Екстремальні задачі для інтегралів від невід'ємних функцій

Abstract

We investigate the best approximations eσ(f) of the integrals of functions of the spaces Lp(A,dμ) by the integrals of rank σ. We find the exact values of the upper bounds of these approximations in the case where the function f is a product of two functions, one of which is fixed and the other one changes in some subset of Lp(A,dμ). We also obtain, in terms of approximations eσ(•), the necessary and sufficient conditions for a function of the space Lp(A,dμ) to belong to the space Ls(A,dμ), 0 < p, s < ∞.