A Haar-type theory of best 𝐿₁-approximation with constraints

Kroó, András; Schmidt, Darrell

doi:10.1090/S0002-9947-1992-1062190-X

A Haar-type theory of best $L_ 1$-approximation with constraints
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by András Kroó and Darrell Schmidt

Trans. Amer. Math. Soc. 331 (1992), 301-319

DOI: https://doi.org/10.1090/S0002-9947-1992-1062190-X

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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 $\|f\|_w = \int _a^b {|f|w dx}$ 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 $\|\;\cdot \;\|_w$-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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