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We prove that if $c_{jk}\to 0$ as max(|j|,|k|) → ∞, and ${\sum }_{\left|j\right|=0\pm }^{\infty }{\sum }_{\left|k\right|=0\pm }^{\infty }{\theta \left(\right|j|}^{\top }{\left)\vartheta \right(\left|k\right|}^{\top }\left)\right|{\Delta }_{12}{c}_{jk}|<\infty$, then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and ${\iint }_{T²}|s_{mn}(x,y)-f(x,y)\left|·\right|\varphi \left(x\right)\psi \left(y\right)|dxdy\to 0$ as min(m,n) → ∞, where f(x,y) is the limiting function of the rectangular partial sums $s_{mn}(x,y)$, (ϕ,θ) and (ψ,ϑ) are pairs of type I. A generalization of this result concerning L¹-convergence is also established. Extensions of these results to double series of orthogonal functions are also considered. These results can be extended to n-dimensional case. The aforementioned results generalize work of Balashov [1], Boas [2],...

It is shown that under certain conditions on $c_{jk}$, the rectangular partial sums $s_{mn}(x,y)$ converge uniformly on $T^2$. These conditions include conditions of bounded variation of order (1,0), (0,1), and (1,1) with the weights |j|, |k|, |jk|, respectively. The convergence rate is also established. Corresponding to the mentioned conditions, an analogous condition for single trigonometric series is ${\sum }_{\left|k\right|=n}^{\infty }\left|\Delta c_k\right|=o(1/n)$ (as n → ∞). For O-regularly varying quasimonotone sequences, we prove that it is equivalent to the condition: $n{c}_n=o\left(1\right)$ as...

Let s* denote the maximal function associated with the rectangular partial sums $s_{mn}(x,y)$ of a given double function series with coefficients $c_{jk}$. The following generalized Hardy-Littlewood inequality is investigated: ${\left|\right|s*\left|\right|}_{p,\mu }\le C_{p,\alpha ,\beta }{{\Sigma }_{j=0}^{\infty }{\Sigma }_{k=0}^{\infty }{\left(j̅\right)}^{p-\alpha -2}{\left(k̅\right)}^{p-\beta -2}{\left|c_{jk}\right|}^p}^{1/p}$, where ξ̅=max(ξ,1), 0 < p < ∞, and μ is a suitable positive Borel measure. We give sufficient conditions on $c_{jk}$ and μ under which the above Hardy-Littlewood inequality holds. Several variants of this inequality are also examined. As a consequence, the ||·||p,μ-convergence property of $s_{mn}(x,y)$...

Let $f_c(x,y)\equiv {\sum }_{j=1}^{\infty }{\sum }_{k=1}^{\infty }{a}_{jk}(1-cosjx)(1-cosky)$ with $a_{jk}\ge 0$ for all j,k ≥ 1. We estimate the integral ${\int }_0^{\pi }{\int }_0^{\pi }{x}^{\alpha -1}{y}^{\beta -1}\varphi \left(f_c(x,y)\right)dxdy$ in terms of the coefficients $a_{jk}$, where α, β ∈ ℝ and ϕ: [0,∞] → [0,∞]. Our results can be regarded as the trigonometric analogues of those of Mazhar and Móricz [MM]. They generalize and extend Boas [B, Theorem 6.7].