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

Abstract

Let $Y^n,$ , 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 $L_\infty$. In the space $Y^n,$ consider the class $\mathcal {Q}^n$ of functions, for which the $L_\infty$-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, , by bounded operators on the class $\mathcal {Q}^n$ and the optimal calculation of the differentiation operator $D^k$ on functions from the class $\mathcal {Q}^n$ 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 $L_\infty$-norm of its derivative of order n.

This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).