Abstract:
In classical inverse theorems of the constructive theory of functions, structure characteristics of a function are described in terms of the rate of its approximation. Mostly, proofs of the invers theorem are based on Bernstein's idea of expanding a function in a series involving the polynomials of best approximation for this finction. In the present paper, Bernstein's approach is modified by replasing sums with integrals. It turns out that the inequalities are then deduced form identities of Frullani integrals type. The arguments are of fairly general nature, which makes it possible to obtain analogs of inverse theorems for functionals on abstract Banach (and even quasi-normed) spaces. Abstract results are used to deduce inverse theorems in specific function spaces, including weighted spaces, with specific constants.
Citation:
O. L. Vinogradov, “On constants in abstract inverse theorems of approximation theory”, Algebra i Analiz, 34:4 (2022), 22–46; St. Petersburg Math. J., 34:4 (2023), 573–589
\Bibitem{Vin22}
\by O.~L.~Vinogradov
\paper On constants in abstract inverse theorems of approximation theory
\jour Algebra i Analiz
\yr 2022
\vol 34
\issue 4
\pages 22--46
\mathnet{http://mi.mathnet.ru/aa1823}
\transl
\jour St. Petersburg Math. J.
\yr 2023
\vol 34
\issue 4
\pages 573--589
\crossref{https://doi.org/10.1090/spmj/1770}
Linking options:
https://www.mathnet.ru/eng/aa1823
https://www.mathnet.ru/eng/aa/v34/i4/p22
This publication is cited in the following 3 articles:
A. O. Leont'eva, “Bernstein Inequality for the Riesz Derivative of Order 0<α<1 of Entire Functions of Exponential Type in the Uniform Norm”, Math. Notes, 115:2 (2024), 205–214
O. L. Vinogradov, “Direct and inverse theorems of approximation theory in Lebesgue spaces with Muckenhoupt weights”, Ufa Math. J., 15:4 (2023), 42–61
O. L. Vinogradov, “Direct and inverse theorems of approximation theory in Banach function spaces”, St. Petersburg Math. J., 35:6 (2024), 907–928
Citation in format AMSBIB
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