Vyas, Rajendra G. Small gaps Fourier series and generalized variations. (English) Zbl 1241.42004 Adv. Pure Appl. Math. 3, No. 2, 223-230 (2012). 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 \((p(n)\uparrow \infty)\). Cited in 2 Documents 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 Keywords:Fourier series with small gaps; Fourier coefficients; \(\Lambda\)BV \((p(n)\uparrow \infty)\); \(\beta\)-absolute convergence of Fourier series PDFBibTeX XMLCite \textit{R. G. Vyas}, Adv. Pure Appl. Math. 3, No. 2, 223--230 (2012; Zbl 1241.42004) Full Text: DOI