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Best approximants in modular function spaces. (English) Zbl 0718.41049

The paper is dealing with existence of elements of best approximation in modular spaces. Modular spaces are natural generalizations of \(L_ p\), \(p>0\), Orlicz, Lorentz and Köthe spaces [see W. M. Kozlowski, Modular function spaces 122 (1988; Zbl 0661.46023)]. Let \(\Sigma\) be a \(\sigma\)-algebra of subsets of the set X, \(Y\subseteq X\) and \({\mathcal B}\) a \(\sigma\)-algebra of subsets of Y such that \({\mathcal B}\subseteq \Sigma | Y\). A typical result is: If \(Y=\cup^{\infty}_{k=1}A_ k\), where \(A_ k\in {\mathcal B}\) are all atoms then \(f\in L_{\rho}(X,\Sigma)\) has a \(\rho\)-best approximant in the subspace \(L_{\rho}(Y,\Sigma)\) whenever it is bounded on each \(A_ k\). A similar result holds for the best approximation with respect to the \(\rho\)-norm \(\| \cdot \|_{\rho}\) corresponding to the semimodular \(\rho\). The cases of a countably set X and of a orthogonally additive modulus \(\rho\) are considered separately in Sections 3 and 4, respectively. The uniqueness question is also considered.

MSC:

41A80 Remainders in approximation formulas
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)

Citations:

Zbl 0661.46023
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