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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11548)

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.

Keywords

Functions with exactly or approximately given Fourier transform Kolmogorov inequality Optimal differentiation method

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Authors and Affiliations

• Vitalii V. Arestov
• 1
• 2
1. 1.Ural Federal UniversityEkaterinburgRussia
2. 2.Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of SciencesEkaterinburgRussia