PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
1994 | 108 | 2 | 177-190
Tytuł artykułu

Weighted integrability and L¹-convergence of multiple trigonometric series

Autorzy
Treść / Zawartość
Warianty tytułu
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].
Twórcy
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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv108i2p177bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.