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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On an inequality of Osgood, Phillips and Sarnak

Author: Harold Widom
Journal: Proc. Amer. Math. Soc. 102 (1988), 773-774
MSC: Primary 42A05
MathSciNet review: 929019
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Abstract: In their work [4, 5] on isospectral families for the Laplacian Osgood, Phillips and Sarnak needed and proved the following inequality: For any real-valued function $ \phi $ belonging to the Sobolev space $ {H_{1/2}}( = {W^{1/2,2}})$ of the unit circle and satisfying the side condition $ \int {{e^\phi }{e^{i\theta }}d\theta = 0} $ one has

$\displaystyle \log \frac{1} {{2\pi }}\int {{e^\phi }d\theta - \frac{1} {{2\pi }... ...a \leq \frac{1} {2}\sum\limits_{k = 1}^\infty {k\vert\hat \phi } (k){\vert^2}} $

where $ \hat \phi (k)$ is the $ k$th Fourier coefficient of $ \phi $. Without the side condition the factor $ \frac{1} {2}$ does not appear on the right side and the inequality is the first Lebedev-Milin inequality [1, §5.1]. The purpose of this note is to show that these are the first two of a series of inequalities which follow quickly from some theorems of G. Szegö on Toeplitz determinants.

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Article copyright: © Copyright 1988 American Mathematical Society