On two basic theorems in the theory of almost periodic functions.

Abstract

In this study we offer proofs for two basic results in the theory of almost periodic functions; the propositions in question are Bohr's Approximation Theorem and the Uniqueness Theorem for almost periodic functions. By using the analytic representation theory of positive definite sequences, we carry out a special type of Diophantine analysis and derive from a part of this a well-known theorem concerning almost periods. We then deduce Bohr's theorem directly from the theorem concerning almost periods. The noteworthy feature in this step is that we avoid considering the so-called limit periodic functions. In the presentation of the uniqueness theorem, we have followed H. Weyl and based our proof on the theory of compact normal operators in a pre-Hilbert space. The eigenvalue problem which arises in this way, is resolved in terms of elementary linear algebra. At the end of our discussion an interpretation of these two theorems in terms of the notions pertaining to the theory of Banach algebras as briefly indicated. In the quest for a self-contained presentation we were obliged to include in our discussions several results belonging to the firmly established portions of the theory in question.