A Generalized Translation Operator Generated by the Sinc Function on an Interval

V. V. Arestov ; M. V. Deikalova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 27-48

Résumé

We discuss the properties of the generalized translation operator generated by the system of functions $\mathfrak{S}=\{{(\sin k\pi x)}/{(k\pi x)}\}_{k=1}^\infty$ in the spaces $L^q=L^q((0,1),{\upsilon})$, $q\ge 1$, on the interval $(0,1)$ with the weight $\upsilon(x)=x^2$. We find an integral representation of this operator and study its norm in the spaces $L^q$, $1\le q\le\infty$. The translation operator is applied to the study of Nikol'skii's inequality between the uniform norm and the $L^q$-norm of polynomials in the system $\mathfrak{S}$.

Détail

Keywords: generalized translation, sinc function, inequality of different metrics.

Bibliographie

[1] Vatson G.N., Teoriya besselevykh funktsii, Chast pervaya, IL, M., 1949, 798 pp.

[2] Beitmen G., Erdeii A.I., Vysshie transtsidentnye funktsii, v. 2, Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1966, 295 pp.

[3] Vladimirov V.S., Uravneniya matematicheskoi fiziki, Nauka, M., 1981, 512 pp. | MR

[4] Levitan B.M., “Razlozheniya po funktsiyam Besselya v ryady i integraly Fure”, Uspekhi mat. nauk, 6:2 (1951), 102–143 | MR | Zbl

[5] Babenko A.G., “Tochnoe neravenstvo Dzheksona — Stechkina v prostranstve $L^2(\mathbb{R}^m)$”, Tr. In-ta matematiki i mekhaniki UrO RAN, 5 (1998), 183–198 | Zbl

[6] Platonov S.S., “Garmonicheskii analiz Besselya i priblizhenie funktsii na polupryamoi”, Izv. RAN. Ser. Mat., 71:5 (2007), 149–196 | DOI | MR | Zbl

[7] Liu Y., “Best $L^2$-approximation of function on $[0,1]$ with the weight $x^{2\nu+1}$”, Tr. mezhdunar. letn. mat. shk. S. B. Stechkina po teorii funktsii, Izd-vo TulGU, Tula, 2007, 180–190

[8] Abilov V.A., Abilova F.V., Kerimov M.K., “Nekotorye voprosy priblizheniya funktsii summami Fure — Besselya”, Zhurn. vychisl. matematiki i mat. fiziki, 53:7 (2013), 1051–1057 | DOI | MR | Zbl

[9] Arestov V., Babenko A., Deikalova M., Horváth A., “Nikol'skii inequality between the uniform norm and integral norm with Bessel weight for entire functions of exponential type on the half-line”, Anal. Math., 44:1 (2018), 21–42 | DOI | MR | Zbl

[10] Arestov V.V., Deikalova M.V., “Ob odnom obobschennom sdvige i sootvetstvuyuschem neravenstve raznykh metrik”, Tr. In-ta matematiki i mekhaniki UrO RAN, 28:4 (2022), 40–53 | DOI

[11] Arestov V., Deikalova M., “On one inequality of different metrics for trigonometric polynomials”, Ural Math. J., 8:2 (2022), 25–43 | DOI | MR

[12] Levitan B.M., “Primenenie operatorov obobschennogo sdviga k lineinym differentsialnym uravneniyam vtorogo poryadka”, Uspekhi mat. nauk, 4:1 (1949), 1–107

[13] Grei E., Metyuz G.B., Funktsii Besselya ikh prilozheniya k fizike i mekhanike, IL, M., 1953, 372 pp.

[14] Danford N., Shvarts Dzh.T., Lineinye operatory. Obschaya teoriya, Editorial URSS, M., 2004, 896 pp.

[15] Natanson I.P., Teoriya funktsii veschestvennoi peremennoi, uchebn. posobie, Lan, SPb., 1999 | MR

[16] Stein I., Veis G., Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974, 333 pp.

