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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Regularity properties of the element of closest approximation

Author: Harold S. Shapiro
Journal: Trans. Amer. Math. Soc. 181 (1973), 127-142
MSC: Primary 42A08; Secondary 41A50
MathSciNet review: 0320606
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Abstract: Given an element $ f \epsilon {L^p}(T),1 < p < \infty $, and a closed translation invariant subspace $ S$ of $ {L^p}(T)$, we investigate the regularity (smoothness) properties of the element of $ S$ which is closest to $ f$. The regularity of this element is in general less than that of $ f$. The problem reveals a surprising connection with a hitherto unstudied class of extremal Fourier multipliers.

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Keywords: Best approximation, $ {L^p}$ modulus of continuity, regularity, metric projection, translation invariant space, Fourier multiplier
Article copyright: © Copyright 1973 American Mathematical Society