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Small gaps Fourier series and generalized variations. (English) Zbl 1241.42004

Summary: Suppose \(f \in L[-\pi,\pi]\) has a Fourier series \(\sum_{k=-\infty}^{\infty}\hat f(n_k)e^{in_kx} (n_{-k}=-n_k)\) with small gaps \(n_{k+1}-n_k\geq q \geq 1 \) for all \(k \geq 0\). Here, by applying the Wiener-Ingham result for finite trigonometric sum with ‘small’ gaps, we estimate the order of magnitude of the Fourier coefficients and obtain a sufficient condition for the convergence of the series \(\sum_{k \in \mathbb Z} |\hat{f}(n_k)|^{\beta}\) \((0 <\beta\leq 2)\) if \(f\) is locally of class \(\Lambda\)BV \((p(n)\uparrow \infty)\).

MSC:

42A15 Trigonometric interpolation
42A24 Summability and absolute summability of Fourier and trigonometric series
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A45 Functions of bounded variation, generalizations
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