DOCSLIB.ORG

Spherical and Spheroidal Harmonics: Examples and Computations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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 2 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- 2 8 tion with ϕ [0; 5 π] (right) ....................... 47 2 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 2 f0:4; 0:6; 0:9; 1:2; 1:5g. ..... 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
Recommended publications
  • Ladder Operators for Lam\'E Spheroconal Harmonic Polynomials

    Ladder Operators for Lam\'E Spheroconal Harmonic Polynomials

    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 074, 16 pages Ladder Operators for Lam´eSpheroconal Harmonic Polynomials? Ricardo MENDEZ-FRAGOSO´ yz and Eugenio LEY-KOO z y Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: [email protected] URL: http://sistemas.fciencias.unam.mx/rich/ z Instituto de F´ısica, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: eleykoo@fisica.unam.mx Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012 http://dx.doi.org/10.3842/SIGMA.2012.074 Abstract. Three sets of ladder operators in spheroconal coordinates and their respective actions on Lam´espheroconal harmonic polynomials are presented in this article. The poly- nomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lam´epolynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers n1 and n2 counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components L^x, L^y, L^z of the angular momentum connect pairs of the four species of poly- nomials of a chosen kind and angular momentum. The third set of operators, the cartesian componentsp ^x,p ^y,p ^z of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities.
  • Quaternion Zernike Spherical Polynomials

    Quaternion Zernike Spherical Polynomials

    MATHEMATICS OF COMPUTATION Volume 84, Number 293, May 2015, Pages 1317–1337 S 0025-5718(2014)02888-3 Article electronically published on August 29, 2014 QUATERNION ZERNIKE SPHERICAL POLYNOMIALS J. MORAIS AND I. CAC¸ AO˜ Abstract. Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this develop- ment. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quater- nionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R3 and R4). (R)QZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second or- der homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. (R)QZSPs are new in literature and have some consequences that are now under investigation. 1. Introduction 1.1. The Zernike spherical polynomials. The complex Zernike polynomials (ZPs) have long been successfully used in many different fields of optics.
  • Laplacian Eigenmodes for the Three Sphere

    Laplacian Eigenmodes for the Three Sphere

    Laplacian eigenmodes for the three sphere M. Lachi`eze-Rey Service d’Astrophysique, C. E. Saclay 91191 Gif sur Yvette cedex, France October 10, 2018 Abstract The vector space k of the eigenfunctions of the Laplacian on the 3 V three sphere S , corresponding to the same eigenvalue λk = k (k + 2 − k 2), has dimension (k + 1) . After recalling the standard bases for , V we introduce a new basis B3, constructed from the reductions to S3 of peculiar homogeneous harmonic polynomia involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of S3 and SO(4), we are able to write explicitely the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility to select those functions of k which remain invariant under V a chosen rotation of SO(4). When the rotation is an holonomy transfor- mation of a spherical space S3/Γ, this gives a method to calculates the eigenmodes of S3/Γ, which remains an open probleme in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a forthcoming paper, we present the derivation for dodecahedral space. 1 Introduction 3 The eigenvalues of the Laplacian ∆ of S are of the form λk = k (k + 2), + − k where k IN . For a given value of k, they span the eigenspace of ∈ 2 2 V dimension (k +1) . This vector space constitutes the (k +1) dimensional irreductible representation of SO(4), the isometry group of S3.
  • A User's Guide to Spherical Harmonics

    A User's Guide to Spherical Harmonics

    A User's Guide to Spherical Harmonics Martin J. Mohlenkamp∗ Version: October 18, 2016 This pamphlet is intended for the scientist who is considering using Spherical Harmonics for some appli- cation. It is designed to introduce the Spherical Harmonics from a theoretical perspective and then discuss those practical issues necessary for their use in applications. I expect great variability in the backgrounds of the readers. In order to be informative to those who know less, without insulting those who know more, I have adopted the strategy \State even basic facts, but briefly." My dissertation work was a fast transform for Spherical Harmonics [6]. After its publication, I started receiving questions about Spherical Harmonics in general, rather than about my work specifically. This pamphlet aims to answer such general questions. If you find mistakes, or feel that important material is unclear or missing, please inform me. 1 A Theory of Spherical Harmonics In this section we give a development of Spherical Harmonics. There are other developments from other perspectives. The one chosen here has the benefit of being very concrete. 1.1 Mathematical Preliminaries We define the L2 inner product of two functions to be Z hf; gi = f(s)¯g(s)ds (1) R 2π whereg ¯ denotes complex conjugation and the integral is over the space of interest, for example 0 dθ on the R 2π R π 2 p 2 circle or 0 0 sin φdφdθ on the sphere. We define the L norm by jjfjj = hf; fi. The space L consists of all functions such that jjfjj < 1.
  • 32 FA15 Abstracts

