On the absolute summability of lacunary Fourier series

N.V. Patel and V.M. Shah

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

Classification (MSC2000): 42A28

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