A generalization of the prolate spheroidal wave functions
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- by Ahmed I. Zayed PDF
- Proc. Amer. Math. Soc. 135 (2007), 2193-2203 Request permission
Abstract:
Many systems of orthogonal polynomials and functions are bases of a variety of function spaces, such as the Hermite and Laguerre functions which are orthogonal bases of $L^2(-\infty , \infty )$ and $L^2(0,\infty ),$ and the Jacobi polynomials which are an orthogonal basis of a weighted $L^2(-1,1).$ The associated Legendre functions, and more generally, the spheroidal wave functions are also an orthogonal basis of $L^2(-1,1).$ The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property. They are an orthogonal basis of both $L^2(-1,1)$ and a subspace of $L^2(-\infty , \infty ),$ known as the Paley-Wiener space of bandlimited functions. They also satisfy a discrete orthogonality relation. No other system of classical orthogonal functions is known to possess this strange property. This raises the question of whether there are other systems possessing this property. The aim of the article is to answer this question in the affirmative by providing an algorithm to generate such systems and then demonstrating the algorithm by a new example.References
- John P. Boyd, Approximation of an analytic function on a finite real interval by a bandlimited function and conjectures on properties of prolate spheroidal functions, Appl. Comput. Harmon. Anal. 15 (2003), no. 2, 168–176. MR 2007058, DOI 10.1016/S1063-5203(03)00048-4
- Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. III, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. Based, in part, on notes left by Harry Bateman. MR 0066496
- Carson Flammer, Spheroidal wave functions, Stanford University Press, Stanford, California, 1957. MR 0089520
- Kedar Khare and Nicholas George, Sampling theory approach to prolate spheroidal wavefunctions, J. Phys. A 36 (2003), no. 39, 10011–10021. MR 2024509, DOI 10.1088/0305-4470/36/39/303
- H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The dimension of the space of essentially time- and band-limited signals, Bell System Tech. J. 41 (1962), 1295–1336. MR 147686, DOI 10.1002/j.1538-7305.1962.tb03279.x
- H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell System Tech. J. 40 (1961), 65–84. MR 140733, DOI 10.1002/j.1538-7305.1961.tb03977.x
- S. Saitoh, Integral transforms, reproducing kernels and their applications, Pitman Research Notes in Mathematics Series, vol. 369, Longman, Harlow, 1997. MR 1478165
- Saburou Saitoh, Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, vol. 189, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1988. MR 983117
- Saburou Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer. Math. Soc. 89 (1983), no. 1, 74–78. MR 706514, DOI 10.1090/S0002-9939-1983-0706514-9
- David Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983), no. 3, 379–393. MR 710468, DOI 10.1137/1025078
- David Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty. IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J. 43 (1964), 3009–3057. MR 181766, DOI 10.1002/j.1538-7305.1964.tb01037.x
- D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty. I, Bell System Tech. J. 40 (1961), 43–63. MR 140732, DOI 10.1002/j.1538-7305.1961.tb03976.x
- H. Volkmer, Spheroidal Wave Functions, in Handbook of Mathematical Functions, Nat. Bureau of Stds., Applied Math. Series 2004.
- Gilbert G. Walter, Differential operators which commute with characteristic functions with applications to a lucky accident, Complex Variables Theory Appl. 18 (1992), no. 1-2, 7–12. MR 1157019, DOI 10.1080/17476939208814523
- Gilbert G. Walter and Xiaoping Shen, Wavelets based on prolate spheroidal wave functions, J. Fourier Anal. Appl. 10 (2004), no. 1, 1–26. MR 2045522, DOI 10.1007/s00041-004-8001-7
- Gilbert G. Walter and Xiaoping A. Shen, Sampling with prolate spheroidal wave functions, Sampl. Theory Signal Image Process. 2 (2003), no. 1, 25–52. MR 2002855, DOI 10.1007/BF03549384
- H. Xiao, V. Rokhlin, and N. Yarvin, Prolate spheroidal wavefunctions, quadrature and interpolation, Inverse Problems 17 (2001), no. 4, 805–838. Special issue to celebrate Pierre Sabatier’s 65th birthday (Montpellier, 2000). MR 1861483, DOI 10.1088/0266-5611/17/4/315
- Ahmed I. Zayed, Advances in Shannon’s sampling theory, CRC Press, Boca Raton, FL, 1993. MR 1270907
Additional Information
- Ahmed I. Zayed
- Affiliation: Department of Mathematical Sciences, DePaul University, Chicago, Illiniois 60614
- Email: azayed@condor.depaul.edu
- Received by editor(s): October 20, 2005
- Received by editor(s) in revised form: March 27, 2006
- Published electronically: March 2, 2007
- Communicated by: Carmen C. Chicone
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 2193-2203
- MSC (2000): Primary 33C47, 44A05; Secondary 42C05, 33C45
- DOI: https://doi.org/10.1090/S0002-9939-07-08739-4
- MathSciNet review: 2299497