    32 FA15 Abstracts

    32 FA15 Abstracts IP1 metric simple exclusion process and the KPZ equation. In Vector-Valued Nonsymmetric and Symmetric Jack addition, the experiments of Takeuchi and Sano will be and Macdonald Polynomials briefly discussed. For each partition τ of N there are irreducible representa- Craig A. Tracy tions of the symmetric group SN and the associated Hecke University of California, Davis algebra HN (q) on a real vector space Vτ whose basis is [email protected] indexed by the set of reverse standard Young tableaux of shape τ. The talk concerns orthogonal bases of Vτ -valued polynomials of x ∈ RN . The bases consist of polyno- IP6 mials which are simultaneous eigenfunctions of commuta- Limits of Orthogonal Polynomials and Contrac- tive algebras of differential-difference operators, which are tions of Lie Algebras parametrized by κ and (q, t) respectively. These polynomi- als reduce to the ordinary Jack and Macdonald polynomials In this talk, I will discuss the connection between superin- when the partition has just one part (N). The polynomi- tegrable systems and classical systems of orthogonal poly- als are constructed by means of the Yang-Baxter graph. nomials in particular in the expansion coefficients between There is a natural bilinear form, which is positive-definite separable coordinate systems, related to representations of for certain ranges of parameter values depending on τ,and the (quadratic) symmetry algebras. This connection al- there are integral kernels related to the bilinear form for lows us to extend the Askey scheme of classical orthogonal the group case, of Gaussian and of torus type. The mate- polynomials and the limiting processes within the scheme.
  • Orthogonal Functions: the Legendre, Laguerre, and Hermite Polynomials

    Orthogonal Functions: the Legendre, Laguerre, and Hermite Polynomials

    ORTHOGONAL FUNCTIONS: THE LEGENDRE, LAGUERRE, AND HERMITE POLYNOMIALS THOMAS COVERSON, SAVARNIK DIXIT, ALYSHA HARBOUR, AND TYLER OTTO Abstract. The Legendre, Laguerre, and Hermite equations are all homogeneous second order Sturm-Liouville equations. Using the Sturm-Liouville Theory we will be able to show that polynomial solutions to these equations are orthogonal. In a more general context, finding that these solutions are orthogonal allows us to write a function as a Fourier series with respect to these solutions. 1. Introduction The Legendre, Laguerre, and Hermite equations have many real world practical uses which we will not discuss here. We will only focus on the methods of solution and use in a mathematical sense. In solving these equations explicit solutions cannot be found. That is solutions in in terms of elementary functions cannot be found. In many cases it is easier to find a numerical or series solution. There is a generalized Fourier series theory which allows one to write a function f(x) as a linear combination of an orthogonal system of functions φ1(x),φ2(x),...,φn(x),... on [a; b]. The series produced is called the Fourier series with respect to the orthogonal system. While the R b a f(x)φn(x)dx coefficients ,which can be determined by the formula cn = R b 2 , a φn(x)dx are called the Fourier coefficients with respect to the orthogonal system. We are concerned only with showing that the Legendre, Laguerre, and Hermite polynomial solutions are orthogonal and can thus be used to form a Fourier series. In order to proceed we must define an inner product and define what it means for a linear operator to be self- adjoint.
  • Radial Basis Functions

    Radial Basis Functions

    Acta Numerica (2000), pp. 1–38 c Cambridge University Press, 2000 Radial basis functions M. D. Buhmann Mathematical Institute, Justus Liebig University, 35392 Giessen, Germany E-mail: [email protected] Radial basis function methods are modern ways to approximate multivariate functions, especially in the absence of grid data. They have been known, tested and analysed for several years now and many positive properties have been identified. This paper gives a selective but up-to-date survey of several recent developments that explains their usefulness from the theoretical point of view and contributes useful new classes of radial basis function. We consider particularly the new results on convergence rates of interpolation with radial basis functions, as well as some of the various achievements on approximation on spheres, and the efficient numerical computation of interpolants for very large sets of data. Several examples of useful applications are stated at the end of the paper. CONTENTS 1 Introduction 1 2 Convergence rates 5 3 Compact support 11 4 Iterative methods for implementation 16 5 Interpolation on spheres 25 6 Applications 28 References 34 1. Introduction There is a multitude of ways to approximate a function of many variables: multivariate polynomials, splines, tensor product methods, local methods and global methods. All of these approaches have many advantages and some disadvantages, but if the dimensionality of the problem (the number of variables) is large, which is often the case in many applications from stat- istics to neural networks, our choice of methods is greatly reduced, unless 2 M. D. Buhmann we resort solely to tensor product methods.
  • Spherical-Harmonic Tensors Jm

