − 1)c2 + dml(c)+ dml (c) = 0, (2m + 2k − 1)(2m + 2k + 3) k (2m + 2k − 3)(2m + 2k − 1) k−2 (5.9) where k ∈ Z≥0. Further complicated analysis [24, Chapter 3] based on (5.9) leads
to procedures to approximate λml(c). A common memthod to approximate λml(c)
is the so-called Bouwkamp procedure [12, 23, 24]. Once we have the approximation
ofλml(c), the D constants can be calculated subsequently from (5.9). We shall skip
these details here.
Note that different normalization schemes are used for the D constants1 [ , Chapter
21, page 755]. The Meixner and Schäfke scheme [43, page 286] is to have the same
m normalization factor as Pl (η) (recall the formula (4.15)), that is, ∫ 1 2 2 (l + m)! [Sml(c, η)] dη = − . (5.10) −1 2l + 1 (l m)!
The solutions of the ODE (5.5) are the radial functions Rml(c, ξ) of degree l
and order m, for 1 ≤ ξ ≤ ∞, corresponding to the same λml(c) as the angular
functions Sml(c, η) do, where l = m, m + 1, m + 2, ··· . The radial function Rml(c, ξ)
is usually expanded by the spherical Bessel functions, which are the solutions of
the Helmholtz equation solved in the spherical coordinates (see [46, page 621-622] [4,
Chapter 14.7] [23] for more details about the spherical Bessel functions). To justify
this expansion, changing the variable by ζ = cξ and writing R(ξ) = (ξ2 − 1)m/2g(ζ)
52 in (5.5), we obtain
∂2g(ζ) ∂g(ζ) ( ) (ζ2 − c2) + 2(m + 1)ζ + ζ2 + µ(µ + 1) − λ g(ζ) = 0. ∂ζ2 ∂ζ
When c → 0, the above equation becomes
∂2g(ζ) ∂g(ζ) ( ) ζ2 + 2(m + 1)ζ + ζ2 + µ(µ + 1) − λ g(ζ) = 0. ∂ζ2 ∂ζ
Let g(ζ) = ζ−mh(ζ), and then we obtain that
∂2h(ζ) ∂h(ζ) ( ) ζ2 + 2ζ + ζ2 − λ h(ζ) = 0. (5.11) ∂ζ2 ∂ζ
Equation (5.11) is exactly the spherical Bessel equation, satisfied by the spher-
ical Bessel functions jl(ζ), which can be represented using the Rayleigh formula [4,
page 703] ( ) 1 d l j (ζ) = (−1)lζl j (ζ), l ζ dζ 0
sin ζ where j0(ζ) = ζ . Hence, for small value of c, we can define the radial spheroidal harmonics as ∑∞ 2 − m/2 −m ′ ml Rml(c, ξ) = (ξ 1) (cξ) ak (c)jm+k(cξ), k=0,1 where similar to the case of angular function (5.8), the prime ( ’ ) symbol after the sum means that the summation is over only even values of k when l − m is even, and over only odd values of k when l − m is odd.
More details about this expansion can be found in equation (238)-(240) of both [13,
page 299] and [59, page 41] (Note that [59] reprinted the material of [13], and included
data and tables for spheroidal functions additionally).
Note that when c → 0, the radial spheroidal harmonics reduces to the spherical
ζ Bessel functions, limc→0 Rml(c, c ) = jl(ζ) (See Falloon’s Master thesis [23, Chapter 2.2, page 44] and Thompson [63]).
53 Similar to the case of angular functions, we can derive the recurrence relations
ml satisfied by the coefficients ak (c) [23, Appendix 4.3]. A popular normalization
ml scheme [23, Chapter 2.2] for ak (c) is to have
as cξ→∞ Rml(c, ξ) −−−−−→ jl(cξ), (5.12)
and note also that
as cξ→∞ 1 1 j (cξ) −−−−−→ sin(cξ − lπ). l cξ 2
This normalization scheme has been adopted by [24, page 31] [13, page 300] [43][1,
Chapter 21, page 755]. See also the summary in [22, 63].
Detailed algorithms for calculating the spheroidal wave functions can be referred
to [2,9,22–24,36,50,64,65] and the references therein. We shall not dive into further
details.
The spheroidal wave functions Sml(c, η) and Rml(c, ξ) are implemented in the
Mathematica by the functions SpheroidalPS[l, m, c, η] and SpheroidalS1[l, m, c, ξ], respectively. The angular functions are normalized using the Meixner and Schäfke scheme. However, the Mathematica built-in radial functions often yields the wrong values. In fact, as commented by Noah Graham [29], the Mathematica built-in func-
tions seem to be based on P. E. Falloon’s package [22, 23], since they share many of
the same bugs. Noah Graham modified Falloon’s package to optimize the code and fix
many bugs [18]. The calculation speed is much faster than the Mathematica built-in
function by 2-100 times. Note that Graham’s modified functions use the same name
as the Mathematica built-in functions, but adjust the order of the last two arguments, which are called by SpheroidalPS[l, m, η, c] and SpheroidalS1[l, m, ξ, c].
