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## Studia Mathematica

1994 | 108 | 2 | 177-190
Tytuł artykułu

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

Autorzy
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EN
Abstrakty
EN
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].
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
177-190
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-02-02
Twórcy
autor
Bibliografia
• [1] L. A. Balashov, Series with respect to the Walsh system with monotone coefficients, Sibirsk. Mat. Zh. 12 (1971), 25-39 (in Russian).
• [2] R. P. Boas, Integrability Theorems for Trigonometric Transforms, Springer, Berlin, 1967.
• [3] C.-P. Chen, L¹-convergence of Fourier series, J. Austral. Math. Soc. Ser. A 41 (1986), 376-390.
• [4] C.-P. Chen, Integrability and L¹-convergence of multiple trigonometric series, preprint.
• [5] C.-P. Chen, L¹-convergence of multiple Fourier series, submitted.
• [6] C.-P. Chen and P.-H. Hsieh, Pointwise convergence of double trigonometric series, J. Math. Anal. Appl. 172 (1993), 582-599.
• [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.
• [9] M. M. H. Marzuq, Integrability theorem of multiple trigonometric series, J. Math. Anal. Appl. 157 (1991), 337-345.
• [10] F. Móricz, Convergence and integrability of double trigonometric series with coefficients of bounded variation, Proc. Amer. Math. Soc. 102 (1988), 633-640.
• [11] F. Móricz, On the integrability and L¹-convergence of double trigonometric series, Studia Math. 98 (1991), 203-225.
• [12] F. Móricz, Double Walsh series with coefficients of bounded variation, Z. Anal. Anwendungen 10 (1991), 3-10.
• [13] F. Móricz and F. Schipp, On the integrability and L¹-convergence of double Walsh series, Acta Math. Hungar. 57 (1991), 371-380.
• [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.