Inequalities of the Kolmogorov type for norms of Riesz derivatives of multivariate functions and some of their applications

摘要

We obtained new exact inequalities that estimate the L_∞-norm of the Riesz derivative D~α f of a function f defined on R~m in terms of the uniform norm of the function itself and the L_s-norm of the function acted by the Laplace operator. On a class of functions f such that ||Δf||_s ≤ 1, we solved the problem of approximation of an unbounded operator D~α by bounded ones and the problem of optimal recovery of the operator D~α on elements of this class given with known error.