Real Analysis Exchange

Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series.

Ushangi Goginava and György Gát

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Abstract

The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh-Fourier series is of type $(H^{\#},L_{1})$ provided that the supremum in the maximal operator is taken over some special indices. The set of Walsh polynomials is dense in $ H^{\#}$, so by the well-known density argument we have that $t_{2^{n},2^{m}}f\left( x^{1},x^{2}\right) \rightarrow f\left( x^{1},x^{2}\right) $ a. e. as $m,n\rightarrow \infty$ for all $f\in H^{\#} (\supset L\log^{+} L$). We also prove the sharpness of this result. Namely, For all measurable function $\delta :[0,+\infty) \to [0,+\infty) , \, \lim_{t\to\infty}\delta(t)=0$ we have a function $f$ such as $f\in \llogld (L)$ and the two-dimensional Nörlund logarithmic means do not converge to $f$ a.e. (in the Pringsheim sense) on $I^2$.

Article information

Source
Real Anal. Exchange, Volume 31, Number 2 (2005), 447-464.

Dates
First available in Project Euclid: 10 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.rae/1184104037

Mathematical Reviews number (MathSciNet)
MR2265786

Zentralblatt MATH identifier
1103.42017

Subjects
Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Keywords
double Walsh-Fourier series logarithmic means a. e. convergence and divergence

Citation

Gát, György; Goginava, Ushangi. Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series. Real Anal. Exchange 31 (2005), no. 2, 447--464. https://projecteuclid.org/euclid.rae/1184104037


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References

  • G. Gát, Pointwise Convergence of the Cesàro Means of Double Walsh Series, Ann. Univ. Sci. Budapest Sect. Comp., 16 (1996), 173–184.
  • G. Gát, On the divergence of the $(C,1)$ means of double Walsh-Fourier series, Proc. Amer. Math. Soc., 128, no. 6 (2000), 1711–1720.
  • G. Gát, U. Goginava, G. Tkebuchava, Convergence in measure of Logarithmic means of double Walsh-Fourier series, Georgian Math. J., to appear.
  • G. Gát, U. Goginava, G. Tkebuchava, Convergence of logarithmic means of multiple Walsh-Fourier series, Anal. Theory Appl., to appear.
  • G. Gát, U. Goginava, G. Tkebuchava, Convergence in measure of Logarithmic means of quadratical partial sums of double Walsh-Fourier series, J. Math. Anal. Appl. (to appear).
  • R. Getsadze, On the divergence in measure of multiple Fourier series (in Russian), Some problems of functions theory, 4 (1988), 84–117.
  • R. Getsadze, On the convergence in measure of Nörlund logarithmic means of multiple orthogonal Fourier series, East Journal on Approximations, 11 (3) (2006), 237–256.
  • U. Goginava, Almost everywhere convergence of subsequence of logarithmic means of Walsh-Fourier series, Acta Math. Acad. Paed. Nyiregyh., 21 (2005), 169–175.
  • U. Goginava, G. Tkebuchava, Convergence of subsequence of partial sums and logarithmic means of Walsh-Fourier series, Acta Sci. Math. (Szeged), to appear.
  • B. I. Golubov, A. V. Efimov, and V. A. Skvortsov, Series and transformations of Walsh, Nauka, Moscow, 1987 (Russian), English transl., Kluwer Acad. publ., 1991.
  • G. H. Hardy, Divergent series, Oxford, (1949), 141–142.
  • A. N. Kolmogorov, Sur les functions harmoniques conjugees and les series de Fouries, Fund. Math., 17 (1925), 23–28.
  • I. Marcinkiewicz, and A. Zygmund, On the summability of double Fourier series, Fund. Math., 32 (1939), 112–132.
  • G. Morgenthaller, Walsh-Fourier series, Trans. Amer. Math. Soc., 84, 2 (1957), 472–507.
  • F. Móricz, F. Schipp, and W. R. Wade, Cesàro summability of double Walsh-Fourier series, Trans Amer. Math. Soc., 329 (1992), 131–140.
  • A. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc., 34 (1932), 241–279.
  • F. Schipp, W.R. Wade, P. Simon, and J. Pál, Walsh series: an introduction to dyadic harmonic analysis, Adam Hilger, Bristol and New York, 1990.
  • P. Simon, Cesàro summability with respect to two-parameter Walsh system, Monatsh. Math., 131 (2000), 321–334.
  • G. Tkebuchava, On multiple Fourier, Fourier-Walsh and Fourier-Haar series in nonreflexive separable Orlicz space, Bull. Georg. Acad. Sci., 149, 2 (1994), 1–3.
  • G. Tkebuchava, Subsequence of partial sums of multiple Fourier and Fourier-Walsh series, Bull. Georg. Acad. Sci, 169 2(2004), 252-253.
  • W. R. Wade, Recent development in the theory of Walsh series, Intern. J. Math. Sci., 5, 4 (1982), 625–673.
  • C. Watary, On generalized Fourier-Walsh series, Tōhōku Math. J., 10, 3 (1968), 211–241.
  • L. V. Zhizhiashvili, Some problems of multidimensional harmonic analysis, Tbilisi, TGU, 1996 (Russian).