Methods of Approximation Theory

This monograph can be regarded as a result of the activity of many mathematicians of the 20th century in the field of classical Fourier series and the theory of approximation of periodic functions, beginning with H. Lebesgue, D. Jackson, and S. N. Bernstein. The key point of the monograph is the classification of periodic functions introduced by the author and developed methods that enable one to solve, within the framework of a common approach, traditional problems of approximation theory for large collections of periodic functions, including, as particular cases, the well-known Weyl - Nagy and Sobolev classes as well as classes of functions defined by convolutions with arbitrary summable kernels. The developed methods enable one to solve problems of approximation theory not only in the periodic case but also in the case where objects of approximation are functions locally integrable on the entire axis and functions defined by Cauchy-type integrals in domains of the complex plane bounded by rectifiable Jordan curves. The main results are fairly complete and are presented in the form of either exact or asymptotically exact equalities. Most results of the monograph represent the latest achievements, which have not yet been published in existing monographs. First of all, this refers to problems of regularity and saturation of linear processes of summation and the convergence rate of Fourier series in different metrics, approximation by interpolation polynomials, approximation of locally integrable functions by entire functions of exponential type, and approximation of Cauchy integrals in Jordan domains by Faber polynomials. The monograph also contains entirely new results aimed at the construction of approximation theory in general linear spaces. The present monograph is, in many respects, a store of knowledge accumulated in approximation theory by the beginning of the third millennium and serving for its further development.