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Approximation of classes of periodic functions of several variables by nuclear operators. (English. Russian original) Zbl 0723.41018
Math. Notes 47, No. 3, 248-254 (1990); translation from Mat. Zametki 47, No. 3, 32-41 (1990).
V. N. Temyakov [Dokl. Akad. Nauk SSSR, 267, 314-317 (1982; Zbl 0524.41013)] proved that for \(1<p\leq q<\infty\) the Fourier operator gives the best approximation of the classes \(\tilde H^ r_ p\) and \(\tilde W^ r_ p\) in the space \(\tilde L_ q\) among all operators of orthogonal projection, and also among a class of operators, wider than that of orthogonal projections, vid. among the class of certain linear operators. In the present paper the author gives other conditions on linear operators, under which the Fourier operator gives the order- optimal approximation. He also gives a theorem on the approximation of the Besov class \(B^ r_{p,\theta}\) in the space \(\tilde L_ q(1<q<p<2)\) by nuclear operators and so determines the ortho-projection widths of the classes \(B^ r_{p,\theta}\) and \(\tilde H^ r_ p\).

41A35 Approximation by operators (in particular, by integral operators)
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