# Weighted integrability and L¹-convergence of multiple trigonometric series

Studia Mathematica (1994)

- Volume: 108, Issue: 2, page 177-190
- ISSN: 0039-3223

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topChen, 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 -

## References

top- [1] L. A. Balashov, Series with respect to the Walsh system with monotone coefficients, Sibirsk. Mat. Zh. 12 (1971), 25-39 (in Russian). Zbl0224.42010
- [2] R. P. Boas, Integrability Theorems for Trigonometric Transforms, Springer, Berlin, 1967. Zbl0145.06804
- [3] C.-P. Chen, L¹-convergence of Fourier series, J. Austral. Math. Soc. Ser. A 41 (1986), 376-390. Zbl0642.42005
- [4] C.-P. Chen, Integrability and L¹-convergence of multiple trigonometric series, preprint. Zbl0795.42007
- [5] C.-P. Chen, L¹-convergence of multiple Fourier series, submitted.
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- [7] C.-P. Chen, F. Móricz and H.-C. Wu, Pointwise convergence of multiple trigonometric series, ibid., to appear.
- [8] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414. Zbl0036.03604
- [9] M. M. H. Marzuq, Integrability theorem of multiple trigonometric series, J. Math. Anal. Appl. 157 (1991), 337-345. Zbl0729.42012
- [10] F. Móricz, Convergence and integrability of double trigonometric series with coefficients of bounded variation, Proc. Amer. Math. Soc. 102 (1988), 633-640. Zbl0666.42004
- [11] F. Móricz, On the integrability and L¹-convergence of double trigonometric series, Studia Math. 98 (1991), 203-225. Zbl0724.42015
- [12] F. Móricz, Double Walsh series with coefficients of bounded variation, Z. Anal. Anwendungen 10 (1991), 3-10. Zbl0748.42010
- [13] F. Móricz and F. Schipp, On the integrability and L¹-convergence of double Walsh series, Acta Math. Hungar. 57 (1991), 371-380. Zbl0746.42016
- [14] F. Móricz, F. Schipp and W. R. Wade, On the integrability of double Walsh series with special coefficients, Michigan Math. J. 37 (1990), 191-201. Zbl0714.42017
- [15] F. Schipp, W. R. Wade and P. Simon, Walsh Series, An Introduction to Dyadic Harmonic Analysis, Akadémiai Kiadó, Budapest, 1990. Zbl0727.42017
- [16] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. 12 (1913), 41-70. Zbl44.0300.03
- [17] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1959. Zbl0085.05601

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