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A class of I. M. Vinogradov’s series and its applications in harmonic analysis. (English) Zbl 0815.42003
Gonchar, A. A. (ed.) et al., Progress in approximation theory. An international perspective. Proceedings of the international conference on approximation theory, Tampa, South Florida, USA, March 19-22, 1990. New York: Springer-Verlag. Springer Ser. Comput. Math. 19, 353-402 (1992).
The Vinogradov series are of the form \[ \sum_ n \widehat f(n) e(n^ r x_ r+\cdots+ nx_ 1), \] where the \(\widehat f(n)\) are the Fourier coefficients of a function \(f\) of period 1 which is Lebesgue integrable over the period. Here the convergence of the series is understood in the sense of the principal value. For instance, if \(f(x)= 1/2- \{x\}\), then \(\widehat f(0)= 0\) and \(\widehat f(n)= (2\pi in)^{-1}\) if \(n\neq 0\), and the author calls the related Vinogradov series a discrete Hilbert transform; it plays an important role in the theory.
The autor presents a comprehensive survey, representing largely his own recent work, of applications of Vinogradov series. Here some examples of the topics discussed: Lebesgue constants related to partial sums of Fourier series and uniform convergence of Fourier series (under the assumption that the Fourier coefficients vanish if \(n\) does not lie in a given sequence), solutions of the Schrödinger equation, incomplete Gaussian sums, quantum chaos.
For the entire collection see [Zbl 0764.00001].
Reviewer: M.Jutila (Turku)

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
11L15 Weyl sums
42A20 Convergence and absolute convergence of Fourier and trigonometric series
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces