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International Theoretical Science Journal
Volume 62 >>> № 2 MARCH — APRIL 2026
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DOI 10.34229/KCA2522-9664.26.2.14
UDC 519.8

D.V. Shapovalov
Lesya Ukrainka Volyn National University, Lutsk, Ukraine,
 shapovalovdv@ukr.net

Yu.I. Kharkevich
Lesya Ukrainka Volyn National University, Lutsk, Ukraine,
 kharkevich.juriy@gmail.com


ON INTEGRAL REPRESENTATIONS FOR THE QUANTITIES
OF APPROXIMATIONOF FUNCTIONS OF THE LIPSHITZ CLASS
BY OPERATORS OF THE JACOBI–POISSON TYPE

Abstract. Jacobi–Poisson type operators as a method of summing Fourier series constructed using orthogonal polynomials play an important role in mathematical modeling problems and optimal solution theory. The work is devoted to the issue of approximation of functions, which are given on the segment [-1; 1] and satisfy the Lipschitz condition on it, by their Jacobi–Poisson type operators constructed using a system of orthogonal Jacobi polynomials. In particular, at each point x of the segment [-1; 1], we have established integral representations for the exact upper bounds of the deviations of Jacobi–Poisson type operators from functions of the class Lip[-1; 1] α for all 0<α ≤1. Solving many problems in both the theory of function approximation and systems analysis ultimately boils down to studying certain integral representations of the corresponding quantities. For this purpose, integral representations of the exact upper bounds of the deviations of Jacobi–Poisson type operators from functions of the Lipshitz class are established in the work.

Keywords: optimal solution theory, Jacobi–Poisson type operators, Lipschitz class, integral representation.


full text

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