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Transactions of the American Mathematical Society

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



A Haar-type theory of best $ L\sb 1$-approximation with constraints

Authors: András Kroó and Darrell Schmidt
Journal: Trans. Amer. Math. Soc. 331 (1992), 301-319
MSC: Primary 41A29; Secondary 41A52
MathSciNet review: 1062190
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Abstract: A general setting for constrained $ {L^1}$-approximation is presented. Let $ {U_n}$ be a finite dimensional subspace of $ C[a,b]$ and $ L$ be a linear operator from $ {U_n}$ to $ {C^r}(K)\;(r = 0,1)$ where $ K$ is a finite union of disjoint, closed, bounded intervals. For $ \upsilon,u \in {C^r}(K)$ with $ \upsilon < u$, the approximating set is $ {\tilde U_n}(\upsilon,u) = \{ p \in {U_n}:\upsilon \leq Lp \leq u\;{\text{on}}\;K\} $ and the norm is $ \Vert f\Vert _w = \int_a^b {\vert f\vert w\,dx} $ where $ w$ a positive continuous function on $ [a,b]$. We obtain necessary and sufficient conditions for $ {\tilde U_n}(\upsilon,u)$ to admit unique best $ \Vert\;\cdot\;\Vert _w$-approximations to all $ f \in C[a,b]$ for all positive continuous $ w$ and all $ \upsilon,u \in {C^r}(K)\;(r = 0,1)$ satisfying a nonempty interior condition. These results are applied to several $ {L^1}$-approximation problems including polynomial and spline approximation with restricted derivatives, lacunary polynomial approximation with restricted derivatives, and others.

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Keywords: $ {L_1}$-approximation with constraints, $ {L^0}-A$-spaces, $ {L^1}-A$-spaces
Article copyright: © Copyright 1992 American Mathematical Society