On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

• Autorzy: G. Gát
• Języki publikacji: EN

Abstrakty

EN

Let G be the Walsh group. For $f ∈ L^1(G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_n$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ*f ≔ sup_n |σ_n f|.$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥σ*f∥_1 ≤ c∥|f|∥_H$, where H is the Hardy space on the Walsh group.

Kategorie tematyczne

• 43A75: Analysis on specific compact groups
• 40G05: Ces\`aro, Euler, N\"orlund and Hausdorff methods
• 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1998
• Tom: 130
• Numer: 2
• Strony: 135-148

Daty

wydano

1998

otrzymano

1997-06-07

poprawiono

1997-11-26

Twórcy

autor

G. Gát

• Department of Mathematics, Bassenyei College, P.O. Box 166, H-44000 Nyíregyháza, Hungary

Bibliografia

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• [SCH2] F. Schipp, Pointwise convergence of expansions with respect to certain product systems, Anal. Math. 2 (1976), 63-75.
• [SWS] F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: an Introduction to Dyadic Harmonic Analysis, Adam Higler, Bristol and New York, 1990.
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• [SK2] F. Schipp, $L_1$ convergence of Fourier series with respect to the Walsh-Kaczmarz system, Vestnik Moskov. Univ. Ser. Mat. Mekh. 1981, no. 6, 3-6 (in Russian).
• [WY] W. S. Young, On the a.e. convergence of Walsh-Kaczmarz-Fourier series, Proc. Amer. Math. Soc. 44 (1974), 353-358.