    Spherical-Harmonic Tensors Jm

    Spherical-harmonic tensors Francisco Gonzalez Ledesma1 and Matthew Mewes2 1Department of Physics, Florida State University, Tallahassee, Florida 32306, USA 2Physics Department, California Polytechnic State University, San Luis Obispo, California 93407, USA The connection between spherical harmonics and symmetric tensors is explored. For each spher- ical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an orthonormal angular-momentum eigenbasis for symmetric tensors of any rank. The relationship between the spherical-harmonic tensors and spin- weighted spherical harmonics is derived. The results facilitate the spherical-harmonic expansion of a large class of tensor-valued functions. Several simple illustrative examples are discussed, and the formalism is used to derive the leading-order effects of violations of Lorentz invariance in Newtonian gravity. I. INTRODUCTION Spherical harmonics Yjm provide an orthonormal basis for scalar functions on the 2-sphere and have numerous applications in physics and related fields. While they are commonly written in terms of the spherical-coordinate polar angle θ and azimuthal angle φ, spherical harmonics can be expressed in terms of cartesian coordinates, which is convenient in certain applications. The cartesian versions involve rank-j symmetric trace-free tensors jm [1–5]. These form a basis for traceless tensors and provide a link between functions on the sphere and symmetricY traceless tensors in three dimensions. This work builds on the above understanding in several ways. We first develop a new method for calculating the scalar spherical harmonics Yjm in terms of components of the direction unit vector n = sin θ cos φ ex + sin θ sin φ ey + cos θ ez .
  • Generalizations and Specializations of Generating Functions for Jacobi, Gegenbauer, Chebyshev and Legendre Polynomials with Definite Integrals

    Generalizations and Specializations of Generating Functions for Jacobi, Gegenbauer, Chebyshev and Legendre Polynomials with Definite Integrals

    Journal of Classical Analysis Volume 3, Number 1 (2013), 17–33 doi:10.7153/jca-03-02 GENERALIZATIONS AND SPECIALIZATIONS OF GENERATING FUNCTIONS FOR JACOBI, GEGENBAUER, CHEBYSHEV AND LEGENDRE POLYNOMIALS WITH DEFINITE INTEGRALS HOWARD S. COHL AND CONNOR MACKENZIE Abstract. In this paper we generalize and specialize generating functions for classical orthogo- nal polynomials, namely Jacobi, Gegenbauer, Chebyshev and Legendre polynomials. We derive a generalization of the generating function for Gegenbauer polynomials through extension a two element sequence of generating functions for Jacobi polynomials. Specializations of generat- ing functions are accomplished through the re-expression of Gauss hypergeometric functions in terms of less general functions. Definite integrals which correspond to the presented orthogonal polynomial series expansions are also given. 1. Introduction This paper concerns itself with analysis of generating functions for Jacobi, Gegen- bauer, Chebyshev and Legendre polynomials involving generalization and specializa- tion by re-expression of Gauss hypergeometric generating functions for these orthog- onal polynomials. The generalizations that we present here are for two of the most important generating functions for Jacobi polynomials, namely [4, (4.3.1–2)].1 In fact, these are the first two generating functions which appear in Section 4.3 of [4]. As we will show, these two generating functions, traditionally expressed in terms of Gauss hy- pergeometric functions, can be re-expressed in terms of associated Legendre functions (and also in terms of Ferrers functions, associated Legendre functions on the real seg- ment ( 1,1)). Our Jacobi polynomial generating function generalizations, Theorem 1, Corollary− 1 and Corollary 2, generalize the generating function for Gegenbauer polyno- mials.
  • An Algebraic Approach to Harmonic Polynomials on S3

