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Integrability of trigonometric series III
Published online by Cambridge University Press: 17 April 2009
Abstract
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.
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- Copyright © Australian Mathematical Society 1971
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