On the $L_1$-convergence of Fourier series

S. Fridli

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Weighted integrability and L¹-convergence of multiple trigonometric series

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We prove that if $c_{j k} \to 0$ as max(|j|,|k|) → ∞, and $\sum_{| j | = 0 \pm}^{\infty} \sum_{| k | = 0 \pm}^{\infty} {θ (| j |}^{⊤} {) ϑ (| k |}^{⊤}) | Δ_{12} c_{j k} | < \infty$ , then f(x,y)ϕ(x)ψ(y) ∈ L¹(T²) and $\iint_{T ²} | s_{m n} (x, y) - f (x, y) | \cdot | ϕ (x) ψ (y) | d x d y \to 0$ as min(m,n) → ∞, where f(x,y) is the limiting function of the rectangular partial sums $s_{m n} (x, y)$ , (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],...