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On the absolute summability of lacunary Fourier series. (English) Zbl 0569.42005

Let \(f\in L[-\pi,\pi]\) and let its Fourier series \(\sigma\) (f) be lacunary. The absolute convergence of \(\sigma\) (f) when f satisfies a Lipschitz condition of order \(\alpha\), \(0<\alpha <1\), only at a point and when \(\{n_ k\}\) satisfies the gap condition \(n_{k+1}-n_ k\leq An_ k^{\beta}k^{\gamma}\) \((0<\beta <1\), \(\gamma\geq 0)\) is obtained by Patadia and Shah when \(\alpha \beta +\alpha \gamma >(1-\beta)/2\). Here we study the absolute summability of \(\sigma\) (f) when \(\alpha \beta +\alpha \beta \leq (1-\beta)/2\).

MSC:

42A20 Convergence and absolute convergence of Fourier and trigonometric series
42A55 Lacunary series of trigonometric and other functions; Riesz products
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