A Clifford Construction of Multidimensional Prolate Spheroidal Wave Functions

Hamed Baghal Ghaffari Jeffrey A. Hogan Joseph D. Lakey School of Math and Phys Sci School of Math and Phys Sci Department of Mathematical Sciences University of Newcastle University of Newcastle New Mexico State University Australia Australia Las Cruces, NM 88003-8001 [email protected] [email protected] USA [email protected]

Abstract—We investigate the construction of multidimensional embedded in the Clifford algebra Rm by identifying the point prolate spheroidal wave functions using techniques from Clifford m Pm x = (x1, ··· , xm) ∈ R with the 1-vector x = j=1 ejxj. analysis. The prolates are defined to be eigenfunctions of a certain It should be noted that [λ] is the scalar part of the Clifford differential operator and we propose a method for computing 0 these eigenfunctions through expansions in Clifford-Legendre number λ. The product of two 1-vectors splits up into a polynomials. It is shown that the differential operator commutes scalar part and a 2-vector (also called the bivector, part): Pm with a time-frequency limiting operator defined relative to balls xy = −hx, yi + x ∧ y where hx, yi = j=1 xjyj and in n-dimensional Euclidean space. P x ∧ y = iLaplace operator in the sense that constructed by definition of a new Sturm-Liouville differential 2 operator. ∆m = −∂x. (2) Clifford analysis is a means through which many of the Definition 2. A left (resp. right) monogenic homogeneous fundamental theorems and techniques of complex analysis can m polynomial Pk of degree k (k ≥ 0) in R is called a left be lifted to higher dimensions (see [3]). In this paper we (resp. right) solid inner spherical monogenic of order k. The study the higher-dimensional prolate spheroidal wave func- set of all left (resp. right) solid inner spherical monogenics of tions (PSWFs) thorough the lens of Clifford analysis. + + order k will be denoted by Ml (k), respectively Mr (k). II.CLIFFORD ANALYSIS Pm + Lemma 1. Let x = j=0 xjej. For Pk ∈ Ml (k) and s ∈ N Let {e1, . . . , em} be the standard basis for m-dimensional the following fundamental formula holds: m euclidean space R . The non-commutative multiplication in  s−1 m s −sx Pk for s even the Clifford algebra Rm built over R is governed by the rules ∂x[x Pk] = s−1 −(s + 2k + m − 1)x Pk for s odd. e2 = −1 j = 1, ··· , m j For the proof, the reader is referred to [3]. m eiej = −ejei i 6= j. The Rm-valued inner product of the functions f, g : R → Rm is given by A canonical base for Rm is obtained by considering for any ordered set A = {j , j , ··· , j } ⊂ {1, ··· , m} = M, the Z 1 2 h hf, gi = f(x)g(x)dV (x), element eA = ej1 ej2 ··· ejh , e∅ = 1. For example, each m λ ∈ R2, may be written as λ = λ0 + λ1e1 + λ2e2 + λ12e1e2, R P m where λi ∈ R. The conjugation λ of λ = A λAeA ∈ Rm where dV is the Lebesgue measure on R . The associated P 2 is given by λ = A λAeA where ej = −ej, e¯∅ = e∅, and norm is given by kfk = [hf, fi]0. The unitary right Clifford- m αβ = βα for all α, β ∈ Rm. The Euclidean space R is module of Clifford algebra-valued measurable functions on Rm for which kfk2 < ∞ is a right Hilbert Clifford-module IV. RESULTS m which we denote by L2(R , Rm). The multi-dimensional In this section, we start the process of building multi- Fourier transform F is given by dimensional prolate spheroidal wave functions. 1 Z Ff(ξ) = exp(−ihx, ξi)f(x)dV (x) Theorem 5. The Clifford-Legendre polynomials admit the m/2 (3) (2π) m R following explicit representations: 1 m N for f ∈ L (R , Rm) and may be extended unitarily to 22N (2N)! X N 2 m C0 (P )(x) = L (R , Rm). 2N k N! j j=0 Theorem 2. (Clifford-Stokes theorem) Let f, g ∈ C1(Ω). Γ(j + k + m + N)  Then for each compact set C ⊂ Ω, one has × 2 (−1)j|x|2jP (x) (9) Γ(j + k + m ) k Z Z 2 f(x)n(x)g(x) dσ(x) = [(f∂x)g + f(∂xg)] dV (x) and ∂C C 2N+1 N   0 2 (2N + 1)! X N where n(x) is the outward pointing unit normal on ∂C and C (Pk)(x) = − 2N+1 N! j dσ is the surface area measure on ∂C. j=0 m  Γ(j + k + 2 + N + 1) j 2j Proof. For the proof see [3]. m (−1) |x| xPk(x) . (10) Γ(j + k + 2 + 1) III.LEGENDRE POLYNOMIALS Proof. By Lemma 1, Definition 3. Let B(1) be the closed unit ball in m and 2 2j 2j−2 R ∂x(x Pk(x)) = 2j(2j + 2k + m − 2)x Pk(x), α ∈ R with α > −1. Then the operator Dα is defined on and continuous functions f : B(1) → Rm by 2 2j+1 2j−1 2 −α 2 α+1 ∂x(x Pk(x)) = 2j(2j + 2k + m)x Pk(x). Dαf(x) = (1 + x ) ∂x((1 + x ) f(x)). (4)