54 Figure 5.3: Examples of prolate spheroidal wave functions: angular functions (left) and radial functions (right)
We first plot several angular and radial functions inFig. 5.3 using Graham’s modified functions, where we choose c = 5, m = 10, and l = 10, 11, ··· , 15. To have a better comparison between angular functions of different degrees l, we further normalize the angular functions by its L2 norm, that is, by the square root of the ¯ number obtained in (5.10). Hence, the re-normalized function (denoted by Sml(c, η)) satisfy ∫ 1 [ ] ¯ 2 Sml(c, η) dη = 1, −1 and their plots are shown in Fig. 5.4.
55 ¯ Figure 5.4: Renormalized angular functions Sml(5, η)
Figure 5.5: R10,10(5, ξ) obtained from Mathematica built-in function (left) and Gra- ham’s modified function (right)
We then compare one radial function calculated by both Mathematica’s built- in function and Graham’s modified function in Fig. 5.5. It can be seen that the
Mathematica’s built-in function gives the wrong value around ξ = 10.2. In fact, the function values within ξ ∈ [10.1655, 10.2504] are identically 0 in the left of Fig. 5.5.
56 Finally, we show the density plots of the spheroidal harmonics Rml(c, ξ)Sml(c, η)Φ(ϕ)
on the ellipsoid (5.3) with a = 1 and ξ = cosh 1 = 1.54308 in Fig. 5.6. The density is
given by R3,4(3, ξ)S3,4(3, η) cos 3ϕ for the left figure and by R3,4(3, ξ)S3,4(3, η) cos 3ϕ +
R2,4(3, ξ)S2,4(3, η) cos 2ϕ for the right figure.
Figure 5.6: Mathematica density plot for R3,4(3, ξ)S3,4(3, η) cos 3ϕ on the left and R3,4(3, ξ)S3,4(3, η) cos 3ϕ + R2,4(3, ξ)S2,4(3, η) cos 2ϕ on the right
57 Chapter 6: Application Example
In this chapter, we will demonstrate using the spheroidal wave functions to cal-
culate the eigenvalue distributions of the Gaussian unitary ensemble (GUE) (see for
example, Mehta’s book [41], as well as [3,35]. The reason why we are interested in the
eigenvalue distribution of GUE is that they are possibly related to the zeros of the
Riemann zeta functions according to the Hilbert and Pólya (HP) conjectures. The
HP conjectures say that the zeros of the Riemann zeta function correspond to eigen- values of a positive linear operator. The confirmation of the HP conjectures would
imply the Riemann Hypothesis (RH), which says that the Riemann zeta function
has its zeros only at the negative integers (called trivial zeros) and complex numbers
1 with real part 2 (called non-trivial zeros). The RH is of central interest to number theorists because it would imply results about the distribution of prime numbers.
Heuristically, for the HP conjectures, if such a positive linear operator exists, its eigenvalues might behave like the limit of a sequence of random matrices. Therefore, the zeros of the Riemann zeta function might be distributed like the eigenvalues of a large random matrix. Certain analytical results on the zeros of the Riemann zeta function demonstrate consistence with the eigenvalue distribution of the GUE. For more details about the HP conjecture, please refer to Odlyzko [48] and the reference
therein.
58 In the following, we will compare distributions of the normalized spacings between
the non-trivial zeros of the Riemann zeta function with those of normalized spacings
between the eigenvalues of the GUE. For distributions of the (normalized) spacings be-
tween zeros of the Riemann zeta function, we use the existing database (see Odlyzko’s
homepage [49] and Hiary [31,32]) to calculate the empirical distribution. For distribu- tions of the (normalized) spacings between eigenvalues of GUE, we directly calculate the distribution from analytic formulas (see Mehta [41, page 132]). These analytic
formulas are in terms of the so-called Linear prolate functions (see Slepian and
Pollak [57], [44]), which are special cases of prolate spheroidal wave functions of order
m = 0. We shall briefly review some relevant results in the following.
6.1 Linear Prolate Function
The linear prolate functions fl(x) are defined as eigenfunctions of the integral equation ∫ 1 sin (πt(x − y)/2) γlfl(x) = − fl(y)dy, (6.1) −1 π(x y)
+ where γl is the corresponding eigenvalues of the integral operator, and t ∈ R is a
parameter. Moreover, we shall normalize fl(x) such that they satisfy orthonormality
condition ∫ 1 fk(x)fr(x)dx = δkr. (6.2) −1 Note the above normalization is different from Slepian and Pollak [57, page 58, eq.
(28)]. In addition, in the notation of [57, page 45, eq. (11)], we will have T = 2,
Ω = πt/2.
The γl and fl(x) can be renumbered so that 1 ≥ γ0 ≥ γ1 ≥ · · · ≥ 0. It can be
shown that the integral in (6.1) is actually a finite Fourier transform. In fact, it equals
59 (see Terras [62, page 55-56] and [57]) ∫ ∫ c 1 1 ixw −iyw BT fl(x) = e fl(y)e dydw, (6.3) 2π −c −1 where c = πt/2 and { f(x), if |x| ≤ 1, T f(x) = 0, if |x| > 1, ∫ 1 c w Bf(x) = fˆ( )eixwdw, 2π −c 2π and fˆ is the Fourier transformation of f (see the definition in the appendixA.2 ( )).
