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Abstract

We prove that if $c_{jk} → 0$ as max(|j|,|k|) → ∞, and $∑^∞_{|j|=0±} ∑^∞_{|k|=0±} θ(|j|^⊤)ϑ(|k|^⊤)|Δ_{12}c_{jk}| < ∞$, then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and $∬_{T²} |s_{mn}(x,y) - f(x,y)|·|ϕ(x)ψ(y)|dxdy → 0$ 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], Chen [3,4,5], Marzuq [9], Móricz [11], Móricz-Schipp-Wade [14], and Young [16].