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The Gibbs Phenomenon for [S, αn] Means and [T, αn] Means

Published online by Cambridge University Press:  20 November 2018

V. Swaminathan
V. Swaminathan*
Affiliation:
Department of Mathematics, University of Kerala, Kariavattom, Trivandrum, India
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The Gibbs phenomenon of the Fourier series for different summability methods has been investigated by various authors. In this note, we study the same for the [S, αn] method of summability introduced by Meir and Sharma [3]. The corresponding result for the [T, αn] method of summability due to Powell [5] can be worked out in exactly the same way.

The elements cnk of the [S, αn] matrix are defined by the relations:

1

where 0<αn<1. The [S, αn] matrix is regular if and only if

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 2 , 01 June 1976 , pp. 209 - 211
Copyright
Copyright © Canadian Mathematical Society 1976

References

1. Ishiguro, K., Zur Gibbschen Erscheinung für das Kreisverfahren. Math. Z. 76, 288-294 (1961).Google Scholar
2. Ishiguro, K., Über das Sα-Verfahren bei Fourier-Reihen. Math. Z. 80, 4–11 (1962).10.1007/BF01162364CrossRefGoogle Scholar
3. Meir, A. and Sharma, A., A generalization of the Sa-summation method. Proc. Camb. Phil. Soc. 67, 61–66 (1970).10.1017/S0305004100057091CrossRefGoogle Scholar
4. Miracle, C. L., The Gibbs phenomenon for Taylor means and for [F, dn] means. Can. J. Math. 12, 660–673 (1960).10.4153/CJM-1960-059-2CrossRefGoogle Scholar
5. Powell, R. E., The T(rn) summability transform. J. Analyse Math. 20, 289–304 (1967).10.1007/BF02786677CrossRefGoogle Scholar