Classification and Approximation of Periodic Functions

The chapters are split into sections, which, in turn, are split into subsections enumerated by two numbers: the first stands for the number of the section while the second for the number ofthe subsection itself. The same numeration is used for all kinds of statements and formulas. If we refer to statements or formulas in other chapters, we use triple numeration where the first number stands for the chapter and the other two have the same sense. The results presented in this book were discussed on the seminars at the Institute of Mathematics of Ukrainian Academy ofSciences, at the Steklov Mathematical Institute of the Academy of Sciences of the USSR, at Moscow and Tbilisi State Universities. I am deeply grateful to the heads of these seminars Professors V. K. Dzyadyk, N. P. Kor­ neichuk, S. B. Stechkin, P. L. U1yanov, and L. V. Zhizhiashvili as well as to the mem­ bers ofthese seminars that took an active part in the discussions. In TRODUCTIon It is well known for many years that every 21t -periodic summable function f(x) can be associated in a one-to-one manner with its Fourier series (1. 1) Slfl where I It = - f f(t)cosktdt 1t -It and I It - f f(t)sinktdt. 1t -It Therefore, if for approximation of a given function f(·), it is necessary to construct a sequence ofpolynomials Pn (.