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Transactions of the American Mathematical Society

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Band-limited functions: $ L\sp p$-convergence


Authors: Juan A. Barceló and Antonio Córdoba
Journal: Trans. Amer. Math. Soc. 313 (1989), 655-669
MSC: Primary 42A38; Secondary 33A55, 44A15
DOI: https://doi.org/10.1090/S0002-9947-1989-0951885-1
MathSciNet review: 951885
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Abstract: We consider the set $ {B_p}(\Omega)$ (functions of $ {L^p}({\mathbf{R}})$ whose Fourier spectrum lies in $ [ - \Omega , + \Omega ]$). We prove that the prolate spheroidal wave functions constitute a basis of this space if and only if $ 4/3 < p < 4$. The result is obtained as a consequence of the analogous problem for the spherical Bessel functions. The proof rely on a weighted inequality for the Hilbert transform.


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  • [1] A. Askey and S. Wainger, Mean convergence of expansion in Laguerre and Hermite series, Amer. J. Math. 87 (1965), 695-708. MR 0182834 (32:316)
  • [2] C. Flamer, Spherical wave functions, Stanford Univ. Press, Stanford, Calif., 1957.
  • [3] H. L. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty II, Bell Systems Tech. J. 40 (1961), 65-84. MR 0140733 (25:4147)
  • [4] -, Prolate spheroidal wave functions, Fourier analysis and uncertainty III, Bell Systems Tech. J. 41 (1962), 1295-1335. MR 0147686 (26:5200)
  • [5] J. Newman and W. Rudin, On mean convergence of orthogonal series. Proc. Amer. Math. Soc. 3 (1952), 219-222. MR 0047811 (13:936b)
  • [6] H. Pollard, The mean convergence of orthogonal series. I, Trans. Amer. Math. Soc. 62 (1947), 387-403. MR 0022932 (9:280d)
  • [7] E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11. MR 676801 (84i:42032)
  • [8] D. Slepian, On bandwidth, Proc. IEEE 64 (1976), 292-300. MR 0462765 (57:2738)
  • [9] -, Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983), 379-393. MR 710468 (84i:94016)
  • [10] D. Slepian, and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertainty I. Bell Systems Tech. J. 40 (1961) 43-64. MR 0140732 (25:4146)
  • [11] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton N.J., 1971. MR 0304972 (46:4102)
  • [12] J. A. Straton, P. M. Morse, L. J. Chu, J. D. Little and F. J. Corbato, Spherical wave functions, Wiley, New York, 1956.
  • [13] G. Szego, Orthonormal polynomials, 4th ed., Amer. Math. Soc. Colloq. Publ., Amer. Math. Soc., Providence, R.I., 1975.
  • [14] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, 1966. MR 1349110 (96i:33010)

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DOI: https://doi.org/10.1090/S0002-9947-1989-0951885-1
Article copyright: © Copyright 1989 American Mathematical Society

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