Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

Finite Fourier self-transforms


Authors: A. Fedotowsky and G. Boivin
Journal: Quart. Appl. Math. 30 (1972), 235-254
MSC: Primary 42A60
DOI: https://doi.org/10.1090/qam/422988
MathSciNet review: 422988
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Abstract: This paper considers the integral equation

$\displaystyle \lambda \gamma \left( {t'} \right) = {\left( {2\pi } \right)^{ - ... ..._T {\gamma \left( t \right)} \exp \left( {i\omega \cdot t} \right)dt d\omega } $

as well as a more general one wherein the Fourier kernels are weighted. When $ \Omega $ and $ T$ are $ N$-dimensional spherical domains, the eigenfunctions of the integral equation are generalized prolate spheroidal functions for which a new nomenclature is proposed. Many properties of the eigenfunctions are developed and summarized. Because of the importance of these functions in Fourier transform theory, old as well as new properties are included.

References [Enhancements On Off] (What's this?)

  • [1] D. Slepian and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty--I, Bell Syst. Tech. J. 40, 43-64 (1961) MR 0140732
  • [2] H. J. Landau and H. O. Pollak, Prolate wave functions, Fourier analysis and uncertainty--II, Bell Syst. Tech. J. 40, 65-84 (1961) MR 0140733
  • [3] 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 Syst. Tech. J. 41, 1295-1337 (1962) MR 0147686
  • [4] D. Slepian, Prolate spheroidal wave functions, Fourier analysis and uncertainty--IV: Extensions to many dimensions; generalized prolate spheroidal functions, Bell Syst. Tech. J. 43, 3009-3057 (1964) MR 0181766
  • [5] D. Rhodes, On some double orthogonality properties of the spheroidal and Mathieu functions, J. Math. and Phys. 44, 52-65 (1965); errata, ibid. 44, 411 (1965) MR 0181765
  • [6] A. Erdelyi, Higher transcendental functions, Vol. II, McGraw-Hill, New York, 1953 (p. 233)
  • [7] J. C. Hurtley, Hyperspheroidal functions, in Proceedings of the symposium on quasioptics, Polytechnic Press of Brooklyn, New York, 1969 (p. 367)
  • [8] J. A. Morrison, Eigenfunctions of the finite Fourier transform operator over a hyperellipsoidal region, J. Math. and Phys. 44, 245-254 (1965) MR 0184052
  • [9] J. A. Morrison, On the eigenfunctions corresponding to the band-pass kernel in the case of degeneracy, Quart. Appl. Math. 21, 13-19 (1963) MR 0145306
  • [10] K. Gottfried, Quantum mechanics, Vol. I, W. A. Benjamin, New York, 1966 (p. 93)
  • [11] J. H. II. Chalk, The optimum pulse shape for pulse communications, Proc. I. E. E. 87, 88 (1950)
  • [12] G. Boivin and G. Lansraux, Maximum of the factor of encircled energy, Can. J. Phys. 39, 158 (1961) MR 0127145
  • [13] D. Slepian, Analytical solution of two apodization problems, J. Opt. Soc. Am. 55, 1110-15 (1965)
  • [14] D. R. Rhodes, The optimum line source for the best mean-square approximation to a given radiation pattern, I.E.E.E. Trans. AP-11, 440-446 (1963)
  • [15] G. Toraldo Di Francia, Degrees of freedom of an image, J. Opt. Soc. Am. 59, 799 (1969)
  • [16] B. R. Frieden, The extrapolating pupil, image synthesis, and some thought applications, J. Opt. Soc. Am. 9, 2489-2496 (1970)
  • [17] Y. Itoh, Evaluation of aberration using the general spheroidal wave functions, J. Opt. Soc. Am. 60, 10 (1970) MR 0273901
  • [18] L. A. Weinstein, Open resonators and open waveguides, The Golem Press, Boulder, 1969
  • [19] D. Slepian, Asymptotic expansions for prolate spheroidal wave functions, J. Math. Phys. (2) 44, 99-140 (1965) MR 0179392
  • [20] H. J. Landau, Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math. 117, 37-52 (1967) MR 0222554
  • [21] D. R. Rhodes, On spheroidal functions, J. Res. Bur. Standards 74B, 187-209 (1970) MR 0271425
  • [22] S. Bergman and M. Shiffer, Kernel functions and elliptic differential equations, Academic Press, New York, 1953
  • [23] A. Fedotowsky and G. Boivin, Inversion of microwave scattering data, Can. J. Phys. 49, 3082 (1971) MR 0334695

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DOI: https://doi.org/10.1090/qam/422988
Article copyright: © Copyright 1972 American Mathematical Society

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