Whittaker's cardinal function in retrospect
Authors:
J. McNamee, F. Stenger and E. L. Whitney
Journal:
Math. Comp. 25 (1971), 141154
MSC:
Primary 41A30; Secondary 65R05
MathSciNet review:
0301428
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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 PaleyWiener theorem. The cardinal function and the centraldifference 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.
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 E. T. Whittaker, "On the functions which are represented by the expansions of the interpolation theory," Proc. Roy. Soc. Edinburgh, v. 35, 1915, pp. 181194.
 [2]
 J. M. Whittaker, "On the cardinal function of interpolation theory," Proc. Edinburgh Math. Soc., Ser. 1, v. 2, 1927, pp. 4146.
 [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. 535560.
 [5]
 H. Nyquist, "Certain topics in telegraph transmission theory," Trans. Amer. Inst. Elec. Engrg., v. 47, 1928, pp. 617644.
 [6]
 C. E. Shannon, "A mathematical theory of communication," Bell System Tech. J., v. 27, 1948, pp. 379423, 623656. MR 10, 133. MR 0026286 (10:133e)
 [7]
 I. J. Schoenberg, "Cardinal interpolation and spline functions," J. Approximation Theory, v. 2, 1969, pp. 167206. MR 0257616 (41:2266)
 [8]
 A. F. Timan, Theory of Approximation of Functions of a Real Variable, Fizmatgiz, Moscow, 1960; English transl., Internat. Series of Monographs in Pure and Appl. Math., vol. 34, Macmillan, New York, 1963. MR 22 #8257; MR 33 #465. MR 0117478 (22:8257)
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 N. E. Nörlund, Vorlesungen über Differenzenrechung, Springer, Berlin, 1924.
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 N. E. Nörlund, "Sur les formules d'interpolation de Stirling et de Newton," Ann. Sci. École Norm. Sup., v. 39, 1922, pp. 343403.
 [11]
 H. O. Pollak, "Energy distribution of bandlimited functions whose samples on a half line vanish," J. Math. Anal. Appl., v. 2, 1961, pp. 299322. MR 0133208 (24:A3042)
 [12]
 T. J. Bromwich, An Introduction to the Theory of Infinite Series, 2nd ed., Macmillan, London, 1926.
 [13]
 R. V. Churchill, Fourier Series and Boundary Value Problems, McGrawHill, New York, 1941. MR 2, 189. MR 0003251 (2:189d)
 [14]
 G. K. Warmbrod, "The distributional finite Fourier transform," SIAM J. Appl. Math, v. 17, 1969, pp. 930956. MR 40 #4760. MR 0251533 (40:4760)
 [15]
 E. C. Titchmarsh, The Theory of Fourier Integrals, Clarendon Press, Oxford, 1954.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197103014280
PII:
S 00255718(1971)03014280
Keywords:
Cardinal function,
interpolation,
quadrature,
complex
Article copyright:
© Copyright 1971
American Mathematical Society
