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Approximation of functions with bounded mixed derivative. (Priblizhenie funktsij s ogranichennoj smeshannoj proizvodnoj). (Russian) Zbl 0625.41028
Trudy Matematicheskogo Instituta im. V. A. Steklova, Tom 178. Moskva: “Nauka”. 112 p. R. 1.40 (1986).
The main purpose of this monograph is to study the approximations of periodic functions of many variables. The first chapter is devoted to the study of Bernstein’s and Jackson-Nikol’skij’s inequalities, and to the connections between the best approximations in various metrics. The second chapter deals with approximations by means of trigonometric polynomials of the functions from \(W^ r_{q,\alpha}\) and \(H^ r_ q\) in the metric \(L_ p\), \(1<q\leq p<\infty\) and by means of linear methods. Next, in the third chapter, the widths of \(W^ r_{q,\alpha}\) and \(H^ r_ q\) in \(L_ p\), \(1\leq q\leq p<\infty\), and some extremal problems are studied. In the fourth, and closing, chapter the author presents some generalizations of the Hardy-Littlewood theorem and approximations of periodic functions of several variables by means of bilinear forms.
Reviewer: J.Albrycht

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41-02 Research exposition (monographs, survey articles) pertaining to approximations and expansions
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
42A10 Trigonometric approximation
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy