Positive Definite Functions and Bochner’s Theorem

Abstract

Continuous (and not necessarily periodic) positive definite functions of a real variable were seemingly first studied by Bochner who, by using the existing theory of Fourier integrals, established for them a fundamental representation theorem now known by his name and which is the analogue for the group R of 9.2.8. These positive definite functions were not seen in their true perspective until some ten or fifteen years had elapsed. Then, as a result of the birth and growth of the theory of commutative Banach algebras and the applications of this theory to harmonic analysis on locally compact Abelian groups (see 11.4.18(3)), the central position of the Bochner theorem came to be appreciated. The developments in this direction were due largely to the Russian mathematicians Gelfand and Raikov, who enlarged still more the role played by positive definite functions by noticing their intimate relationship with the theory of representations of (not necessarily Abelian) locally compact groups. A similar path was hewn, independently, almost simultaneously, and from a slightly different point of view, by the French mathematicians H. Cartan and Godement; see [B], pp. 220 ff. It is now true to say that a considerable portion of our function-theoretical knowledge of locally compact groups rests upon a study of positive definite functions on such groups.