From their investigation of the band-limited functions, the authors in [57] dis-
covered that the eigenfunctions fl(x) of the integral operator in (6.1) are in fact
spheroidal wave functions. More specifically, they are the so-called linear prolate
functions which satisfies d2u du (1 − x2) − 2x + (λ − c2x2)u = 0, (6.4) dx2 dx where c = πt/2 as in (6.3). We immediately see that both spheroidal differential
equations (5.5) and (5.6) reduce to (6.4) when m = 0. Therefore, fl(x) are in fact
the (prolate) spheroidal wave functions S0l(c, x) and R0l(c, x), l = 0, 1, ··· . In fact, as
discussed at the begining of Section 5.3, when m = 0, S0l(c, x) differs from R0l(c, x)
only by a real scale factor (see also [57, page 57]).
On the other hand, if the radial functions are normalized in the way of (5.12), then it can be shown that [57][24, page 51] [43, page 312] [44] ∫ 1 2c 2 sin c(x − y) [R0l(c, 1)] S0l(c, x) = − S0l(c, y)dy. π −1 π(x y)
The above equation shows that S0l(c, x) is a solution of the integral equation (6.1)
corresponding to the eigenvalue 2c γ (c) = [R (c, 1)]2 , l = 0, 1, 2, ··· (6.5) l π 0l 60 Note that the Sturm–Liouville eigenvalues λ = λ0l(c) should not be confused with
γl(c). Also, notice that Rml is defined on ξ ∈ [1, +∞) and Sml is defined on η ∈ [−1, 1].
See the discussion at the begining of Section 5.3.
Furthermore, define ul(c) such that ∫ 1 2 2 2 [ul(c)] = [S0l(c, x)] dx = , (6.6) −1 2l + 1 where the last equality is by (5.10). Then we normalize the angular function to obtain
1 fl(x) = fl(c, x) = S0l(c, x), ul(c)
so that fl(c, x) satisfies (6.2). Recall the series expansion of Sml(c, x) in (5.8), we can
further have that ∑∞ 1 ′ 0l fl(c, x) = dk (c)Pk(η), (6.7) ul(c) k=0,1 which is expanded by the Legendre polynomials. Then it can be seen immediately
that fl(c, x) is an odd (or even) function if l is an odd (or even) integer.
6.2 GUE Eigenvalue Spacing Distribution
Let p(k, u) denote the probability density of the k-th normalized neighboring spacings (according to Wigner’s semi-circle law; see Mehta’s book [41, Chapter 4]) of
the GUE eigenvalues, i.e., the probability that the (normalized) difference between an
eigenvalue of the GUE and the (k + 1)th smallest eigenvalue of those that are larger
than it lies in the interval (a, b) is ∫ a p(k, u)du. b
61 Then as shown in [42, page 335] [41, page 132], the probability density function p(k, t)
is given by ( ) ( ) ∏∞ ∑ k∏+2 −2 γji p(k, t) = 4t (1 − γn) 1 − γj n=0 (j) i=1 i ∑k+2 − − − − fjl (1)fjr ( 1) (fjl (1)fjr ( 1) fjl ( 1)fjr (1)) , l,r=1 where (j) denotes the set of arbitrary indices 0 ≤ j1 < ··· < jk+2. Using the odd/even
property of fl as demonstrated through (6.7), the above formula can be equivalently written as ( ) ∏∞ ∑ −2 p(k, t) = 4t (1 − γn) V (j1, ··· , jk+2), (6.8) n=0 (j) where ( ) k∏+2 γ ∑ ··· ji 2 2 V (j1, , jk+2) = 4 fjl (1) fjr (1) . (6.9) 1 − γj i=1 i 1≤l 132]: ( ) ∞ ∞ ∞ ∏ ∑ γ ∑ γ −2 − 2s 2 2k+1 2 p(0, t) = 16t (1 γn) f2s(1) f2k+1(1). (6.10) 1 − γ2s 1 − γ2k+1 n=0 s=0 k=0 But note that numerically formula (6.10) is as efficient as (6.8) when k = 0. Using the radial and angular spheroidal functions, we can calculate γl(c) and fl(c, x), which can be further used to calculate the spacing distribution of the GUE eigenvalues via (6.8) or (6.10). Note that for the infinite product in (6.8) and (6.10), ∏ N0 we can approximate it by calculating a finite product n=0, where N0 is the truncation point. Correspondingly, we approximate the infinite sum over all (j) in (6.8) by a finite sum over those (j) which satisfy jk+2 ≤ N0. And similarly, we approximate the infinite { ⌊ ⌋} ∈ ··· N0 sums over all non-negative s and k in (6.10) by finite sums over s 0, 1, , 2 62 { ⌊ ⌋} − ∈ ··· N0 1 ⌊·⌋ and k 0, 1, , 2 , respectively, where is the floor function. Note that the bottleneck of an accurate computation of p(k, t) lies in the infinite summation part. We will discuss this later in equation (6.11). Since the eigenvalue γl(c) decreases to 0 very fast as n increases, generally N0 = 7 can produce very accurate results as compared to the empirical distribution of the k-th neighboring spacings of the Riemann zeta zeros with k ≤ 4. 