Weighted integrability and L¹-convergence of multiple trigonometric series
Weighted integrability and L¹-convergence of multiple trigonometric series
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 functi