Best Approximation of a Differentiation Operator on the Set of Smooth Functions with Exactly or Approximately Given Fourier Transform

摘要

Let Y~n, n ≥ 2, be the set of continuous bounded functions on the numerical axis with the following two properties: (1) the Fourier transform of a function is a function of bounded variation on the axis (in particular, a summable function); (2) a function is n — 1 times continuously differentiable, its derivative of order n — 1 is locally absolutely continuous, and the nth order derivative is bounded, more exactly, belongs to the space L_∞ In the space Y~n, consider the class Q~n of functions, for which the Loo-norm of the nth order derivative is bounded by a constant, for example, by 1. The following two approximation problems are discussed: the best approximation of the differentiation operator D~k of order k, 1 ≤ k < n, by bounded operators on the class Q~n and the optimal calculation of the differentiation operator D~k on functions from the class Q~n under the assumption that their Fourier transform is given with a known error in the space of functions of bounded variation, in particular, in the space L of functions summable on the axis. In interrelation with these two problems, we discuss the exact Kolmogorov type inequality in the space Y~n between the uniform norm of the Kth order derivative of a function, the variation of the Fourier transform of the function, and the L_∞-norm of its derivative of order n.