Some problems of approximation of periodic functions by trigonometric polynomials in $L_2$

Abstract

The paper is consists from two parts. In first part summarizes the
review of findings on best approximation of periodic functions by
trigonometric polynomials in Hilbert space $$L_{2}:=L_{2}[0,2\pi].$$
The sharp inequalities between the best approximation and averaged
with given weights modulus of continuity of mth order values rth
derivatives of functions and analogues for some modified modulus of
continuity presented.

In second part, some new sharp Jackson-Stechkin type inequalities
for characteristics of smoothness studied by K. V. Runovski and more
detail by S. B. Vakarchuk and V. I. Zabutnaya are proposed. The sharp
result on joint approximation of function and successive derivatives
for some classes of functions defined by modulus of smoothness
obtained.