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XXIII.—On the Cardinal Function of Interpolation-Theory

Published online by Cambridge University Press:  15 September 2014

W. L. Ferrar
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Extract

The function which is discussed in the greater part of this paper was introduced by Professor E. T. Whittaker in vol. xxxv (1915), pp. 181–194 of these Proceedings. A one-valued function, analytic in the finite part of the plane save possibly for poles, is taken and a table made of the values of f(x) for x = a, a + w, a — w, a + 2w, a — 2w, … The function

called the Cardinal Function, is then shown to be the simplest function satisfying the table of values so obtained, in that it is a function which, when analysed into its periodic constituents, has no rapid oscillations.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 45 , Issue 3 , 1926 , pp. 269 - 282
Copyright
Copyright © Royal Society of Edinburgh 1926

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References

page 269 note * References to this in the present paper are denoted by (I).

page 271 note * This fact was pointed out, but not discussed fully, in a paper by Brown, T. Arnold, Proc. Edin. Math. Soc., xxxiv (1915–16)Google Scholar.

page 271 note † yr is assumed to be bounded as r→∞.

page 273 note * Bromwich, Theory of Infinite Series (1908), p. 205.

page 273 note † Cf. N. Nielsen, Handbuch der Theorie der Gammafunktion (B. G. Teubner, Leipzig, 1906), p. 127. In further references this work is denoted by (II).

page 274 note * K. Ogura, Tôhoku Math. Journal, vol. xvii, p. 141, has shown that Newton's formula is also expressible as an integral, viz. -1<R(z)<0 where ϕ(t) = f 0 - tδf + t 2δ2f 1-…. The present integral is, essentially, an expression for the Gauss formula in terms of the tabulated values.

page 280 note * An Introduction to the Theory of Infinite Series (1908), p. 123.

page 280 note † Carmichael, R. D., Bull. Amer. Math. Soc. (2), vol. xxv (1918–19), p. 113Google Scholar; or T. H. Gronwall, Comptes Rendus (June 1914), vol. clviii, p. 1664.

page 281 note * I hope to publish shortly a note dealing with this point.

page 281 note † Norlünd, , Annales Sci. de l'Ecole Normale (3), vol. xxxix (1922), p 365Google Scholar.

page 281 note ‡ Dougall, , Proc. Edin. Math. Soc., vol. xviii (1900), p. 78Google Scholar, has used this theorem to obtain some useful expansions in the theory of spherical harmonics.