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trigonometric series; Hardy-Littlewood inequality for functions in $H^{p}$; Bernstein-Zygmund inequalities for the derivative of trigonometric polynomials in $L^{p}$-metric for $0<p<1$; necessary conditions for the convergence in $L^{p}$-metric
We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the $L^{p}$-metric, where $0<p<1$. The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in $H^{p}$ and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the $L^{p}$-metric, where $0<p<1$.
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