On the Constants in Inverse Theorems for the First-Order Derivative

The known proofs of the inverse theorems in the theory of approximation by trigonometric polynomials and entire functions of exponential type are based on S.N. Bernstein’s idea to expand the function in a series with respect to the functions of its best approximation. In this paper, a new method to...

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Bibliographic Details
Published in:Vestnik, St. Petersburg University. Mathematics Vol. 54; no. 4; pp. 334 - 344
Main Author: Vinogradov, O. L.
Format: Journal Article
Language:English
Published: Moscow Pleiades Publishing 01.10.2021
Springer Nature B.V
Subjects:
Analysis
Approximation
Constants
Derivatives
Entire functions
Functions (mathematics)
Identities
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Theorems
inverse theorems
conjugate function
ISSN:1063-4541, 1934-7855
Online Access:Get full text
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Summary:The known proofs of the inverse theorems in the theory of approximation by trigonometric polynomials and entire functions of exponential type are based on S.N. Bernstein’s idea to expand the function in a series with respect to the functions of its best approximation. In this paper, a new method to prove inverse theorems is proposed. Sufficiently simple identities are established that immediately lead to the aforementioned inverse theorems, with the constants being improved. This method can be applied to derivatives of any order—not necessarily integer—as well as (with certain modifications) to the estimates of some other functionals via their best approximations. In this paper, the case of the first-order derivative of the function itself and of its trigonometrically conjugate function is considered.
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ISSN:1063-4541
1934-7855
DOI:10.1134/S1063454121040208