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, 0In 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
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
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
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.

The main part of the project was carried out with the research group from the University of Lübeck. Jointly with the research group from the Centre de Recerca Matemàtica, Barcelona, we solved a series of problems concerning inequalities for moduli of smoothness. Jointly with the research group from the Bar-Ilan University, we obtained new results concerning Fourier multipliers and families of multiplier operators.

1. Simultaneous approximation of functions and their derivatives. We showed that if a function belongs to the Sobolev space W_1^r and its derivative belongs to the Besov space B_(p,p)^(1/p-1), then this function and its derivative of order r can be approximated simultaneously by the corresponding methods in Lp, 0
2. New inequalities for moduli of smoothness. A new type of direct inequalities in which the modulus of smoothness of a function is estimated from above via a certain integral with the modulus of smoothness of the derivative of the considered function was established. The obtained results were used to derive a description of a class of functions with an optimal rate of decay of moduli of smoothness in Lp, 0
3. Fourier multipliers and families of multiplier operators. The new Gagliardo-Nirenberg-type inequalities for the weighted Besov spaces with various parameters were obtained. As a corollary, the authors derived new sufficient conditions ensuring that the Fourier transform of a given function belongs to the spaces Lp for 0The results of the project were presented at important international conferences and seminars and disseminated in top mathematical journals (5 published and 2 accepted papers) and in open access repositories such as arXiv and ReseachGate.

The research undertaken significantly advanced the state of the art. Thus, in the Banach spaces, the problems related to the simultaneous approximation of functions and their derivatives have been intensively studied. At the same time, little was known about such approximation in Lp, 0The successful solution of the problems proposed in the project led to the development of previously known and new methods and tools in approximation theory and Fourier analysis. In particular, the results regarding the simultaneous approximation of functions and their derivatives provide new tools for measuring the smoothness of functions in non-linear approximations and can be used to study approximate solutions of certain classes of differential equations. The sharp Ulyanov-type inequalities and inequalities for moduli of smoothness of functions and their derivatives yield up-to-date methods for proving new embedding theorems for quasi-normed Sobolev, Lipschitz, and Besov spaces. The results on Fourier multipliers provide powerful tools for investigation in approximation theory, harmonic analysis, the theory of function spaces, and partial differential equations.

Periodic Reporting for period 1 - AFFMA (Approximation of Functions and Fourier Multipliers and their applications)

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