Open AccessOn the constants in the inverse theorems for the first derivative
Author(s) -
O. L. Vinogradov,
AUTHOR_ID
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/spbu01.2021.401
Subject(s) - mathematics , inverse , inverse trigonometric functions , inverse function , exponential function , simple (philosophy) , trigonometric functions , proofs of trigonometric identities , derivative (finance) , function (biology) , constant (computer programming) , mathematical proof , pure mathematics , mathematical analysis , polynomial , computer science , philosophy , geometry , epistemology , evolutionary biology , linear interpolation , economics , financial economics , bicubic interpolation , biology , programming language
The known proofs of the inverse theorems of the theory of approximation by trigonometric polynomials and by functions of exponential type are based on the idea of S. N. Bernstein to expand a function in a series containing its functions of best approximation. In this paper, a new method to establish the inverse theorems is introduced. We establish simple identities that immediately imply the inverse theorems mentioned and, moreover, with better constants. This method can be applied to derivatives of arbitrary order (not necessarily an integer one) and (with certain modifications) to estimates of some other functionals in terms of best approximations. In this paper, the case of the first derivative of a function itself and of its trigonometrically conjugate is considered.