Open AccessApproximation by entire functions on a countable set of continua. The inverse theorem
Author(s) -
Olga V. Silvanovich,
N. A. Shirokov
Publication year - 2021
Publication title -
vestnik sankt-peterburgskogo universiteta. matematika. mehanika. astronomiâ/vestnik sankt-peterburgskogo universiteta. seriâ 1, matematika, mehanika, astronomiâ
Language(s) - English
Resource type - Journals
eISSN - 2587-5884
pISSN - 1025-3106
DOI - 10.21638/spbu01.2021.405
Subject(s) - mathematics , countable set , inverse , equioscillation theorem , smoothness , approximation theory , minimax approximation algorithm , function approximation , discrete mathematics , disjoint sets , pure mathematics , mathematical analysis , orthogonal polynomials , classical orthogonal polynomials , gegenbauer polynomials , geometry , machine learning , artificial neural network , computer science
The approximation theory contains many statements where the rate of approximation of a function by polynomials, rational functions and so on is measured with the help of a scale. The statements where some points on the relevant scale are associated with the smoothness of the approximated function are called direct theorems of the theory of approximation. The statements where the smoothness of the approximated function is derived from the points on the scale of approximation by polynomials, rational functions etc., are called inverse theorems of the theory of approximation. The class of functions is constructively discribed in terms of the rate of approximation by polynomials, rational functions etc., if the direct theorems correspond to the inverse theorems, i. e. the smoothness of the approximated function and the points on the scale of approximation have one-to-one correspondence for the class under consideration. The authors have stated earlier the direct theorem concerning approximation by entire functions of exponential type. We considered a set of functions defined on the countable set of mutually disjoint continua and found the rate of their approximation by those entire functions. The present paper contains the inverse theorem to the menthioned above direct one.