6.3 Simulation Results We will compare the spacing distribution of Riemann zeta zeros with that of the GUE prediction (6.8). We first compare the nearest neighbor spacing distri- butions. For the GUE prediction, we directly solve for the p(0, t) at the points ∏ ··· ∞ t = 0.05, 0.1, , 3, where note that we approximate the infinite product n=0 with ∏ N0 the finite product n=0 with N0 = 4, and correspondingly approximate the infinite ∑ ∑ sum with in equation (6.8). For the empirical data, we used 104 (j) (j), jk+2≤N0 zeros above the height 267653395647 (the imaginary part of the zero) from [49] and 107 zeros above the height 1028 from [31]. They correspond to the left and right figures in Fig. 6.1 respectively. According to [48], the spacing are normalized as d − d d δ = n+1 n ln n , n 2π 2π where dn is the imaginary part of the n-th zeros of the Riemann zeta function. Note γn that there is a typo in eq. (2.7) of [48], where (log γn)/2π should be log( 2π ) instead, and the log function there is actually ln and the γn notation is the dn in this thesis. We can see from Fig. 6.1 that the numerical data appear to be a reasonable match for the GUE prediction. As the height and the number of zeros increase, the fit of the empirical distribution to the GUE prediction improves. 63 1.2 Empirical data 1.0 GUE prediction 0.8 0.6 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 6.1: Probability density of the nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). Next, we compare the distribution of the next-nearest neighbor spacing δ(2) = δn + δn+1 with the GUE predicted distribution p(1, t). The values of the distribution p(1, t) at points t = 0.05, 0.1, ··· , 4 are approximated using (6.8) with the truncation 4 point N0 = 4. The same 10 zeros from [49] are used to plot the empirical distribution in the left figure of Fig. 6.2. For the empirical distribution plotted in the right, we use 107 zeros near the height of 1028 from [31]. Similar to the case of the nearest neighbor spacing distribution, we see that the match between the empirical distribution of the next nearest neighbor spacing distribution and the GUE prediction improves as the height and the number of zeros increase. (3) The distribution of 3rd nearest neighbor spacing δ := δn+δn+1+δn+2 is compared against p(2, t) in Fig. 6.3 at the discrete points t = 1.05, 1.1, ··· , 5. We approximate 4 p(2, t) by evaluating formula (6.8) with the truncation point N0 = 6. The same 10 zeros from [49] are used to plot the empirical distribution in the left figure of Fig. 6.2. For the empirical distribution plotted in the right, we use 107 zeros near the height of 1028 from [31]. Again, we observe a better match between the empirical data and 64 1.0 1.0 Empirical data Empirical data 0.8 GUE prediction 0.8 GUE prediction 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 1 2 3 4 0 1 2 3 4 Figure 6.2: Probability density of the next-nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). the GUE prediction as the height and the number of zeros increase. However, the right figure shows that the discrepancy between the two densities is more noticeable around δ(3) = 3 when compared to the the previous plots. (4) Finally, the distribution of 4th nearest neighbor spacing δ := δn + δn+1 + δn+2 + δn+3 is compared against p(3, t) in Fig. 6.4 at the discrete points t = 2.05, 1.1, ··· , 6. Again, we p(2, t) by evaluating formula (6.8) with the truncation point N0 = 7. In addition, the data sets for the zeros of the Riemann zeta function are the same as those used in the last experiment. We can see a even larger discrepancy in the peak values around δ(4) = 4 shown in the right figure than the previous comparison plots. Furthermore, we plot the mismatch between the empirical data and the GUE prediction in Fig. 6.5 and 6.6 for k = 0, 1, 2, 3 respectively (i.e., empirical data minus the GUE prediction). In particular, we plot in the left of Fig. 6.5 the difference between different numbers of zeros from [31] and the GUE prediction of p(0, t) with N0 = 4 . It can be observed that as the number of zeros increases, the fit between the GUE prediction and empirical data improves. In the right of Fig. 6.5, the mismatch 65 corresponds to the right figure in Fig. 6.2. Moreover, it is worth mentioning that in the last two experiments, we also tried to evaluate p(2, t) with N0 = 5 and p(3, t) with N0 = 5, 6, respectively (comparison plots are not shown here). For the case of k = 3, we observe a significant improvement of the accuracy of evaluating p(k, t) and better fit to the empirical data when N0 increases