Almost Periodic Functions (Rare, 1954)

ALMOST PERIODIC FUNCTIONS

by A. S. Besicovitch

This important summary by a well-known mathematician covers the theory of almost periodic functions created by Harold Bohr. It examines Bohr’s own work, as well as newer, shorter, and more elementary proofs than Bohr’s, and also demonstrates extensions of the theory beyond the class of uniformly continuous functions to which Bohr’s work was limited. The contributions of Wiener, Weyl, de la Vallee Poussin, Stepanoff, and Bochner are examined, while the author’s own work on the piecewise continuous case is also included.

The first portion of this book establishes basic theorems of uniformly a.p. functions, including Bohr’s original work, and de la Vallee Poussin’s ingenious short proof based on classical theory of purely periodic functions. It considers such matters as summation of Fourier series of u.a.p. functions by partial sums, the BochnerFejer summation of u.a .p. functions, particular cases of Fourier series, and u.a.p. functions of two variables.

The second portion of this work covers generalizations and extensions of the original theory, discussing relaxation of continuity restrictions, auxiliary theorems and formulae, the Parseval equation and the Riese-Fischer theorem, and similar matters. The third chapter discusses analytic a.p. functions, including results in the location of singularities, behavior at infinity, and convergence of series. It opens a way to study a wide class of trigonometric series of the general type and of exponential series (Dirichlet series).

“For those interested in the concrete and calculational aspects of theory,” APPLIED MECHANICS REVIEW. “A clear, concise, reasonably self-contained treatment of theory fundamentals,” DESIGN NEWS.”