Approximate recovery of periodic functions of several variables. (English. Russian original) Zbl 0635.46025

Math. USSR, Sb. 56, 245-261 (1987); translation from Mat. Sb., Nov. Ser. 128(170), No. 2, 256-268 (1985).
It is shown that for every positive integer M there exists a system of points \(\xi_ 1,...,\xi_ M\in [0,2\pi]\) and a linear continuous operator \(T_ M\), with values \(T_ M(f)\) consistent with \(f(\xi_ i)\) for \(i=1,...,M\) such that for every function \(f\in H_ p^{\bar r}\), \(r>1/p\), \(1\leq p\leq \infty\), there holds the inequality \[ \| f-T_ M(f)\| \leq C\quad M^{-r}(\log M)^{(r+1)(\nu -1)}, \] where \(\nu\) is the number of smallest coordinate of the vector \(\bar r\) and \(C>0\) is independent of f.
Reviewer: J.Musielak


46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
41A35 Approximation by operators (in particular, by integral operators)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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