Norm convergence of logarithmic means on unbounded Vilenkin groups

György Gát; Ushangi Goginava

doi:10.1215/17358787-2017-0031

April 2018 Norm convergence of logarithmic means on unbounded Vilenkin groups

György Gát, Ushangi Goginava

Banach J. Math. Anal. 12(2): 422-438 (April 2018). DOI: 10.1215/17358787-2017-0031

Abstract

In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces $X(G)$ for every $f\in X(G)$ , where by $X(G)$ we denote either the class of continuous functions with supremum norm or the class of integrable functions.

References

1.

[1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, and A. I. Rubinshteĭn, Multiplicative Systems of Functions and Harmonic Analysis on Zero-dimensional Groups, Ehlm, Baku, 1981.[1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, and A. I. Rubinshteĭn, Multiplicative Systems of Functions and Harmonic Analysis on Zero-dimensional Groups, Ehlm, Baku, 1981.

2.

[2] M. Avdispahić, Concepts of generalized bounded variation and the theory of Fourier series, Int. J. Math. Math. Sci. 9 (1986), no. 2, 223–244. 0595.42012 10.1155/S0161171286000285[2] M. Avdispahić, Concepts of generalized bounded variation and the theory of Fourier series, Int. J. Math. Math. Sci. 9 (1986), no. 2, 223–244. 0595.42012 10.1155/S0161171286000285

3.

[3] M. Avdispahić and N. Memić, On the Lebesgue test for convergence of Fourier series on unbounded Vilenkin groups, Acta Math. Hungar. 129 (2010), no. 4, 381–392. 1274.43006 10.1007/s10474-010-0023-9[3] M. Avdispahić and N. Memić, On the Lebesgue test for convergence of Fourier series on unbounded Vilenkin groups, Acta Math. Hungar. 129 (2010), no. 4, 381–392. 1274.43006 10.1007/s10474-010-0023-9

4.

[4] M. Avdispahić and M. Pepić, On summability in $L_{p}$-norm on general Vilenkin groups, Taiwanese J. Math. 4 (2000), no. 2, 285–296.[4] M. Avdispahić and M. Pepić, On summability in $L_{p}$-norm on general Vilenkin groups, Taiwanese J. Math. 4 (2000), no. 2, 285–296.

5.

[5] M. Avdispahić and M. Pepić, Summability and integrability of Vilenkin series, Collect. Math. 51 (2000), no. 3, 237–254.[5] M. Avdispahić and M. Pepić, Summability and integrability of Vilenkin series, Collect. Math. 51 (2000), no. 3, 237–254.

6.

[6] I. Blahota and G. Gát, Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups, Anal. Theory Appl. 24 (2008), no. 1, 1–17.[6] I. Blahota and G. Gát, Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups, Anal. Theory Appl. 24 (2008), no. 1, 1–17.

7.

[7] A. V. Efimov, On certain approximation properties of periodic multiplicative orthonormal systems, Mat. Sb. (N.S.) 69 (1966), 354–370.[7] A. V. Efimov, On certain approximation properties of periodic multiplicative orthonormal systems, Mat. Sb. (N.S.) 69 (1966), 354–370.

8.

[8] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. 0036.03604 10.1090/S0002-9947-1949-0032833-2[8] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414. 0036.03604 10.1090/S0002-9947-1949-0032833-2

9.

[9] G. Gát, Cesàro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory 124 (2003), no. 1, 25–43.[9] G. Gát, Cesàro means of integrable functions with respect to unbounded Vilenkin systems, J. Approx. Theory 124 (2003), no. 1, 25–43.

10.

[10] G. Gát and U. Goginava, Uniform and $L$-convergence of logarithmic means of Walsh-Fourier series, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 497–506.[10] G. Gát and U. Goginava, Uniform and $L$-convergence of logarithmic means of Walsh-Fourier series, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 2, 497–506.

11.

[11] U. Goginava, On the uniform convergence of Walsh-Fourier series, Acta Math. Hungar. 93 (2001), no. 1–2, 59–70. 0992.42012 10.1023/A:1013865315680[11] U. Goginava, On the uniform convergence of Walsh-Fourier series, Acta Math. Hungar. 93 (2001), no. 1–2, 59–70. 0992.42012 10.1023/A:1013865315680

12.

