On the Cardinal Function of Interpolation Theory

J. M. Whittaker

doi:10.1017/S0013091500007318

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1. The cardinal function is the interpolation function

which takes the values αr at the points α + rw. Its principal properties were discovered by Professor Whittaker, amongst others that

(A) When C (x) is analysed into periodic constituents by Fourier's integral-theorem, all constituents of period less than 2w are absent.

References

^{page 41 note 1} Proc. Roy. Soc. Edin., XXXV (1915), p. 181.Google Scholar

^{page 42 note 1} ibid. XLV (1925), p. 275. See also a later paper by the same author (ibid. XLVI (1926), p. 323) where further references to the literature are given.

^{page 42 note 2} Proc. Land. Math. Soc. (2) 6, 255. Theorem A.Google Scholar

^{page 43 note 1} This section has been rewritten in accordance with the valuable suggestions of Mr W. L. Ferrar, who kindly read the paper in manuscript.

^{page 44 note 1} CfHobson, . Functions of a Real Variable (2nd Ed.), p. 576. That (α) implies the existence of a 0(x) of the form (β) such that C(γ) = αγ is in fact the Riesz-Fischer theorem.Google Scholar

^{page 45 note 1} By a theorem due to Gromvall. Cfde la Vallee, Poussin in The Bice Institute Pamphlet XII (1925), p. 117.Google Scholar

^{page 45 note 2} Hobson, . loc. cit., p. 184.Google Scholar

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Copson, E. T. 1928. On Hardy's Theory of m-Functions. Proceedings of the Edinburgh Mathematical Society, Vol. 1, Issue. 2, p. 129.

Copson, E. T. 1931. XVIII.—The Definite Integrals of Interpolation Theory. Proceedings of the Royal Society of Edinburgh, Vol. 50, Issue. , p. 220.

Weston, J. D. 1949. The cardinal series in Hilbert space. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 45, Issue. 3, p. 335.

McNamee, J. Stenger, F. and Whitney, E. L. 1971. Whittaker’s cardinal function in retrospect. Mathematics of Computation, Vol. 25, Issue. 113, p. 141.

Schoenberg, I. J. 1974. Cardinal interpolation and spline functions VII. The behavior of cardinal spline interpolants as their degree tends to infinity. Journal d'Analyse Mathématique, Vol. 27, Issue. 1, p. 205.

Lundin, L. R. 1980. A cardinal function method of solution of the equation Δ𝑢=𝑢-𝑢³. Mathematics of Computation, Vol. 35, Issue. 151, p. 747.

Stenger, Frank 1981. Numerical Methods Based on Whittaker Cardinal, or Sinc Functions. SIAM Review, Vol. 23, Issue. 2, p. 165.

Burchard, H. G. and Höllig, K. 1985. N-Width and Entropy of $H_p$-Classes in $L_q ( - 1,1)$. SIAM Journal on Mathematical Analysis, Vol. 16, Issue. 2, p. 405.

Higgins, J. R. 1985. Five short stories about the cardinal series. Bulletin of the American Mathematical Society, Vol. 12, Issue. 1, p. 45.

Venturino, Ezio 1986. On solving singular integral equations via a hyperbolic tangent quadrature rule. Mathematics of Computation, Vol. 47, Issue. 175, p. 159.

Schoenberg, I. J. 1988. I. J. Schoenberg Selected Papers. p. 111.

Schoenberg, I. J. 1988. I.J. Schoenberg Selected Papers. p. 517.

Zayed, Ahmed I. 1991. On Kramer’s Sampling Theorem Associated with General Sturm-Liouville Problems and Lagrange Interpolation. SIAM Journal on Applied Mathematics, Vol. 51, Issue. 2, p. 575.

Butzer, P. L. and Stens, R. L. 1992. Sampling Theory for not Necessarily Band-Limited Functions: A Historical Overview. SIAM Review, Vol. 34, Issue. 1, p. 40.

Fornberg, Bengt and Sloan, David M. 1994. A review of pseudospectral methods for solving partial differential equations. Acta Numerica, Vol. 3, Issue. , p. 203.

Nees, Manuela 1996. Two-sided deconvolution for bandlimited kernels. Numerical Functional Analysis and Optimization, Vol. 17, Issue. 9-10, p. 903.

Hoffman, D.K. Wei, G.W. Zhang, D.S. and Kouri, D.J. 1998. Shannon–Gabor wavelet distributed approximating functional. Chemical Physics Letters, Vol. 287, Issue. 1-2, p. 119.

Madych, W. R. 1999. Analysis of Divergence. p. 477.

Butzer, P. L. Schmeisser, G. and Stens, R. L. 2001. Nonuniform Sampling. p. 17.

Meijering, E. 2002. A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proceedings of the IEEE, Vol. 90, Issue. 3, p. 319.

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