+ Therefore, when n = 2N + 1 is odd, we heve Definition 4. Let n ∈ N, and let Pk ∈ Ml (k) be fixed. 0 C0 (P )(x) = ∂2N+1[(1 + x2)2N+1P (x)] Then we define Clifford-Legendre polynomials Cn,m(Pk)(x) 2N+1 k x k as follows: 2N+1   2N+1 X 2N + 1 2j = ∂ [ |x| Pk(x)] 0 x j Cn(Pk)(x) = D0D1 ··· D+n−1(Pk(x)). (5) j=0 2N+1 It is shown in [3] that X 2N + 1 = [ ∂2N−1[(−2j)(−(2j − 1 j x 0 0 j=0 Cn(Pk)(x) = Cn,k(x)Pk(x) (6) 2j−2 0 s + 2k + m − 1))x Pk(x)]] with C ∈ Pn = span {x : s ∈ , s ≤ n, x ∈ }, n,k Rm N R(m) 2N+1 X 2N + 1 the space of polynomials having degree less than or equal to = [ [(2j)(2j + 2k + m − 2)(2j − 2) n C0 k j . n,k has real coefficients depending on and takes value j=0 0 1 in Rm ⊕ Rm. 2N−3 2j−5 (2j + 2k + m − 4)∂x x xPk(x)]] Theorem 3. (Rodrigues’ Formula) The Legendre polynomials 2N+1 X 2N + 1 C0(P )(x) are also determined by = [− 22N−1[(j)(j − 1) ··· (j − (N − 1))] n k j j=N 0 n 2 n C (Pk)(x) = ∂ ((1 + x ) Pk(x)) (7) m m n x × [(j + k + − 1) ··· (j + k + − (N − 1))] 2 2 Proof. For the proof see [3]. 2 2j−2N × ∂xx xPk(x)] Next we note that the Clifford-Legendre polynomials are eigenfunctions of a second order differential equation. N+1 X 2N + 1 (j + N)! Theorem 4. For all n, k ∈ N, there exists a real constant = − 22N+1[ j + N (j − 1)! C(0, n, k) such that j=0 (j + k + m + N − 1)! D ∂ (C0(P )(x)) = C(0, n, k)C0(P )(x), 2 2j−2 0 x n k n k × m ]x xPk(x) (j + k + 2 − 1)! or equivalently, 2N+1 N   m 2 (2N + 1)! X N Γ(j + k + 2 + N + 1) 2 0 0 = − m ∂ C (Pk)(x) − 2x∂xC (Pk)(x) N! j Γ(j + k + + 1) x n n j=0 2 0 −C(0, n, k)Cn(Pk)(x) = 0. (8) j 2j (−1) |x| xPk(x). Proof. For the proof see [3]. The proof is similar when n is even. for real constants an,k, bn,k and cn,k. Substituting (17) into (16) we find that f is an eigenfunction of L with eigenvalue The representation above may be used to provide a Bonnet- c χ if and only if b = (b ) is an eigenvector of a multilinear type formula for the Clifford-Legendre polynomials, i.e., a i,k,j (0) mapping A which is tri-diagonal in the i-variable: formula that expresses xCn (x) as a linear combination of (0) (0) Cn−1(x) and Cn+1(x). Theorem 6. [Bonnet formula for Clifford-Legendre polynomi- als] Ab = bχ. (a) If n is odd, 0 0 0 xC2n+1(Pk)(x) = αn,kC2n+2(Pk)(x)+βn,kC2n(Pk)(x), (11) Computation of these eigenvectors, and substitution into (15) m −m 2(2n+1)( 2 +n+k) gives eigenfunctions of Lc. where αn,k = m , βn,k = m , 4( 2 +2n+k+1) ( 2 +2n+k+1) (b) If n is even, The connections between the PSWFs and the eigenfunctions of a multi-dimensional time-frequency limiting operator is now xC0 (P )(x) = α0 C0 (P )(x)+β0 C0 (P )(x), 2n k n,k 2n+1 k n,k 2n−1 k of interest, in particular, the commutation of these operators.. (12) m 2 0 −( 2 +n+k) 0 4n where αn,k = 2(2n+1)( m +2n+k) , βn,k = ( m +2n+k) . 2 2 Theorem 7. The operator Lc, defined at (14), is self-adjoint. For the proof, the reader is referred to [4]. Proof. With an application of the Clifford-Stokes formula and V. MULTI-DIMENSIONAL PROLATES the observation that (1 − |x|2) = 0 for x ∈ ∂B, we have Definition 8: Given c > 0, we define three operators Lc and 2 Gc on L (B(1), Rm) by Z 2πichx,yi Z Gcf(x) = χB(x) e f(y)dy, (13) 2 hf, Lcgi = f(x)[∂x((1 − |x| )∂xg(x)) B B and + 4π2c2|x|2g(x)]dx 2 2 2 2 Z Lcf(x) = ∂x((1 − |x| )∂xf(x)) + 4π c |x| f(x). (14) 2 = − ((f(x)∂x)(x)(1 − |x| )∂xg(x))dx B Strictly speaking, Lc is defined on a dense subspace of Z 2 ¯ 2 2 2 L (B(1), Rm). The m-dimensional Clifford prolates are de- + 4π c |x| f(x)g(x)dx B fined to be the eigenfunctions of Lc. Here we aim to describe Z 2 an algorithm for their computation and reveal relationships = −{ ((1 − |x| )f(x)∂x)n(x)g(x)dσ ∂B between Lc and certain time-frequency limiting operators Z m 2 defined relative to balls in R . − ([(1 − |x| )(f(x)∂x)]∂x)g(x)dx} d + Let {Y } k be an orthonormal basis for M (k). Then B k,j j=1 l Z the Clifford-Legendre polynomials + 4π2c2|x|2f(x)g(x)dx 0 B {Rn,j,k = Cn(Yk,j); n ≥ 0, k ≥ 0, 0 ≤ j ≤ dk} Z 2 + = (∂x[(1 − |x| )(∂xf(x))])g(x)dx (where dk is the dimension of Ml (k)) form an orthonormal B basis for L2(B(1), m). Suppose then that f as an eigenfunc- Z R + 4π2c2|x|2f(x)g(x)dx tion of Lc, i.e., Lcf = fχ for some χ ∈ Rm. We may write B Z ∞ dk X X 0 = Lcf(x)g(x)dx = hLcf, gi f = Ci (Yk,j)bi,k,j, (15) B i,k=0 j=0 for some constants bi,k,j ∈ Rm. Then X 0 which completes the proof. Lcf(x) = Lc[Ci (Yk,j)(x)]bi,k,j i,k,j X 0 2 2 2 0 = [C(i, k)Ci (Yk,j)(x) + 4π c x Ci (Yk,j(x)]. (16) i,k,j

