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The Lebesgue Constants for (γ, r) Summation of Fourier Series

Published online by Cambridge University Press:  20 November 2018

Lee Lorch  and
Donald J. Newman
Lee Lorch
Affiliation:
University of Alberta Edmonton, Canada
Donald J. Newman
Affiliation:
Yeshiva University, New York, N. Y.
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The (γ, r) summation method, 0 < r < 1, is the "circle method" employed by G. H. Hardy and J. E. Littlewood. It is also known as the Taylor method. Its Lebesgue constants, say L(Tr, n), n = 1, 2, …, were studied by K. Ishiguro [1] in the notation L*(n;1-r). He noted that

1

where Im{z} denotes the imaginary part of the complex number z, and proved that

2

Here

3

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 2 , May 1963 , pp. 179 - 182
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Ishiguro, K., The Lebesgue Constants for (γ, r) summation of Fourier series, Proc. Japan Acad., vol. 36(1960), pp. 470-474.Google Scholar
2. Lorch, Lee and Newman, Donald J., On the [F, dn] summation of Fourier series, Comm. on Pure and Applied Math., vol. 15(1962), pp. 109-118.Google Scholar