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

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On the Constants in Inverse Theorems for the First-Order Derivative. / Vinogradov, O. L.

In: Vestnik St. Petersburg University: Mathematics, Vol. 54, No. 4, 10.2021, p. 334-344.

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Vinogradov, O. L. / On the Constants in Inverse Theorems for the First-Order Derivative. In: Vestnik St. Petersburg University: Mathematics. 2021 ; Vol. 54, No. 4. pp. 334-344.

BibTeX

@article{54abc38174ce4006ad133692aba360dd,

title = "On the Constants in Inverse Theorems for the First-Order Derivative",

abstract = "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{\textquoteright}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.",

keywords = "conjugate function, inverse theorems",

author = "Vinogradov, {O. L.}",

note = "Publisher Copyright: {\textcopyright} 2021, Pleiades Publishing, Ltd.",

year = "2021",

month = oct,

doi = "10.1134/S1063454121040208",

language = "English",

volume = "54",

pages = "334--344",

journal = "Vestnik St. Petersburg University: Mathematics",

issn = "1063-4541",

publisher = "Pleiades Publishing",

number = "4",

}

RIS

TY - JOUR

T1 - On the Constants in Inverse Theorems for the First-Order Derivative

AU - Vinogradov, O. L.

PY - 2021/10

Y1 - 2021/10

N2 - 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.

AB - 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.

KW - conjugate function

KW - inverse theorems

UR - http://www.scopus.com/inward/record.url?scp=85121433969&partnerID=8YFLogxK

U2 - 10.1134/S1063454121040208

DO - 10.1134/S1063454121040208

M3 - Article

AN - SCOPUS:85121433969

VL - 54

SP - 334

EP - 344

JO - Vestnik St. Petersburg University: Mathematics

JF - Vestnik St. Petersburg University: Mathematics

SN - 1063-4541

IS - 4

ER -

ID: 101356464