Abstract
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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This work is supported by the Russian Science Foundation under grant no. 18-11-00055.
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To cite this work: Vinogradov O.L. “On the Constants in the Inverse Theorems for the First-Order Derivative.” Vestnik of St. Petersburg University. Mathematics. Mechanics. Astronomy, 2021, vol. 8 (66), no. 4, pp. 559–571. (In Russian.) https://doi.org/10.21638/spbu01.2021.401.
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Translated by M. Talacheva
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Vinogradov, O.L. On the Constants in Inverse Theorems for the First-Order Derivative. Vestnik St.Petersb. Univ.Math. 54, 334–344 (2021). https://doi.org/10.1134/S1063454121040208
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DOI: https://doi.org/10.1134/S1063454121040208