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A Remark on the Gibbs Phenomenon and Lebesgue Constants for a Summability Method of Melikov

Published online by Cambridge University Press:  20 November 2018

Fred Ustina
Fred Ustina*
Affiliation:
University of Alberta Edmonton
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Let u = ∑uk be a given series and let Melikov [4] has defined the n-th σ- transform of u by

where ε and θ are assumed to be non-negative. This is easily shown to be equivalent to

The method is a generalization of a method used by Kaufman [l], and of another one used by Melikov [5]. It reduces to the (n-l)th (C;l) mean when θ = 0 and ε = 0, and to the n-th (C;l) mean when θ = 1 and ε = 0.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 2 , June 1968 , pp. 301 - 303
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Kaufman, B. L., Proceedings, Chkalovsky Pedagogical Institute, 11 (1957) 243-248.Google Scholar
2. Fejér, L., Lebesgueschen Konstanten und divergente Fourierreihen. J. Reine Angew. Math., 138(1910)22-53.Google Scholar
3. Lorch, L., The principal term in the asymptotic expansion of the Lebesgue constants. Amer. Math. Monthly, 61 (1954) 245-249.Google Scholar
4. Melikov, H.H., A class of summation methods for divergent series. (Russian). Proceedings of the Annual Scientific Conference, Kabardino-Balkarsk State University, 24 (1965) 183-188.Google Scholar
5. Melikov, H.H., Proceedings, Siberian-Ossetic State University, Math, and Natural Science Series, 26 (1964).Google Scholar