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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Convex functions and Fourier coefficients


Author: Hann Tzong Wang
Journal: Proc. Amer. Math. Soc. 94 (1985), 641-646
MSC: Primary 26A51; Secondary 26A24, 42A16
MathSciNet review: 792276
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Abstract: Let $ f$ be a continuous function defined on the interval $ (0,1)$. For $ n = 1,2, \ldots $ and $ 0 < s < t < 1$, denote by $ {a_n}(f;s,t),{b_n}(f;s,t)$ the $ n$th Fourier coefficients of $ f\vert(s,t)$. It is shown that the following statements are equivalent:

(i) $ f$ is strictly convex on $ (0,1)$.

(ii) $ {b_n}(f;s,t) < (2/n\pi )[f(s) - f((s + t))/2]$ for all $ n = 1,2, \ldots $ and whenever $ 0 < s < t < 1$.

(iii) $ {b_n}(f;s,t) > (2/n\pi )[f((s + t)/2) - f(t)]$ for all $ n = 1,2, \ldots $ and whenever $ 0 < s < t < 1$.

If, in addition, $ f$ is twice differentiable, then (i) and the following statement are also equivalent:

(iv) $ {a_n}(f;s,t) > 0$ for all $ n = 1,2, \ldots $ and whenever $ 0 < s < t < 1$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1985-0792276-8
PII: S 0002-9939(1985)0792276-8
Keywords: Convex functions, Fourier coefficients, integral inequality
Article copyright: © Copyright 1985 American Mathematical Society