    An Algebraic Approach to Harmonic Polynomials on S3

    AN ALGEBRAIC APPROACH TO HARMONIC POLYNOMIALS ON S3 Kevin Mandira Limanta A thesis in fulfillment of the requirements for the degree of Master of Science (by Research) SCHOOL OF MATHEMATICS AND STATISTICS FACULTY OF SCIENCE UNIVERSITY OF NEW SOUTH WALES June 2017 ORIGINALITY STATEMENT 'I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.' Signed .... Date Show me your ways, Lord, teach me your paths. Guide me in your truth and teach me, for you are God my Savior, and my hope is in you all day long. { Psalm 25:4-5 { i This is dedicated for you, Papa. ii Acknowledgement This thesis is the the result of my two years research in the School of Mathematics and Statistics, University of New South Wales. I learned quite a number of life lessons throughout my entire study here, for which I am very grateful of.
  • Arxiv:2008.08079V2 [Math.FA] 29 Dec 2020 Hypergroups Is Not Required)

    Arxiv:2008.08079V2 [Math.FA] 29 Dec 2020 Hypergroups Is Not Required)

    HARMONIC ANALYSIS OF LITTLE q-LEGENDRE POLYNOMIALS STEFAN KAHLER Abstract. Many classes of orthogonal polynomials satisfy a specific linearization prop- erty giving rise to a polynomial hypergroup structure, which offers an elegant and fruitful link to harmonic and functional analysis. From the opposite point of view, this allows regarding certain Banach algebras as L1-algebras, associated with underlying orthogonal polynomials or with the corresponding orthogonalization measures. The individual be- havior strongly depends on these underlying polynomials. We study the little q-Legendre polynomials, which are orthogonal with respect to a discrete measure. Their L1-algebras have been known to be not amenable but to satisfy some weaker properties like right character amenability. We will show that the L1-algebras associated with the little q- Legendre polynomials share the property that every element can be approximated by linear combinations of idempotents. This particularly implies that these L1-algebras are weakly amenable (i. e., every bounded derivation into the dual module is an inner deriva- tion), which is known to be shared by any L1-algebra of a locally compact group. As a crucial tool, we establish certain uniform boundedness properties of the characters. Our strategy relies on continued fractions, character estimations and asymptotic behavior. 1. Introduction 1.1. Motivation. One of the most famous results of mathematics, the ‘Banach–Tarski paradox’, states that any ball in d ≥ 3 dimensions can be split into a finite number of pieces in such a way that these pieces can be reassembled into two balls of the original size. It is also well-known that there is no analogue for d 2 f1; 2g, and the Banach–Tarski paradox heavily relies on the axiom of choice [37].
  • Spherical Harmonics and Homogeneous Har- Monic Polynomials

    Spherical Harmonics and Homogeneous Har- Monic Polynomials

    SPHERICAL HARMONICS AND HOMOGENEOUS HAR- MONIC POLYNOMIALS 1. The spherical Laplacean. Denote by S ½ R3 the unit sphere. For a function f(!) de¯ned on S, let f~ denote its extension to an open neighborhood N of S, constant along normals to S (i.e., constant along rays from the origin). We say f 2 C2(S) if f~ is a C2 function in N , and for such functions de¯ne a di®erential operator ¢S by: ¢Sf := ¢f;~ where ¢ on the right-hand side is the usual Laplace operator in R3. With a little work (omitted here) one may derive the expression for ¢ in polar coordinates (r; !) in R2 (r > 0;! 2 S): 2 1 ¢u = u + u + ¢ u: rr r r r2 S (Here ¢Su(r; !) is the operator ¢S acting on the function u(r; :) in S, for each ¯xed r.) A homogeneous polynomial of degree n ¸ 0 in three variables (x; y; z) is a linear combination of `monomials of degree n': d1 d2 d3 x y z ; di ¸ 0; d1 + d2 + d3 = n: This de¯nes a vector space (over R) denoted Pn. A simple combinatorial argument (involving balls and separators, as most of them do), seen in class, yields the dimension: 1 d := dim(P ) = (n + 1)(n + 2): n n 2 Writing a polynomial p 2 Pn in polar coordinates, we necessarily have: n p(r; !) = r f(!); f = pjS; where f is the restriction of p to S. This is an injective linear map p 7! f, but the functions on S so obtained are rather special (a dn-dimensional subspace of the in¯nite-dimensional space C(S) of continuous functions-let's call it Pn(S)) We are interested in the subspace Hn ½ Pn of homogeneous harmonic polynomials of degree n (¢p = 0).