A new classification of periodic functions and their approximation.

*(English)*Zbl 0744.42002
Approximation and function spaces, Proc. 27th Semest., Warsaw/Pol. 1986, Banach Cent. Publ. 22, 437-447 (1989).

[For the entire collection see Zbl 0681.00013.]

In 1983 the author proposed to study some classes of functions (see below). He and his students have obtained the analogues of all the classical approximation theoretical results for these new classes and the present paper is a survey of their achievements.

The classes are defined in terms of multipliers and translations: Let \(f\in L_{2\pi}\) with Fourier coefficients \(a_ k(f)\), \(b_ k(f)\), \(\psi\) a function on the positive integers and \(\beta\) a real number. Suppose that the series \[ \sum^ \infty_{k=1}{1\over\psi(k)} (a_ k(f)\cos(kx+\beta\pi/2)+b_ k(f)\cos(kx+\beta\pi/2)) \] is the Fourier series of some \(f^ \psi_ \beta\), which is called the \((\psi,\beta)\)- derivative of \(f\). Now the smoothness classes in question are defined via this derivative in the usual manner.

The questions that are discussed are as follows: representation of deviations for linear means of Fourier series; approximation by Fourier sums in \(C\) and \(L_ 1\); best trigonometric approximation.

In 1983 the author proposed to study some classes of functions (see below). He and his students have obtained the analogues of all the classical approximation theoretical results for these new classes and the present paper is a survey of their achievements.

The classes are defined in terms of multipliers and translations: Let \(f\in L_{2\pi}\) with Fourier coefficients \(a_ k(f)\), \(b_ k(f)\), \(\psi\) a function on the positive integers and \(\beta\) a real number. Suppose that the series \[ \sum^ \infty_{k=1}{1\over\psi(k)} (a_ k(f)\cos(kx+\beta\pi/2)+b_ k(f)\cos(kx+\beta\pi/2)) \] is the Fourier series of some \(f^ \psi_ \beta\), which is called the \((\psi,\beta)\)- derivative of \(f\). Now the smoothness classes in question are defined via this derivative in the usual manner.

The questions that are discussed are as follows: representation of deviations for linear means of Fourier series; approximation by Fourier sums in \(C\) and \(L_ 1\); best trigonometric approximation.

Reviewer: V.Totik (Szeged)