Weighted integrability and L¹-convergence of multiple trigonometric series

Chen, Chang-Pao. "Weighted integrability and L¹-convergence of multiple trigonometric series." Studia Mathematica 108.2 (1994): 177-190. <http://eudml.org/doc/216048>.

@article{Chen1994,
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].},
author = {Chen, Chang-Pao},
journal = {Studia Mathematica},
keywords = {multiple trigonometric series; rectangular partial sums; Cesàro means; weighted integrability; L¹-convergence; conditions of generalized bounded variation; -convergence; double trigonometric series; double series of orthogonal functions},
language = {eng},
number = {2},
pages = {177-190},
title = {Weighted integrability and L¹-convergence of multiple trigonometric series},
url = {http://eudml.org/doc/216048},
volume = {108},
year = {1994},
}

TY - JOUR
AU - Chen, Chang-Pao
TI - Weighted integrability and L¹-convergence of multiple trigonometric series
JO - Studia Mathematica
PY - 1994
VL - 108
IS - 2
SP - 177
EP - 190
AB - 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].
LA - eng
KW - multiple trigonometric series; rectangular partial sums; Cesàro means; weighted integrability; L¹-convergence; conditions of generalized bounded variation; -convergence; double trigonometric series; double series of orthogonal functions
UR - http://eudml.org/doc/216048
ER -