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On Banach Limit of Fourier Series and Conjugate Series I

Published online by Cambridge University Press:  20 November 2018

S. Dayal
S. Dayal*
Affiliation:
Aligarh University
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Let (xn) be a sequence of real numbers. (xn) corresponds to a number Lim xn called the Banach limit of (xn) satisfying the following conditions:

  1. (1) Lim (axn + byn) = a Lim xn + b Lim yn

  2. (2) If xn ≥ 0 for every n, then Lim xn ≥ 0

  3. (3) Lim xn+1 = Lim xn

  4. (4) If xn = 1 for every n, then Lim xn = 1

The existence of such limits is proved by Banach [1].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 2 , June 1968 , pp. 255 - 262
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Banach, S., Théorie des opérations linéaires, Warzawa (1932).Google Scholar
2. Lorentz, G.G., A contribution to the theory of divergent series. Acta Mathematica 80 (1948) 167-190.Google Scholar
3. Zygmund, A., Trigonometric series. Cambridge University Press, New York, 1959.Google Scholar