from 5 to 6, but very little improvement from 6 to 7, as seen from the right figure in Fig. 6.6. For the case of k = 2, the improvement from N0 = 5 to N0 = 6 is not significant, as can be seen from the left of Fig. 6.6. The mismatch plots in Fig. 6.6 also imply that the highest accuracy of the computed p(k, t) for k = 2, 3 is about 0.001. Lastly, we note that for the case of k = 1, 2, 3, the mismatch plots demonstrate similar patterns that are related to the probability density functions. However, using larger data sets might reduce such differences. The exact evaluation of p(k, t) is when N0 → ∞. We can roughly estimate the numerical complexity by increasing N0 when evaluate p(k, t) approximately. Assum- ing the complexity of evaluating γl(c) and fl(c) are constant for different value of l and c. Then the increase of the computation in (6.8) is approximately proportional ( ) N to the increasing summands indexed by (j), which has a total number of 0 . k + 2 Therefore, the increase from N0 to N1 would increase the computation complexity by ( ) /( ) ∏N1 ∏N1 i + 1 i i + 1 = , (6.11) k + 2 k + 2 i − k − 1 i=N0 i=N0 times. However, since the default precision of the spheroidal function package is relatively very high, and the bottle neck to improve the precision of p(k, t) lies in the evaluation of the formula (6.8), one can reduce the overall computational complexity by setting a lower numerical precision inside the spheroidal functions package. Alternatively, 66 since the evaluations around the peak values of p(k, t) are prone to larger numerical errors, one may consider using high N0 for around the center point and low N0 for the other points. The Mathematica code for the above computation (of the k = 0 case) is included in Appendix D. 1.0 1.0 Empirical data Empirical data 0.8 GUE prediction 0.8 GUE prediction 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 1 2 3 4 5 1 2 3 4 5 Figure 6.3: Probability density of the 3rd nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). Figure 6.4: Probability density of the 4th nearest neighbor spacing. Solid line: GUE prediction. Scatter plot: empirical data based on 104 zeros above the height 267653395647 (left) and 107 zeros above the height 1028 (right). 67 Figure 6.5: Difference of the probability densities between the empirical dataand GUE prediction. 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Wang, “A review of prolate spheroidal wave functions from the perspective of spectral methods,” Journal of Mathematical Study, vol. 50, no. 2, pp. 101–143, June 2017, global-Science Press. 74 Appendix A: Harmonic Analysis on the Flat Space Rm On the flat space Rm, the distance metric is the usual Euclidean distance, with ∑ ∏ 2 m 2 m the arc length element ds := i=1 dxi , the volume measure dµ = i=1 dxi, and the Laplacian defined by ∑m ∂ ∆ := , (A.1) ∂x2 i=1 i whose eigenfunctions are {e2πix·y, ∀y ∈ Rm}, i.e., 2πixT y − 2∥ ∥2 2πix·y ∆e = 4π y 2 e , T where ∥·∥2 denotes the Euclidean norm, induced by the inner product x · y := x y. Note that ∆ is translation invariant, meaning that ∀k ∈ Rm and differentiable func- tion f on Rm, we have ∆(k · f) = k · (∆f), where k act as translation such that (k · f)(x) = f(x + k). The harmonic analysis on Rm is usually known as the Fourier analysis. We give a briefly overview of the Fourier analysis, first on Schwartz functions, thenon L1 and L2 functions, and we also comment briefly on Fourier analysis for the generalized functions (also known as distributions). Denote by (Z+)m the set of m-dimensional vectors consisting of non-negative + m integer entries. For a ∈ (Z ) , a = (a1, a2, ··· , am) we shall use the multi-index |a| a a1 a2 ··· am a ∂ | | ··· notation x = x1 x2 xm and D = a1 ··· am with a = a1 + + am. ∂x1 ∂xm 75 Definition 1. The Schwartz space S is the space of all infinitely differentiable Rm → C a b ∈ Z+ m functions f : such that supx x D f(x) is finite for all a, b ( ) . Such functions will be called Schwartz functions. Given f ∈ S, its Fourier transform fˆ is defined by ∫ fˆ(y) = f(x)e−2πix·ydx, (A.2) Rm where x, y ∈ Rm. Observe that this integral converges absolutely for all y ∈ Rm, ∫ ∫ |fˆ(y)| ≤ |f(x)|dx ≤ (1+∥x∥2)−n sup (1+∥x∥2)n|f(x)|dx < ∞, Rm Rm x∈Rm ∥ ∥2 n| | S where note that supx∈Rm (1+ x ) f(x) is finite by the definition of and the con- ∫ ∥ ∥2 −n vergence of the integral of Rm (1+ x ) dx is easy to