Mathematics

Abstract

INTERPOLATION by finite differences is an old subject, going back to the time of Newton, but Prof. Whittaker's treatment will come as a surprise to those familiar with the usual treatises. His object is to deal with new and little-known aspects of the interpolation series, and to discuss them in the light of the modern theory of integral and meromorphic functions (which is briefly summarised in the introduction). After three chapters dealing respectively with series of polynomials, differences and summation, and successive derivates, we come to two chapters concerning what is called the cardinal series. It was discovered by J. F. Steffenson and E. T. Whittaker (the author's father) that this series is closely connected with the well-known Newton-Gauss interpolation series, although at first sight it seems to be quite different. The cardinal series is also related to the theory of Fourier series and integrals, and to Hardy's ‘m-functions'; its interesting properties have been studied by Ferrar, Copson, Polya, Miss Cartwright and others. Interpolatory Function Theory By Prof. J. M. Whittaker. Pp. vi + 107. (Cambridge Tracts in Mathematics and Mathematical Physics, No. 33.) (Cambridge: At the University Press, 1935.) 6s. 6d. net.