Taiwanese Journal of Mathematics

A TAUBERIAN THEOREM FOR UNIFORMLY WEAKLY CONVERGENCE AND ITS APPLICATION TO FOURIER SERIES

Chang-Pao Chen and Meng-Kuang Kuo

Full-text: Open access

Abstract

In 1995, S. Mercourakis introduced the concept of uniformly weakly convergent sequences and characterized such sequences as those with the property that any of its subsequences is Ces`aro-summable. In this paper, we present a Tauberian theorem for such kind of convergence. As a consequence, we prove that the uniformly pointwise convergence and the uniform convergence of a sequence of complex-valued functions coincide under a suitable Tauberian condition. This result affirmatively answers a question raised by S. Mercourakis concerning the Fourier series of a continuous function on the circle group T. In this paper, a result of Banach type is also established for uniformly weakly convergent sequences. Our result generalizes the work of Mercourakis.

Article information

Source
Taiwanese J. Math., Volume 12, Number 5 (2008), 1061-1066.

Dates
First available in Project Euclid: 20 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500574247

Digital Object Identifier
doi:10.11650/twjm/1500574247

Mathematical Reviews number (MathSciNet)
MR2431879

Zentralblatt MATH identifier
1172.40002

Subjects
Primary: 40A30: Convergence and divergence of series and sequences of functions 40E05: Tauberian theorems, general 40G05: Cesàro, Euler, Nörlund and Hausdorff methods

Keywords
uniformly weakly convergence Tauberian conditions uniform convergence of Fourier series

Citation

Chen, Chang-Pao; Kuo, Meng-Kuang. A TAUBERIAN THEOREM FOR UNIFORMLY WEAKLY CONVERGENCE AND ITS APPLICATION TO FOURIER SERIES. Taiwanese J. Math. 12 (2008), no. 5, 1061--1066. doi:10.11650/twjm/1500574247. https://projecteuclid.org/euclid.twjm/1500574247


Export citation

References

  • [1.] N. K. Bary, A Treatise on Trigonometric Series, Vols. I & II. Pergamon Press, New York, 1964.
  • [2.] Chang-Pao Chen and Jui-Ming Hsu, Tauberian theorems for weighted means of double sequences, Anal. Math., 26 (2000), 243-262.
  • [3.] J. Diestel, Sequences and Series in Banach Spaces, Graduate Texts in Math. 92, Springer-Verlag, 1984.
  • [4.] N. Dunford and J. T. Schwartz, Linear Operators, Part I: General Theory, Interscience, New York, 1958.
  • [5.] G. H. Hardy, Divergent Series, Oxford University Press, New York, 1949.
  • [6.] S. Mercourakis, On Cesàro summable sequences of continuous functions, Mathematika 42(1995), no. 1, 87-104.
  • [7.] W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976.
  • [8.] E. C. Titchmarsh, The Theory of Functions, 2nd ed., Oxford University Press, London, 1939; Russian transl. Nauka Moscow, 1980.
  • [9.] P. L. Ul'yanov, The metric theory of functions. (Russian) English transl in Proc. Steklov Inst. Math. 1990, No. 1, 199-244. Probability theory, function theory, mechanics (Russian). Trudy Mat. Inst. Steklov., 182 (1988), 180-223.
  • [10.] K. Yosida, Functional Analysis, 2nd ed., Springer, Berlin-Heidelberg-New York, 1968.
  • [11.] A. Zygmund, Trigonometric Series, Vols. I & II combined, 3rd ed., Cambridge University Press, New York, 2002.