Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

Abstract

For the classB _p ^ρ , 0 ≤ ρ < 1, 1≤p ≤ ∞, of 2π-periodic functions of the form f(t)=u(ρ,t), whereu (ρ,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K_ρ(t) of the convolution f= K_ρ *g, ∥g∥_ρ≤l, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.