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We extend the results of paper of F. Móricz (2010), where necessary conditions were given for the $L^1$-convergence of double Fourier series. We also give necessary and sufficient conditions for the $L^1$-convergence under appropriate assumptions.

It is a classical problem in Fourier analysis to give conditions for a single sine or cosine series to be uniformly convergent. Several authors gave conditions for this problem supposing that the coefficients are monotone, non-negative or more recently, general monotone. There are also results for the regular convergence of double sine series to be uniform in case the coefficients are monotone or general monotone double sequences. In this paper we give new sufficient conditions for the uniformity...

We investigate the convergence behavior of the family of double sine integrals of the form ${\int }_0^{\infty }{\int }_0^{\infty }f(x,y)sinuxsinvydxdy$, where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals ${\int }_{a₁}^{b₁}{\int }_{a₂}^{b₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and $b_j>a_j\ge 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals...

Let a single sine series (*) ${\sum }_{k=1}^{\infty }{a}_{k}sinkx$ be given with nonnegative coefficients $a_k$. If $a_k$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k{a}_k\to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) ${\sum }_{k=1}^{\infty }{\sum }_{l=1}^{\infty }{c}_{kl}sinkxsinly$, even with complex coefficients $c_{kl}$. We also give a uniform...

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 $0<p<1$.