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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.
##### MSC:
 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)