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Wiener’s lemma along primes and other subsequences. (English) Zbl 1414.43001

Wiener’s lemma [N. Wiener and A. Wintner, Am. J. Math. 63, 415–426 (1941; Zbl 0025.06504)] relates the asymptotic behaviour of the Fourier coefficients of a complex Borel measure on the unit circle with its atomic part. Here the question of what conclusions can be drawn in the same spirit from knowledge of the Fourier coefficients along various subsequences of the integers of arithmetic interest (polynomials, primes, polynomials of primes), and these results are then related to ergodic notions like rigidity, return times, and strong sweeping out. In addition some of these ideas are applied to orbits of operators on Banach spaces, extending results of J. A. Goldstein [Bull. Lond. Math. Soc. 25, No. 4, 369–376 (1993; Zbl 0794.47025)] and of J. A. Goldstein and B. Nagy [Ill. J. Math. 39, No. 3, 441–449 (1995; Zbl 0841.47022)].

MSC:

43A05 Measures on groups and semigroups, etc.
43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
47A10 Spectrum, resolvent
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47A35 Ergodic theory of linear operators
37A30 Ergodic theorems, spectral theory, Markov operators
47D06 One-parameter semigroups and linear evolution equations
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