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The Gibbs Phenomenon and Lebesgue Constants for Regular Sonnenschein Matrices

Published online by Cambridge University Press:  20 November 2018

W. T. Sledd
W. T. Sledd*
Affiliation:
University of Kentucky, Michigan State University
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If ψ(x) is a real-valued function which has a jump discontinuity at x = ε and otherwise satisfies the Dirichlet conditions in a neighbourhood of x = ε then {sn(x)} the sequence of partial sums of the Fourier series for ψ(x) cannot converge uniformly at x = ε. Moreover, it can be shown that given τ in [ — π, π] then there is a sequence {tn} such that tn → ε and

This behaviour of {sn(x)} is called the Gibbs phenomenon. If {σn(x)} is the transform of {sn(x)} by a summability method T, and if {σn(x)} also has the property described then we say that T preserves the Gibbs phenomenon.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 14 , 1962 , pp. 723 - 728
Copyright
Copyright © Canadian Mathematical Society 1962

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