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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Whittaker’s cardinal function in retrospect
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by J. McNamee, F. Stenger and E. L. Whitney PDF
Math. Comp. 25 (1971), 141-154 Request permission

Abstract:

This paper exposes properties of the Whittaker cardinal function and illustrates the use of this function as a mathematical tool. The cardinal function is derived using the Paley-Wiener theorem. The cardinal function and the central-difference expansions are linked through their similarities. A bound is obtained on the difference between the cardinal function and the function which it interpolates. Several cardinal functions of a number of special functions are examined. It is shown how the cardinal function provides a link between Fourier series and Fourier transforms, and how the cardinal function may be used to solve integral equations.
References
    E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. Roy. Soc. Edinburgh, v. 35, 1915, pp. 181-194. J. M. Whittaker, "On the cardinal function of interpolation theory," Proc. Edinburgh Math. Soc., Ser. 1, v. 2, 1927, pp. 41-46. J. M. Whittaker, Interpolatory Function Theory, Cambridge, London, 1935. R. V. L. Hartley, "The transmission of information," Bell System Tech. J., v. 7, 1928, pp. 535-560. H. Nyquist, "Certain topics in telegraph transmission theory," Trans. Amer. Inst. Elec. Engrg., v. 47, 1928, pp. 617-644.
  • C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. MR 26286, DOI 10.1002/j.1538-7305.1948.tb01338.x
  • I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approximation Theory 2 (1969), 167–206. MR 257616, DOI 10.1016/0021-9045(69)90040-9
  • A. F. Timan, Teorij pribli+enij funkciĭ deĭstvitel’nogo peremennogo, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian). MR 0117478
  • N. E. Nörlund, Vorlesungen über Differenzenrechung, Springer, Berlin, 1924. N. E. Nörlund, "Sur les formules d’interpolation de Stirling et de Newton," Ann. Sci. École Norm. Sup., v. 39, 1922, pp. 343-403.
  • H. O. Pollack, Energy distribution of band-limited functions whose samples of a half line vanish, J. Math. Anal. Appl. 2 (1961), 299–332. MR 133208, DOI 10.1016/0022-247X(61)90038-5
  • T. J. Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926.
  • Ruel V. Churchill, Fourier Series and Boundary Value Problems, McGraw-Hill Book Co., Inc., New York-London, 1941. MR 0003251
  • G. K. Warmbrod, The distributional finite Fourier transform, SIAM J. Appl. Math. 17 (1969), 930–956. MR 251533, DOI 10.1137/0117082
  • E. C. Titchmarsh, The Theory of Fourier Integrals, Clarendon Press, Oxford, 1954.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 141-154
  • MSC: Primary 41A30; Secondary 65R05
  • DOI: https://doi.org/10.1090/S0025-5718-1971-0301428-0
  • MathSciNet review: 0301428