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The Approximate Symmetric Integral

Published online by Cambridge University Press:  20 November 2018

D. Preiss  and
B. S. Thomson
D. Preiss
Affiliation:
Charles University, Praha 8, Czechoslovakia
B. S. Thomson
Affiliation:
Simon Fraser University, Burnaby, British Columbia
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By a symmetric integral is understood an integral obtained from some kind of symmetric derivation process. Such integrals arise most naturally in the study of trigonometric series and in particular to handle the following problem. Suppose that a trigonometric series

converges everywhere to a function À. It is known that this may occur without À being integrable in any of the more familiar senses so that the series may not be considered as a Fourier series of À; indeed Denjoy [4] has shown that if bnis a sequence of real numbers decreasing to zero but with

+00 then the function À(x) = is not Denjoy-integrable. It is natural to ask then for an integration procedure that can be applied to À in order that the series be the Fourier series of À with respect to this integral.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 3 , 01 June 1989 , pp. 508 - 555
Copyright
Copyright © Canadian Mathematical Society 1989

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