Extreme Values of the Best Approximations of Linear Combinations оf Harmonic Functions
Authors:   Kamianets-Podіlskyi Ivan Ohiienko National University, Sorych Viktor , Sorych Nina ,   Kamianets-Podіlskyi Ivan Ohiienko National University
Published: January 2024

Extreme problems and their practical applications have been under the scrutiny of mathematicians since ancient times. An important step in the development of extreme problems was made by P. L. Chebyshev, who in the 50s of the 19th century laid the foundations of a section of destructive function theory – the theory of approximation. A significant role of the formation of the theory of approximation of functions was played by Carl Weierstrass’s theorem on the convergence to zero of best approximations by polynomials of a continuous function. As is well known, Weierstrass’s theorem is not constructive – it does not contained estimates of the approach speed. Thanks to the work of D. Jackson, S. N. Bernstein, Vallee-Poussin and others, such estimates began to appear in works on approximation theory. At the same time, at the first stages of the development of the theory of approximation, approximations of individual functions were studied. That beginning of a new period, a dipper study of the deviation values of functions from their approximating polynomials, dates back to the 30s and 40s of the 20th century and is associated with the names of A. M. Kolmogorov, S. M. Nikolsky, J. Favard, N. I. Achieser, M. G. Crane and B. Nagy. Thanks to their works, the main emphasis in the theory of approximations is shifted to the study of the best approximations or other approximation characteristics of functions that have certain differential-difference or smoothness properties. In particular, in 1936, J. Favard calculated the exact values of the best uniform approximations by trigonometric polynomials of order no higher than n – 1 on classes of differentiable 2π-periodic functions, whose r-th (r – natural) derivatives are in a unit sphere of the space of essentially bounded functions. The problem of obtaining exact values of the best approximations in uniform and integral metrics for various functional compacts was in sight of many prominent mathematicians of the XX century. General issues related to the study of the best approximation functional: the existence of a polynomial of the best approximation, its characteristic properties, are destribed in detail in many monographs, in particular, for example, in the book by M. P. Korneichuk [1]. In the 80s and 90s of the XX century, O. I. Stepanets (see, [2, section III]) developed a new approach to the classification of periodic functions, which allowed for a fairly fine classification of extremely wide sets of periodic functions. At the same time, the results obtained for these classes are, on the one hand, general, and on the other hand, they give a number of new, hitherto unknown results that were impossible to obtain on previously known classes. Following the approaches to the requirements of function classification, we can consider a linear combination of function classes of a more complex nature. And then the problem of finding the exact values of the upper bounds of the best joint approximations will be reduced to the problem of the best approximation of this composite class corresponding to convolutions with the composite kernel.