For citation:
Sadekova E. H. On the approximation of bounded functions by trigonometric polynomials in Hausdorff metric. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2023, vol. 23, iss. 2, pp. 169-182. DOI: 10.18500/1816-9791-2023-23-2-169-182, EDN: JKUQAS
On the approximation of bounded functions by trigonometric polynomials in Hausdorff metric
The article discusses a method for constructing a spline function to obtain estimates that are exact in order to approximate bounded functions by trigonometric polynomials in the Hausdorff metric. The introduction provides a brief history of approximation of continuous and bounded functions in the uniform metric and the Hausdorff metric. Section 1 contains the main definitions, necessary facts, and formulates the main result. An estimate for the indicated approximations is obtained from Jackson's inequality for uniform approximations. In section 2 auxiliary statements are proved. So, for an arbitrary $2\pi$-periodic bounded function, a spline function is constructed. Then, estimates are obtained for the best approximation, variation, and modulus of continuity of a given spline function. Section 3 contains evidence of the main results and final comments.
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