Regularity properties of the element of closest approximation

Shapiro, Harold S.

doi:10.1090/S0002-9947-1973-0320606-6

Regularity properties of the element of closest approximation
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by Harold S. Shapiro PDF

Trans. Amer. Math. Soc. 181 (1973), 127-142 Request permission

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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