[17] Jackson D., “Certain problems of closest approximation”, Bull. Amer. Math. Soc., 39:12 (1933), 889–906 | DOI | MR | Zbl

[18] Nikolskii S.M., “Neravenstva dlya tselykh funktsii konechnoi stepeni i ikh primenenie v teorii differentsiruemykh funktsii mnogikh peremennykh”, Tr. MIAN SSSR, 38 (1951), 244–278 | Zbl

[19] Nikolskii S.M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Gl. redaktsiya fiz.-mat. literatury izd-va Nauka, M., 1977, 456 pp. | MR

[20] Bernshtein S.N., Ekstremalnye svoistva polinomov, OHTI, M.-L., 1937

[21] Korneichuk N.P., Babenko V.F., Ligun A.A., Ekstremalnye svoistva polinomov i splainov, Naukova dumka, Kiev, 1992, 304 pp. | MR

[22] Milovanović G.V., Mitrinović D.S., Rassias Th.M., Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994, 821 pp. | MR | Zbl

[23] Borwein P., Erdélyi T., Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995 | MR | Zbl

[24] Rahman Q.I., Schmeisser G., Analytic Theory of Polynomials, Oxford Univ. Press, Oxford, 2002, 742 pp. | MR | Zbl

[25] Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A., Neravenstva dlya proizvodnykh i ikh prilozheniya, Naukova dumka, Kiev, 2003, 590 pp.

[26] Bari N.K., “Obobschenie neravenstv S. N. Bernshteina i A. A. Markova”, Izv. AN SSSR. Ser. matematicheskaya, 18:2 (1954), 159–176 | Zbl

[27] Ivanov V.I., “Nekotorye neravenstva dlya trigonometricheskikh polinomov i ikh proizvodnykh v raznykh metrikakh”, Mat. zametki, 18:4 (1975), 489–498 | MR | Zbl

[28] Arestov V.V., “O neravenstve raznykh metrik dlya trigonometricheskikh polinomov”, Mat. zametki, 27:4 (1980), 539–547 | MR | Zbl

[29] Badkov V.M., “Asimptoticheskie i ekstremalnye svoistva ortogonalnykh polinomov pri nalichii osobennostei u vesa”, Tr. MIAN, 198 (1992), 41–88 | Zbl

[30] Babenko V., Kofanov V., Pichugov S., “Comparison of rearrangement and Kolmogorov–Nagy type inequalities for periodic functions”, Approx. Theory, A volume dedicated to Blagovest Sendov, ed. B. Bojanov, DARBA, Sofia, 2002, 24–53 | MR | Zbl

[31] Gorbachev D.V., “Tochnye neravenstva Bernshteina — Nikolskogo dlya polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sb., 22:5 (2021), 58–110 | DOI | MR | Zbl

[32] Arestov V.V., Deikalova M.V., “Neravenstvo Nikolskogo dlya algebraicheskikh polinomov na mnogomernoi evklidovoi sfere”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19:2 (2013), 34–47 | MR

[33] Arestov V., Deikalova M., “Nikol'skii inequality between the uniform norm and $L_q$-norm with ultraspherical weight of algebraic polynomials on an interval”, Comput. Methods Funct. Theory, 15:4 (2015), 689–708 | DOI | MR | Zbl

[34] Arestov V., Deikalova M., “Nikol'skii inequality between the uniform norm and $L_q$-norm with Jacobi weight of algebraic polynomials on an interval”, Analysis Math., 42:2 (2016), 91–120 | DOI | MR | Zbl

[35] Arestov V., Deikalova M., Horváth Á., “On Nikol'skii type inequality between the uniform norm and the integral $q$-norm with Laguerre weight of algebraic polynomials on the half-line”, J. Approx. Theory, 222 (2017), 40–54 | DOI | MR | Zbl

[36] Arestov V.V., “A characterization of extremal elements in some linear problems”, Ural Math. J., 3:2 (2017), 22–32 | DOI | MR | Zbl