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Almost periodic functions in groups. II

1935; American Mathematical Society; Volume: 37; Issue: 1 Linguagem: Inglês

10.1090/s0002-9947-1935-1501777-9

1088-6850

S. Bochner, John von Neumann,

Finite Group Theory Research

The present paper is a continuation of the article by J. von Neumann on Almost periodic functions in a group, I [l].fIts main object is to extend the theory of almost periodicity to those functions having values which are not numbers but elements of a general linear space L. For functions of a real variable this extension was begun by Bochner [2], and then applied by him, see [3], to a problem concerning partial differential equations.Bochner assumed L to be both complete and metric.In the present paper we shall admit more general linear spaces.We shall drop the metric but keep the completeness.Since the usual notion of completeness is based on the notion of metric, it was necessary to establish, for linear spaces, a notion of completeness independent of it.This was done in the preceding note of J. von Neumann [4].The results of this note will be employed throughout, and we observe that, from the very beginning, we shall assume that L is linear with respect to arbitrary complex coefficients, see [4], Appendix I.As in [1 ], the main difficulty to overcome was the definition and the establishment of a mean.This was done in Part I.The definition of a mean remained actually the same as in [l], but the proof of the existence of a mean necessitated a more elaborate argument, although, in broad lines, the argument does not differ essentially.In Part II we deduce the existence and uniqueness of a Fourier expansion for any almost periodic function.It is worth pointing out that the representations occurring in the Fourier expansions of abstract almost periodic functions are the same as for numerical almost periodic functions, only the constant coefficients by which the representations are multiplied are abstract elements instead of numbers.(More than that, if in a linear manifold L different topologies are suitable for our purposes, then even the nature of the coefficients no longer determines the precise nature of abstractness of the almost periodic function.)Thus, roughly speaking, there are no more abstract almost periodic functions than numerical almost periodic functions.In particular, if a group admits of no other numerical almost periodic functions than the constant ones, there exists no non-constant abstract almost periodic function, no matter how general the range-space L may be.