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A Clifford Construction of Multidimensional Prolate Spheroidal Wave Functions

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A Clifford Construction of Multidimensional Prolate Spheroidal Wave Functions 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) 2 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 = i<j eiej(xiyj − xjyi). Note also that if x is a 1 x2 = −hx; xi = −|xj2 I. INTRODUCTION -vector, then . m In 1964 [1], the higher dimensional version of the of prolates Definition 1. Let f : R ! Rm be defined and continuously were studied and constructed. After polar coordinates were differentiable in an open region Ω of Rm. The Dirac operator employed, part of the construction involved determining the @x is defined on such functions by eigenvalues of the differential operator M given by m c X 2 1 2 @xf = ej@xj f: 2 d u du − N 2 2 M (u)(t) = (1 − t ) − 2t + ( 4 − c t )u = 0; j=1 c dt dt t2 (1) We also allow the Dirac operator to act of the right in the the solutions of which form the radial part of the higher Pm sense that f@x = j=1 @xj fej. f is said to be left (resp. dimensional prolates. The operator has a singularity at the right) monogenic on Ω if @xf = 0 (resp. f@x = 0) on Ω. If origin, causing instabilities. Also it is valuable to mention f is left-and right monogenic, we say f is monogenic. The that, in [2], the prolate spheroidal wave functions has been Dirac operator factorises the Laplace 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 2 Ml (k) and s 2 N Let fe1; : : : ; emg 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 = fj ; j ; ··· ; j g ⊂ f1; ··· ; mg = 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 2 R2; may be written as λ = λ0 + λ1e1 + λ2e2 + λ12e1e2; R P m where λi 2 R: The conjugation λ of λ = A λAeA 2 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 2 Rm. The Euclidean space R is module of Clifford algebra-valued measurable functions on Rm for which kfk2 < 1 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 2 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 2 C1(Ω): Γ(j + k + m + N) Then for each compact set C ⊂ Ω; one has × 2 (−1)jjxj2jP (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) jxj 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 2 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 2 N; and let Pk 2 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 = @ [ jxj 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 2 Pn = span fx : s 2 ; s ≤ n; x 2 g, 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 2 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) jxj 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).
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