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Almost-periodic functions in abstract spaces. (English) Zbl 0648.42006
Research Notes in Mathematics, 126. Boston-London-Melbourne: Pitman Advanced Publishing Program. (Distr. by Longman Group UK Limited, Harlow, Essex, England); IX, 133 p.; £15.00 (1985).
This book provides an easy-to-read introduction to the theory of almost periodic (a.p.) functions on \({\mathbb{R}}\) (except for brief sections where functions on \({\mathbb{R}}^ n \)and on more general groups are considered). As indicated by the title, the values of the functions are in a Banach space for most of the book. This added generality does not require much more work over dealing with real or complex valued functions. Topics covered include: (a) various characterizations of almost periodicity; (b) differentiation and integration of a.p. functions; (c) asymptotically a.p. functions; (d) the mean value and Fourier series of a.p. functions; and (e) weakly a.p. functions (whose definition requires that \(t\to x\) *(f(t)) be a.p. for each x * in the dual of the Banach space).
Reviewer: P.Milnes

MSC:
42A75 Classical almost periodic functions, mean periodic functions
43A60 Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to harmonic analysis on Euclidean spaces