Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Whittaker's cardinal function in retrospect

Authors: J. McNamee, F. Stenger and E. L. Whitney
Journal: Math. Comp. 25 (1971), 141-154
MSC: Primary 41A30; Secondary 65R05
MathSciNet review: 0301428
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

  • [1] 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.
  • [2] J. M. Whittaker, "On the cardinal function of interpolation theory," Proc. Edinburgh Math. Soc., Ser. 1, v. 2, 1927, pp. 41-46.
  • [3] J. M. Whittaker, Interpolatory Function Theory, Cambridge, London, 1935.
  • [4] R. V. L. Hartley, "The transmission of information," Bell System Tech. J., v. 7, 1928, pp. 535-560.
  • [5] H. Nyquist, "Certain topics in telegraph transmission theory," Trans. Amer. Inst. Elec. Engrg., v. 47, 1928, pp. 617-644.
  • [6] C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–423, 623–656. MR 0026286,
  • [7] I. J. Schoenberg, Cardinal interpolation and spline functions, J. Approximation Theory 2 (1969), 167–206. MR 0257616
  • [8] A. F. Timan, \cyr Teorij pribli+enij funkciĭdeĭstvitel’nogo peremennogo., Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1960 (Russian). MR 0117478
  • [9] N. E. Nörlund, Vorlesungen über Differenzenrechung, Springer, Berlin, 1924.
  • [10] N. E. Nörlund, "Sur les formules d'interpolation de Stirling et de Newton," Ann. Sci. École Norm. Sup., v. 39, 1922, pp. 343-403.
  • [11] 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 0133208,
  • [12] T. J. Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926.
  • [13] Ruel V. Churchill, Fourier Series and Boundary Value Problems, McGraw-Hill Book Company, Inc., New York and London, 1941. MR 0003251
  • [14] G. K. Warmbrod, The distributional finite Fourier transform, SIAM J. Appl. Math. 17 (1969), 930–956. MR 0251533,
  • [15] E. C. Titchmarsh, The Theory of Fourier Integrals, Clarendon Press, Oxford, 1954.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 41A30, 65R05

Retrieve articles in all journals with MSC: 41A30, 65R05

Additional Information

Keywords: Cardinal function, interpolation, quadrature, complex
Article copyright: © Copyright 1971 American Mathematical Society