see. One of the properties of the Fourier transform is ˆ fˆ(x) = f(−x), (A.3) which is a restatement of the well-known Fourier inversion formula ∫ f(x) := fˆ(y)e2πix·ydy. (A.4) Rm ∫ Define the inner product for f , g ∈ S by (f, g) = fg¯ with the Lebesgue integral, where g¯ is the complex conjugate of g. Then the Parseval identity says that (f, g) = ˆ 1/2 (f, gˆ), ∀f, g ∈ S. Setting ∥f∥2 = (f, f) , we have the Plancherel identity ∥f∥2 = ˆ ∥f∥2. Moreover, the Fourier transform of the Schwartz functions gives a continuous bijective mapping from vector space S to itself (see for example Friedlander and Joshi [25, page 96]). ∞ Rm Denote by Cc ( ) the space of compactly supported infinitely differentiable func- ∞ Rm S ∞ Rm tions. Since Cc ( ) is a subset of the Schwartz space , and Cc ( ) is dense in 76 the space L2(Rm) of Lebesgue square integrable functions (see for example Stein and Shakarchi [58, Lemma 3.1, page 222]), it follows that S is dense in L2(Rm). Note that integral in the definition (A.2) for f ∈ L2(Rm) is not necessarily convergent. The Fourier transform for functions in L2(Rm) is then defined as the L2 limit (see Terras’s book [62, page 11]), ∫ lim f(x)e−2πix·ydx. →∞ ρ ∥x∥≤ρ The convergence of the above limit in L2 can be easily shown by the L2 dominated convergence theorem (see for example Simon [56, page 247]). The inverse Fourier transform of L2(Rm) functions can be calculated by a similar limiting process (see for example Dym and McKean [17, page 132]). If f ∈ L1(Rm) is a Lebesgue absolutely integrable function, then by the Lebesgue dominated convergence theorem, we have the Fourier transform ∫ fˆ(y) = f(x)e−2πix·ydx, Rm converges absolutely. However, the Fourier transform fˆ need not be in L1(Rm). For example, the indicator function of [−1, 1]m is integrable, but its Fourier transform ∫ ∏ −2πix·y m −1 [−1,1]m e dx = i=1(πyi) sin(2πyi) is not. Hence, it is a more delicate process to recover an L1 function from its Fourier transform. More details in that case can be found in Terras’s book [62, page 12], as well as Dym and McKean’s book [17, page 103]. Now we look at the Fourier transform of distributions. Let D be the space of test functions f : Rm 7→ C such that f and all partials of all orders of f are continuous and vanish off a bounded set. Then a distribution T is a continuous linear functional T : D 7→ C. The continuity of T depends on the topology on D, which 77 is specified as follows. A sequence of test functions fn ∈ D converges to f ∈ D if m all the fn’s vanish off the same bounded set (i.e., have compact support in R ) and the sequence of the partial derivatives of fn converges uniformly to the corresponding partial of the limit function f, i.e., for all multi-index a ∈ Zm, a sup |D (fn − f)| → 0, as n → ∞. x∈Rm For example, suppose g is a locally integrable function; i.e., g is Lebesgue in- tegrable over every bounded Lebesgue measurable subset of Rm. Then g defines a distribution Tg via ∫ Tg(f) = g(x)f(x)dx, for any f ∈ D, where dx is the usual Lebesgue measure on Rm. ′ We shall denote by D the space of the distributions. ∫ ∫ ˆ ˆ It can easily show that ∀f, g ∈ S, fg = fgˆ. In other words, Tg(f) = Tgˆ(f). This suggests that we might attempt to define the Fourier transform of distributions as Tˆ(f) = T (fˆ) for all f ∈ D. However, this will imply that f must be identically zero, since if f ∈ D and f ≠ 0 then we must have fˆ ∈ / D. Instead, we define the Fourier transform on a subset of D′, namely the tempered distributions. A distribution T is said to be a tempered distribution if it extends to a continuous linear functional on the space S of Schwartz functions. Here, the continuity notion defined over S is stronger than that defined over the general test functions in D. The convergence of a sequence of Schwartz functions gn to a Schwartz function g means that all the a b m sequences x D (gn − g) converge to 0 uniformly on R for all multi-indices a, b in (Z+)m. It can be shown that if g ∈ S, then gˆ ∈ S. Therefore, the definition of the Fourier transform of distribution Tˆ(g) = T (ˆg), ∀g ∈ S is well defined. If we consider 78 ∫ the case where T is induced by a locally integrable function f, i.e., T (g) = fg, then the definition of Fourier transform of distributions coincides with the definition of Fourier transform of ordinary functions (for example, Schwartz functions), namely, Tˆ(g) = T (ˆg) ∫ = f(x)ˆg(x)dx ∫Rm ∫ = f(x) e−2πix·yg(y)dydx ∫Rm ∫ Rm = f(x)e−2πix·ydx g(y)dy ∫Rm Rm = fˆ(y) g(y)dy. Rm Therefore, we see that the Fourier transform of the distribution T associated with f is the distribution associated with fˆ. For the Fourier inverse of a tempered distribution, it can be proved that T = ˆ − − − − (Tˆ) , where we use the notation T (g) = T (g ) and g (x) = g(−x) for Schwartz functions g. This is analogous to that of Fourier inversion formula for the Schwartz functions (A.3). Fourier inversion provides a spectral resolution of the Laplace operator on Rm. For example, the formula in (A.4) says that the Schwartz function has a representation as an integral of elementary eigenfunctions 2πix·y m ey(x) = e , y, x ∈ R , of the Euclidean Laplacian ∆. That is, if f ∈ S, then we have the spectral resolution ∫ ˆ f(x) = f(y)ey(x)dy, Rm ∫ ˆ where f(y) = (f, ey) = Rm f(u)ey(u)du by (A.2). 79 Appendix B: Harmonic Analysis on the Torus Rm/Zm The harmonic analysis on Rm/Zm is based on the Fourier series expansions. Let f : Rm/Zm → C, so f has a period 1 in each variable. The domain Rm/Zm can be viewed as a product of m circles, or torus. It can also be viewed as a unit hyper cube [0, 1]m ⊂ Rm. Therefore, the distance metric and volume element are inherited from the flat space Rm. In contrast with the symmetric space Rm, Rm/Zm is also compact. When m = 1, R/Z ≈ S1 is a circle. When m = 2, it is like a doughnut. The exponential functions {e2πix·a, ∀a ∈ Zm} are eigenfunctions for the (Euclidean) Laplacian ∆. Definition 2. The Fourier series of f : Rm/Zm → C is ∑ (f, ea)ea(x), a∈Zm 2πix·a m m where ea(x) = e , for x ∈ R , a ∈ Z , and the Fourier coefficients are ∫ (f, ea) = f(y)ea(y)dy. [0,1]m For different classes of functions, their Fourier series expansion converge indiffer- ent senses. We summarize some of these results below from Terras’s book [62, page 32]. 80 Theorem 13. (Properties of Fourier Series) (1) Suppose f ∈ L2(Rm/Zm), then f is the L2-limit of the partial sums of its ∑ | |2 ∥ ∥2 Fourier series. Moreover, we have the Parseval identity: a∈Zm (f, ea) = f 2. ∑ ∈ 2 Rm Zm | | ∞ (2) Suppose f L ( / ) and that a∈Zm (f, ea) < . Then there is a continuous function f˜ on Rm/Zm which is different from f on a set of measure 0 and ∑ ˜ such that f(x) = a∈Zm (f, ea)ea(x). (3) Suppose f : Rm/Zm → C has continuous partial derivatives of all orders less than or equal to k. If k > m/2, then the Fourier series of f converges uniformly and absolutely to f. (4) Let (Snf)(x) be nth partial sum of the Fourier series of f, that is, (Snf)(x) := ∑ |k|≤n(f, ea)ea(x). If f is a piecewise-smooth function with both one-sided derivatives, then it can be shown that ( ) 1 + − lim (Snf)(x) = f(x ) + f(x ) , n→∞ 2 + − where f(x ) = limt→x,t>x f(t) and f(x ) = limt→x,t Part (3) of the above theorem implies that the speed of convergence of the Fourier series increases with the smoothness of the smoothness of the function. The so-called Gibbs phenomenon arises in Part (4) where the partial sums (Snf)(x) always has an overshoot of about 9% close to the discontinuity point [62, Page 34]. 81 Appendix C: Recurrence Formulas Instead of by the definition, in practice, the Legendre polynomials/associated Legendre functions are often conveniently generated by various recurrence formulas. We present some of the formulas in below. More recurrence formulas can be found in [1, 33, 40]. It is convenient to introduce the generating function of the Legendre polynomials. In fact, Legendre polynomials Pl(t) were originally introduced as the coefficients of the expansion [33, page 14-15]: ∞ 1 ∑ = hlP (t). (C.1) (1 − 2th + h2)1/2 l l=0 where h ∈ [0, 1), t ∈ [−1, 1]. The function on the left side of (C.1) is referred to as the generating function of Pl(t). Utilizing (C.1), we can easily show that: Theorem 14. The Legendre polynomials Pl(t) satisfy the following recurrence formula (2l + 1)tPl(t) = (l + 1)Pl+1(t) + lPl−1(t). (C.2) Proof. DifferentiatingC.1 ( ) with respective to h, we have ∞ t − h ∑ = lhl−1P (t). (1 − 2th + h2)3/2 l l=0 82 Multiplying both sides of the above equation by (1 − 2ht + h2)1/2and using (C.1), it follows that ∑∞ ∑∞ l 2 l−1 (t − h)h Pl(t) = (1 − 2th + h )lh Pl(t). l=0 l=1 Regrouping terms by the common factor hl, we end up with ∑∞ l tP0(t) − P1(t) + [(2h + 1)tPl(t) − (l + 1)Pl+1(t) − lPl−1(t)] h = 0. l=1 Thus we proved (C.2) by noting that the coefficient of hl must be 0. Similarly, using the generating function, we can obtain: Theorem 15. The Legendre polynomials Pl(t) satisfy ′ − ′ (2l + 1)Pl(t) = Pl+1(t) Pl−1(t). (C.3) Proof. Differentiating the equationC.1 ( ) with respect to t, we have ∞ h ∑ = hlP ′(t). (1 − 2th + h2)3/2 l l=0 Multiplying both sides of the above equation by (1 − 2ht + h2)1/2 and using (C.1), it yields ∑∞ ∑∞ l+1 − 2 l ′ h Pl(t) = (1 2th + h ) h Pl (t). l=0 l=0 Regrouping terms by the common factor hl+1, we can have ∑∞ ∑∞ ( ) l+1 ′ l+2 l ′ h [Pl(t) + 2tPl (t)] = h + h Pl (t) l=0 l=0 ∑∞ ( ) l+1 ′ ′ = h Pl−1(t) + Pl+1(t) , l=0 ′ ′ where we define P−1(t) = 0 and notice that P0(t) = 0. 83 Since the coefficient of hl must be 0, we immediately have: ′ ′ ′ Pl(t) + 2tPl (t) = Pl−1(t) + Pl+1(t). (C.4) Now take the t derivative of both sides of (C.2), yielding ′ ′ ′ (2l + 1) (tPl (t) + P l(t)) = (l + 1)Pl+1(t) + lPl−1(t). (C.5) ′ Using (C.4) and (C.5) to eliminate the Pl (t) term, we will have (C.3). Based on the above two recurrence formulas (C.2) and (C.3) for the Legendre polynomials, a recurrence formula for the associated Legendre functions can be easily obtained. m Theorem 16. The associated Legendre functions Pl (t) satisfy − m m − m (l + 1 m)Pl+1(t) = (2l + 1)tPl (t) (l + m)Pl−1(t). (C.6) − (m−1) Proof. Differentiate (C.3) m 1 times and (C.2) m times, eliminate the Pl terms, 2 1 m and then multiply the resulted equation by (t − 1) 2 , yielding the desired formula. 84 Appendix D: Mathematica Code This appendix lists the code that is used in the simulation in Chapter 6. (* Load Noah Graham's Mathematica package of spheroidal functions *) Import["http://community.middlebury.edu/~ngraham/Spheroidal6.m"] (* In Slepian's notation, Ω=πt/2, T=2. Hence, c=ΩT/2=πt/2 *) T=2; (* Linear Prolate Functions (LPF) are angular functions taking m=0 *) LinearProlate[n_,t_,z_]:=SpheroidalPS[n,0,z,π t/2] (* Normalization factor for LPF. See Eq. (6.6) of this thesis. *) u2[n_,t_]:=Integrate[LinearProlate[n,t,z]^2,{z,-T/2,T/2}] (* Normalized LPF assumed in Odlyzko's paper. See Eq. (6.7) of this thesis. *) f[n_,t_,z_]:=LinearProlate[n,t,z]/Sqrt[u2[n,t]] (* By (24) (27) in Slepian's paper, the eigenvalue associated to f 2c 2 is given by [R(1)(c,1)] , where R(1)(c,1) is the Radial function. π 0n 0n 2c See Eq. (6.5) of this thesis. Since c=ΩT/2=πt/2, we have =t.*) π (* SpheroidalS1 outputs an imaginary number with 0 imaginary part *) γ[n_,t_]:=t Re[SpheroidalS1[n,0,1,π t/2]]^2 85 (* N0 is the truncation point when evaluate the infinite product and infinite sum in Eq. (6.8) of this thesis. The true value is when N0→ ∞. *) N0=4; (* Define the spacing distribution function. *) (* (k+1)st smallest eigenvalue of those that are larger than it *) k=0; J=Subsets[Range[0,N0],{k+2}]; Ne=Length[J]; (* GUE predicted distribution. See Eq. (6.9) of this thesis *) 16 pk[t_]:=pk[t]= Product[1-γ[m,t],{m,0,N0}]Sum[Product[ t2 γ[J[[v,s]],t]/(1-γ[J[[v,s]],t]),{s,1,k+2}] Sum[f[J[[v,s1]],t,1]^2 f[J[[v,s2]],t,-1]^2 Boole[Mod[J[[v,s1]]+J[[v,s2]],2]==1] Boole[s1 (* Take the nearest neighbor distribution for example *) (* Calculate the GUE predicted distribution *) pdf={}; For[t=0.05,t≤3,t=t+0.05,AppendTo[pdf,N[pk[t]]]]; (*Plot the GUE predicted distribution *) xl=Table[1/20*i,{i,60}]; ltemp={}; For[i=1,i≤60,i++,AppendTo[ltemp,{xl[[i]],pdf[[i]]}]] p1=ListPlot[ltemp,PlotRange →{{0,3},{0,1.2}},Frame→True, Joined→True] 86 (* Empirical distribution *) (* NZ: the number of data read from data set*) NZ=20000000; data=ToExpression@ReadList["1e28.zeros.1000_10001001.txt",Record, NZ,RecordSeparators→" "]; (* Hiary's data separates the integer part and the decimal part *) (* Using Reap and Sow is signifciantly faster than AppendTo *) Zeros=Flatten[Reap[Do[Sow[N[(data[[i+1]]+data[[i]])]], {i,1,NZ-1,2}]][[2]]] (*Normalize the spacing. See Eq. (6.12) of this thesis *) height=10^28; δn={}; δn=Flatten[Reap[Do[Sow[N[(Zeros[[i+1]]-Zeros[[i]]) Log[height+Zeros[[i]])/(2 Pi)]/(2 Pi)]],{i,1,NZ/2-1,1}]][[2]]] (*Draw the histogram of distribution*) D=HistogramDistribution[δn,{Range[0.025,3.025,0.05]}]; p2=DiscretePlot[#[D,x],{x,0.05,3,0.05},PlotLabel→#, PlotStyle→Red]&/@{PDF} (* Compare in the same plot and add legends *) Show[p1,p2,Epilog→Inset[Framed[Column[{PointLegend[{Red}, {" Empirical data"}],LineLegend[{Blue}, {"GUE prediction"}]}],RoundingRadius→5],Scaled[{0.8,0.85}]]] 87