Generalized Hankel shifts and exact Jackson–Stechkin inequalities in L2

Abstract

In this paper, we have solved several extremal problems of the best mean-square approximation of functions f on the semiaxis with a power-law weight. In the Hilbert space L^2 with a power-law weight t^2α+1 we obtain Jackson–Stechkin type inequalities between the value of the E_σ(f)-best approximation of a function f(t) by partial Hankel integrals of an order not higher than σ over the Bessel functions of the first kind and the k-th order generalized modulus of smoothnes ω_k(B^r f, t), where B is a second–order differential operator.