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Integrability of trigonometric series III

Published online by Cambridge University Press:  17 April 2009

Masako Izumi  and
Shin-Ichi Izumi
Masako Izumi
Affiliation:
Department of Mathematics, University of Sherbrooke, Sherbrooke, Quebec, Canada
Shin-Ichi Izumi
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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Abstract

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Ralph P. Boas, Jr, proved the following theorem: Let g be an odd function, integrable on (0, π) and periodic with period 2π, and its Fourier series be Σ bn sinnt. If 0 < r < 1 and bn ≥ 0 b > for all n, then t−rg(t) ∈ L(0, π) if and only if the series Σ bn/n1−r converges. Philip Heywood asked whether the conditions g(t) ∈ L(0, π) and bn ≥ 0 can be replaced by tg(t) ∈ L(0, π) and bn ≥ −A/nr or not. We prove this problem affirmatively.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 5 , Issue 2 , October 1971 , pp. 205 - 219
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Boas, R.P. Jr, “Fourier series with positive coefficients”, J. Math. Anal. Appl. 17 (1967), 463–483.CrossRefGoogle Scholar
[2]Boas, Ralph P. Jr, Integrability theorems for trigonometric transforms (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 38. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[3]Heywood, Philip, “Some theorems on trigonometric series”, Quart. J. Math. Oxford (2) 20 (1969), 465–481.CrossRefGoogle Scholar