Weighted integrability and L¹-convergence of multiple trigonometric series

• Autorzy: Chang-Pao Chen
• Języki publikacji: 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

multiple trigonometric series; rectangular partial sums; Cesàro means; weighted integrability; L¹-convergence; conditions of generalized bounded variation

Kategorie tematyczne

• 42B05: Fourier series and coefficients
• 42A20: Convergence and absolute convergence of Fourier and trigonometric series
• 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1994
• Tom: 108
• Numer: 2
• Strony: 177-190

Daty

wydano

1994

otrzymano

1993-02-02

Twórcy

autor

Chang-Pao Chen

• Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan 30043, R.O.C.

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.
• [15] F. Schipp, W. R. Wade and P. Simon, Walsh Series, An Introduction to Dyadic Harmonic Analysis, Akadémiai Kiadó, Budapest, 1990.
• [16] W. H. Young, On the Fourier series of bounded functions, Proc. London Math. Soc. 12 (1913), 41-70.
• [17] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1959.