Periodic Reporting for period 1 - AFFMA (Approximation of Functions and Fourier Multipliers and their applications)
Période du rapport: 2016-11-01 au 2018-10-31
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.
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.