AFFMA · Approximation of Functions and Fourier Multipliers and their applications

Approximation of Functions and Fourier Multipliers and their applications

The most significant problem in approximation theory and Fourier analysis is the study of relationships between the “smoothness” of a function and the possibility to approximate or to represent it by a combination of “simple” functions (e.g. polynomials, rational functions, splines and others). This conception plays an increasingly important role in mathematics as well as in many branches of applied sciences and engineering. The classical approximation theory is devoted to problems of approximation of functions which are continuous or at least integrable. Therefore, the scale of the spaces Lp with 1≤p≤∞, Fourier series, convolution operators and linear continuous functionals have traditionally been used for measuring the errors of approximation and the smoothness of functions. In recent decades, it has appeared a need to expand the classical approximation theory to the spaces Lp, 0<p<1. Let us emphasize that these spaces are “pathological in nature”. Thus, they are no Banach spaces; the only nonempty convex open set in Lp, 0<p<1, is the entire space; there are no nonzero continuous linear functionals on Lp. Consequently, Fourier series and the classical approximation methods are not applicable. For this reason, in Lp, 0<p<1, even the analogs of some classical results have not been obtained yet. In light of this, the goal of the research in the project AFFMA was to study a series of open problems related to the approximation and smoothness in Lp, 0<p<1. Namely, we considered the following three topics: 1. Simultaneous approximation of functions and their derivatives. The main objectives of this part of the project were the following: to investigate classes of functions and different methods of approximation for which the problem of simultaneous approximation in Lp, 0<p<1, is solvable; to obtain estimates for the errors of the best approximation of functions and their derivatives for particular methods of approximation in Lp for 0<p<1. 2. New inequalities for moduli of smoothness. The first objective of this part was to obtain new inequalities for moduli of smoothness of functions and their derivatives, namely the direct inequalities (upper estimates of the modulus of smoothness of a function via the modulus of smoothness of the derivatives of this function) and the corresponding inverse inequalities in the spaces Lp for 0<p<1. The second objective was to study inequalities of different metrics for moduli of smoothness, namely the so-called Ulyanov-type inequalities. 3. Fourier multipliers and families of multiplier operators. The objective of this part of the project was to obtain sufficient conditions of the boundedness for Fourier multipliers and families of multiplier operators in terms of the simultaneous behaviour of a function and its derivatives in different weighted function classes.

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