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Widths in \(L_ p\) of classes of continuous and differentiable functions and optimal reconstruction of functions and their derivatives. (English. Russian original) Zbl 0541.41020
Sov. Math., Dokl. 20, 229-233 (1979); translation from Dokl. Akad. Nauk SSSR 244, 1317-1321 (1979).
Let X be C[0,1] or \(L_ p=L(0,1)\), \(1\leq p<\infty\), and let \(E(f,F_ N)_ X\) be the best approximation of f by a subspace \(F_ N\) in X with dim \(F_ N=N\). If \(M\subset X\) and \(f\in M\) implies that -\(f\in M\), \(d_ N(M,X)\equiv \inf_{F_ N}\sup_{f\in M}E(f,F_ N)_ X\) and \[ d'\!_ N(M,X)=\inf_{F_ N}\inf_{AX\subset F_ N}\sup_{f\in M}| f-Af|, \] then \(d_ N(M,X)\leq d'\!_ N(M,X)\), where A is a linear operator from X into \(F_ N\). Some inequalities for the approximation of \(f\in H^{\omega}\) by piecewise constant functions and by interpolating polygons are proved, where \(H^{\omega}\) is the set of functions \(f\in C\) such that \(\omega(f,\delta)\leq \omega(\delta), \omega (f,\delta)\) is the modulus of continuity of f(x) and \(\omega\) (\(\delta)\) is a given modulus of continuity. The problem of optimal restoration of functions is discussed.
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)