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On best approximation of infinitely differentiable functions. (English. Russian original) Zbl 0764.41025

Sib. Math. J. 32, No. 5, 733-749 (1991); translation from Sib. Mat. Zh. 32, No. 5(189), 12-28 (1991).
The characteristics of the best approximation \(E(f,B,F)=\inf_{g\in B} \| f-g\|_ F\) of the element \(f\) (an infinitely differentiable function) from the normed space \(F\) by elements of the subspace \(B\subset F\) are studied. The aim is to describe constructively the class of elements \(f\) for which \(\varlimsup_{k\to\infty}(\| A_ k f\|_ F)^{1/\varphi(k)}\leq e\), where \(\{A_ k\}_{k=0}^ \infty\) is a family of linear operators and \(\{\varphi(k)\}_{k=0}^ \infty\) is a given sequence of numbers. Conditions are determined which are imposed on the normed space \(F\), the sequence \(\{\varphi(k)\}_{k=0}^ \infty\), the family of approximating subspaces \(\{B_ n\}_{n=1}^ \infty\) and operators \(\{A_ k\}_{k=0}^ \infty\) to obtain the characterization of the above class in the form \[ \varlimsup_{n\to\infty} (E(f,B_ n,F))^{1/\varphi^*(\gamma \ln (n+1))}\leq e, \] where \(\varphi^*\) is the Legendre-Young transformation of the sequence \(\{\varphi(k)\}_{k=0}^ \infty\), and \(\gamma\) a constant. As applications approximations by trigonometric and algebraic polynomials and exponential integer type functions are described.
Reviewer: V.Burjan (Praha)

MSC:

41A50 Best approximation, Chebyshev systems
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