György Gát Inst. of Math. and Comp. Sci., College of Nyíregyháza, Nyíregyháza, P.O.Box 166., H–4400, Hungary

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The aim of this paper is to prove the a.e. convergence of sequences of the Fejér means of the Walsh–Fourier series of bivariate integrable functions. That is, let such that aj(n+1)≧δsupknaj(n) (j=1,2, n∊ℕ) for some δ>0 and a1(+∞)=a2(+∞)=+∞. Then for each integrable function fL1(I2) we have the a.e. relation . It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejér means of integrable functions which was proved earlier by the author and Weisz [3,8].

  • [1] Zygmund, A. Jessen, B. Marcinkiewicz, J. 1935 Note on the differentiability of multiple integrals Fund. Math. 32 217234.

  • [2] Fine, N. J. 1955 Cesàro summability of Walsh–Fourier series Proc. Nat. Acad. Sci. U.S.A. 41 558591 .

  • [3] Gát, G. 1996 Pointwise convergence of the Cesàro means of double Walsh series Ann. Univ. Sci. Budapest. Rolando Eötvös, Sect. Comput. 16 173184.

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  • [4] Gát, G. 2000 On the divergence of the (C,1) means of double Walsh–Fourier series Proc. Am. Math. Soc. 128 17111720 .

  • [5] Zygmund, J. A. Marcinkiewicz, J. 1939 On the summability of double Fourier series Fund. Math. 32 122132.

  • [6] Móricz, F. Schipp, F. Wade, W. R. 1992 Cesàro summability of double Walsh–Fourier series Trans Amer. Math. Soc. 329 131140 .

  • [7] Schipp, F. Wade, W. R. Simon, P. Pál, J. 1990 Walsh Series: An Introduction to Dyadic Harmonic Analysis Adam Hilger Bristol and New York.

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  • [8] Weisz, F. 1996 Cesàro summability of two-dimensional Walsh–Fourier series Trans. Amer. Math. Soc. 348 21692181 .

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Acta Mathematica Hungarica
Language English
Size B5
Year of
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Founder Magyar Tudományos Akadémia  
H-1051 Budapest, Hungary, Széchenyi István tér 9.
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ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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