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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| Vydáno v: | Vestnik, St. Petersburg University. Mathematics Ročník 54; číslo 4; s. 334 - 344 |
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| Jazyk: | angličtina |
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Moscow
Pleiades Publishing
01.10.2021
Springer Nature B.V |
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| ISSN: | 1063-4541, 1934-7855 |
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| 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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| AbstractList | 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. |
| Author | Vinogradov, O. L. |
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| Cites_doi | 10.1016/B978-0-08-009929-3.50008-7 10.1090/S1061-0022-06-00922-8 10.1090/S0002-9904-1968-11980-9 10.1515/9781400883882 |
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| Copyright | Pleiades Publishing, Ltd. 2021. ISSN 1063-4541, Vestnik St. Petersburg University, Mathematics, 2021, Vol. 54, No. 4, pp. 334–344. © Pleiades Publishing, Ltd., 2021. Russian Text © The Author(s), 2021, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2021, Vol. 8, No. 4, pp. 559–571. |
| Copyright_xml | – notice: Pleiades Publishing, Ltd. 2021. ISSN 1063-4541, Vestnik St. Petersburg University, Mathematics, 2021, Vol. 54, No. 4, pp. 334–344. © Pleiades Publishing, Ltd., 2021. Russian Text © The Author(s), 2021, published in Vestnik Sankt-Peterburgskogo Universiteta: Matematika, Mekhanika, Astronomiya, 2021, Vol. 8, No. 4, pp. 559–571. |
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| References | SterlinM. D.Estimates of constants in inverse theorems of the constructive theory of functionsDokl. Akad. Nauk SSSR197320912961298336196 AkhiezerN. I.Lectures on Approximation Theory1965MoscowNauka BernsteinS. N.Collected Works1952MoscowAcad. Nauk. SSSR ZhukV. V.Approximation of Periodic Functions1982LeningradLeningrad Univ. Press0521.42003 ShapiroH. S.Some Tauberian theorems with applications to approximation theoryBull. Am. Math. Soc.19687450050422507410.1090/S0002-9904-1968-11980-9 BernsteinS. N.Collected Works1954MoscowAcad. Nauk. SSSR DzyadykV. K.Introduction into the Theory of Uniform Approximation of Functions by Polynomials1977MoscowNauka0481.41001 E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, 1970; Mir, Moscow, 1973). G. I. Natanson, “On the estimate of Lebesgue constants of de la Vallee–Poussin sums,” in Geometric Problems of the Theory of Functions and Sets (Kalinin Gos. Univ., Kalinin, 1986), pp. 102–107 [in Russian]. SterlinM. D.On inverse extremal problems of the constructive theory of functionsDokl. Akad. Nauk SSSR1976229550553410218 A. F. Timan, Theory of Approximation of Functions of a Real Variable (GIFML, Moscow, 1960; Pergamon, Oxford, 1963). BariN. K.StechkinS. B.“Best approximations and differential properties of two conjugate functions,” Tr. Mosk. MatO-va.19565483522 VinogradovO. L.Sharp Jackson-type inequalities for approximation of classes of convolutions by entire functions of finite degreeSt. Petersburg Math. J.200617593633217393710.1090/S1061-0022-06-00922-8 S. N. Bernstein (5122_CR3) 1952 N. I. Akhiezer (5122_CR1) 1965 V. V. Zhuk (5122_CR7) 1982 5122_CR9 H. S. Shapiro (5122_CR8) 1968; 74 M. D. Sterlin (5122_CR4) 1976; 229 M. D. Sterlin (5122_CR6) 1973; 209 S. N. Bernstein (5122_CR13) 1954 5122_CR2 V. K. Dzyadyk (5122_CR11) 1977 N. K. Bari (5122_CR5) 1956; 5 O. L. Vinogradov (5122_CR10) 2006; 17 5122_CR12 |
| References_xml | – reference: BernsteinS. N.Collected Works1954MoscowAcad. Nauk. SSSR – reference: ZhukV. V.Approximation of Periodic Functions1982LeningradLeningrad Univ. Press0521.42003 – reference: VinogradovO. L.Sharp Jackson-type inequalities for approximation of classes of convolutions by entire functions of finite degreeSt. Petersburg Math. J.200617593633217393710.1090/S1061-0022-06-00922-8 – reference: DzyadykV. K.Introduction into the Theory of Uniform Approximation of Functions by Polynomials1977MoscowNauka0481.41001 – reference: SterlinM. D.On inverse extremal problems of the constructive theory of functionsDokl. Akad. Nauk SSSR1976229550553410218 – reference: SterlinM. D.Estimates of constants in inverse theorems of the constructive theory of functionsDokl. Akad. Nauk SSSR197320912961298336196 – reference: AkhiezerN. I.Lectures on Approximation Theory1965MoscowNauka – reference: ShapiroH. S.Some Tauberian theorems with applications to approximation theoryBull. Am. Math. Soc.19687450050422507410.1090/S0002-9904-1968-11980-9 – reference: BariN. K.StechkinS. B.“Best approximations and differential properties of two conjugate functions,” Tr. Mosk. MatO-va.19565483522 – reference: BernsteinS. N.Collected Works1952MoscowAcad. Nauk. SSSR – reference: A. F. Timan, Theory of Approximation of Functions of a Real Variable (GIFML, Moscow, 1960; Pergamon, Oxford, 1963). – reference: E. M. Stein, Singular Integrals and Differentiability Properties of Functions (Princeton Univ. Press, Princeton, 1970; Mir, Moscow, 1973). – reference: G. I. Natanson, “On the estimate of Lebesgue constants of de la Vallee–Poussin sums,” in Geometric Problems of the Theory of Functions and Sets (Kalinin Gos. Univ., Kalinin, 1986), pp. 102–107 [in Russian]. – ident: 5122_CR2 doi: 10.1016/B978-0-08-009929-3.50008-7 – volume-title: Introduction into the Theory of Uniform Approximation of Functions by Polynomials year: 1977 ident: 5122_CR11 – volume: 229 start-page: 550 year: 1976 ident: 5122_CR4 publication-title: Dokl. Akad. Nauk SSSR – volume: 17 start-page: 593 year: 2006 ident: 5122_CR10 publication-title: St. Petersburg Math. J. doi: 10.1090/S1061-0022-06-00922-8 – volume-title: Collected Works year: 1954 ident: 5122_CR13 – ident: 5122_CR12 – volume: 209 start-page: 1296 year: 1973 ident: 5122_CR6 publication-title: Dokl. Akad. Nauk SSSR – volume: 74 start-page: 500 year: 1968 ident: 5122_CR8 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1968-11980-9 – volume-title: Collected Works year: 1952 ident: 5122_CR3 – ident: 5122_CR9 doi: 10.1515/9781400883882 – volume: 5 start-page: 483 year: 1956 ident: 5122_CR5 publication-title: O-va. – volume-title: Approximation of Periodic Functions year: 1982 ident: 5122_CR7 – volume-title: Lectures on Approximation Theory year: 1965 ident: 5122_CR1 |
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| SubjectTerms | Analysis Approximation Constants Derivatives Entire functions Functions (mathematics) Identities Mathematical analysis Mathematics Mathematics and Statistics Polynomials Theorems |
| Title | On the Constants in Inverse Theorems for the First-Order Derivative |
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