A double application of the Bonnet formula gives Theorem 8. The operators Lc and Gc commute, i.e., LcGc = 2 0 0 0 GcLc. x Cn(Yk,j(x)) = an,kCn+2(Yk,j(x)) + bn,kCn(Yk,j(x)) 0 + cn,kC (Yk,j(x)), (17) n−2 Proof. By the definitions of Lc and Gc and the self-adjointness of Lc, we have ACKNOWLEDGMENT Z The authors would like to thank the Centre for Computer- 2πic GcLc(f(x)) = χB(x) e [Lcf(y)]dy Assisted Research in Mathematics and its Applications at B Z the University of Newcastle for its continued support. JAH 2 −2πic =χB(x) [∂y((1 − |y| )∂y(e )) is supported by the Australian Research Council through B Discovery Grant DP160101537. Thanks Roy. Thanks HG. +4π2c2|y|2e−2πic]f(y)dy Z REFERENCES =χ (x) [(1 − |y|2)(4π2c2|x|2e−2πic) B [1] D. Slepian, “Prolate spheroidal wave functions, fourier analysis and B uncertainty—iv: extensions to many dimensions; generalized prolate −2πic 2 2 2 −2πic +4πicyxe + 4π c |y| e ]f(y)dy spheroidal functions,” Bell System Technical Journal, vol. 43, no. 6, pp. Z 1 3009–3057, 1964. 2 2 2 2πic 2 2πic [2] J. Zhang, H. Li, L.-L. Wang, and Z. Zhang, “Ball prolate spheroidal wave =χB(x) [4π c |x| (e − 2 2 ∂x(e )) B 4π c functions in arbitrary dimensions,” Applied and Computational Harmonic 2πic 2 2πic Analysis, 2018. −4πicxye + ∂x(e )]f(y)dy Z [3] R. Delanghe, F. Sommen, and V. Soucek, Clifford algebra and spinor- 2 2 2 2πic 2 2 2πic valued functions: a function theory for the Dirac operator. Springer =χB(x) [4π c |x| e − |x| ∂x(e ) Science & Business Media, 2012, vol. 53. B [4] H. Baghal Ghaffari, J. Hogan, and J. Lakey, “Multidimensional prolate 2πic 2 2πic −2x∂xe + ∂x(e )]f(y)dy functions: a clifford approach,” preprint. Z [5] Y. Shkolnisky, “Prolate spheroidal wave functions on a disc—integration 2 2 2πic 2πic and approximation of two-dimensional bandlimited functions,” Applied =χB(x) [(1 − |x| )∂x(e ) − 2x∂x(e ) B and Computational Harmonic Analysis, vol. 22, no. 2, pp. 235–256, 2007. +c2|x|2(e2πic)]f(y)dy = L G (f(x)). [6] H. Xiao, V. Rokhlin, and N. Yarvin, “Prolate spheroidal wavefunctions, c c quadrature and interpolation,” Inverse problems, vol. 17, no. 4, p. 805, 2001.

