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**Approximation of periodic functions.**
*(English)*
Zbl 0899.41001

Computational Mathematics and Analysis Series. Commack, NY: Nova Science Publishers, Inc. x, 419 p. (1993).

From the preface: “The main purpose of this monograph is to study problems of approximation of real functions of several variables. We pay the most attention to traditional questions. The most important problem of approximation theory is, for a fixed class of functions, to find a system of functions that is well adapted for approximating the functions from the class involved. The study of this question plays the dominating role in this book. For this purpose several types of widths of classes of functions are investigated. In Chapter I–III we find the orders of decrease of these quantities for various classes of functions when the dimension of the approximating subspace tends to infinity. “We confine ourselves to the consideration of classes of periodic functions. Moreover, for these classes we do not strive to formulate the results in the most general form. We pay attention to the evolution of the methods of approximation theory as we pass from functions of a single variable to functions of several variables. We attempt to embrace two of the most common and essentially different sets of functional classes: the anisotropic Sobolev and Nikolskij classes and the classes of functions with bounded mixed derivatives or difference. Chapters II and III are devoted to investigating these classes. In addition to traditional questions concerned with the estimation of the widths of these classes, results about universal systems of functions are treated in this book. “Chapter IV deals with a study of optimal cubature formulas for various classes of functions. This area has developed intensively during the last years. It turned out that the construction of optimal cubature formulas for the classes of functions with bounded mixed derivative uses deep number-theoretic methods. In this chapter, together with optimal cubature formulas for a fixed class of functions, we establish universal cubature formulas for the classes considered.” Contents: Introduction; Chapter I. Approximation of functions of one variable; Chapter II. Approximation of functions of several variables in the Sobolov and Nikolskij classes; Chapter III. Approximation of functions with bounded mixed derivative or difference; Chapter IV. Cubature formulas and recovery of periodic functions of several variables.