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],...

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].