∗ ∗ Theorem 9. The operators Gc and Gc Gc commute with Lc. 2 m Let Q, Pc be the orthogonal projections on L (R , Rm) defined by

Qf(x) = χB(1)(x)f(x); Z Pcf(x) = f(y)Kc(x − y) dy m R R 2πichx,yi where Kc(x) = B(c) e dy. The range of Q is the collection of L2 functions that are supported on the ball B(1) 2 of radius 1 and the range of Pc is the collection of L functions whose Fourier transform are supported on the ball B(c) of ∗ radius c. We note that the operator Gc Gc is a multiple of the time-frequency operator QPc and conclude from Theorem 9 that QPc commutes with Lc.

VI.CONCLUSIONAND FUTURE WORK Through consideration of Clifford-Legendre polynomials and proof of an associated Bonnet formula, we developed in this paper sufficient theory to enable the construction of mul- tidimensional Clifford-valued PSWF’s, defined as the eigen- functions of a self-adjoint differential operator Lc involving the Dirac operator. We defined time and frequency projections Q and Pc and showed that the self-adjoint operator QPc commutes with Lc. It is yet to be shown that the PSWFs are also the eigenfunctions of QPc as there is currently insufficient theory surrounding the linear algebra of Clifford modules to assert that commuting self-adjoint operators share a common eigenbasis – as is the case in Hilbert spaces. Approximation properties of these functions will be explored as has been done in one dimension by Shkolnisky [5] and Xiao, Rokhlin and Yarvin [6].