[12] U. Goginava, Uniform convergence of Cesàro means of negative order of double Walsh-Fourier series, J. Approx. Theory 124 (2003), no. 1, 96–108. 1029.42022 10.1016/S0021-9045(03)00134-5[12] U. Goginava, Uniform convergence of Cesàro means of negative order of double Walsh-Fourier series, J. Approx. Theory 124 (2003), no. 1, 96–108. 1029.42022 10.1016/S0021-9045(03)00134-5

13.

[13] J. Pál and P. Simon, On a generalization of the concept of derivative, Acta Math. Acad. Sci. Hungar. 29 (1977), no. 1–2, 155–164. 0345.42011 10.1007/BF01896477[13] J. Pál and P. Simon, On a generalization of the concept of derivative, Acta Math. Acad. Sci. Hungar. 29 (1977), no. 1–2, 155–164. 0345.42011 10.1007/BF01896477

14.

[14] J. Price, Certain groups of orthonormal step functions, Canad. J. Math. 9 (1957), 413–425. 0079.09204 10.4153/CJM-1957-049-x[14] J. Price, Certain groups of orthonormal step functions, Canad. J. Math. 9 (1957), 413–425. 0079.09204 10.4153/CJM-1957-049-x

15.

[15] M. Riesz, Sur un théorème de la moyenne et ses applications, Acta Litt. ac Scient. Univ. Hung. 1 (1923), 114–126. Acta. Sci. Math. (Szeged), 1 (1922), 114-126.[15] M. Riesz, Sur un théorème de la moyenne et ses applications, Acta Litt. ac Scient. Univ. Hung. 1 (1923), 114–126. Acta. Sci. Math. (Szeged), 1 (1922), 114-126.

16.

[16] F. Schipp, On $L_{p}$-norm convergence of series with respect to product systems, Anal. Math. 2 (1976), no. 1, 49–64. MR0428430 10.1007/BF02079907[16] F. Schipp, On $L_{p}$-norm convergence of series with respect to product systems, Anal. Math. 2 (1976), no. 1, 49–64. MR0428430 10.1007/BF02079907

17.

[17] F. Schipp, W. R. Wade, P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990. 0727.42017[17] F. Schipp, W. R. Wade, P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990. 0727.42017

18.

[18] P. Simon, Verallgemeinerte Walsh-Fourierreihen, II. Acta Math. Acad. Sci. Hungar. 27 (1976), no. 3–4, 329–341.[18] P. Simon, Verallgemeinerte Walsh-Fourierreihen, II. Acta Math. Acad. Sci. Hungar. 27 (1976), no. 3–4, 329–341.

19.

[19] O. Szász, On the logarithmic means of rearranged partial sums of Fourier series, Bull. Amer. Math. Soc. 48 (1942), 705–711.[19] O. Szász, On the logarithmic means of rearranged partial sums of Fourier series, Bull. Amer. Math. Soc. 48 (1942), 705–711.

20.

[20] F. T. Wang, On the summability of Fourier series by Riesz’s logarithmic means, II, Tohoku Math. J. 40 (1935), 392–397.[20] F. T. Wang, On the summability of Fourier series by Riesz’s logarithmic means, II, Tohoku Math. J. 40 (1935), 392–397.

21.

[21] K. Yabuta, Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz’s logarithmic means, Tohoku Math. J. (2) 22 (1970), 117–129.[21] K. Yabuta, Quasi-Tauberian theorems, applied to the summability of Fourier series by Riesz’s logarithmic means, Tohoku Math. J. (2) 22 (1970), 117–129.

22.

[22] W.-S. Young, Mean convergence of generalized Walsh-Fourier series, Trans. Amer. Math. Soc. 218 (1976), 311–320. 0327.43009 10.1090/S0002-9947-1976-0394022-8[22] W.-S. Young, Mean convergence of generalized Walsh-Fourier series, Trans. Amer. Math. Soc. 218 (1976), 311–320. 0327.43009 10.1090/S0002-9947-1976-0394022-8

Citation Download Citation

György Gát and Ushangi Goginava "Norm convergence of logarithmic means on unbounded Vilenkin groups," Banach Journal of Mathematical Analysis 12(2), 422-438, (April 2018). https://doi.org/10.1215/17358787-2017-0031

Received: 28 March 2017; Accepted: 17 July 2017; Published: April 2018

Access the abstract

JOURNAL ARTICLE
17 PAGES

DOWNLOAD PDF + SAVE TO MY LIBRARY