Sharp Bernstein Inequalities for Jacobi--Dunkl Operators

O. L. Vinogradov

Matematičeskie zametki, Tome 112 (2022) no. 5, pp. 770-783.

Résumé

We find sharp constants in the Bernstein inequality $$ \|\Lambda_{\alpha,\beta}^rf\|\le M\|f\| $$ for the Jacobi–Dunkl differential-difference operator $$ \Lambda_{\alpha,\beta}f(x) =f'(x)+\frac{A'_{\alpha,\beta}(x)}{A_{\alpha,\beta}(x)} \frac{f(x)-f(-x)}{2}\,. $$ Here $n,r\in\mathbb N$, $f$ is a trigonometric polynomial of degree $\le n$, the norm is uniform, $\alpha,\beta\ge -1/2$, and $A_{\alpha,\beta}(x)=(1-\cos x)^\alpha(1+\cos x)^\beta|{\sin x}|$ is the Jacobi weight. In the spaces $L_p$ with Jacobi weight, upper bounds are obtained.

Détail

Keywords: Bernstein inequality, Jacobi–Dunkl operator
Mots-clés : sharp constant.

Bibliographie

[1] N. I. Akhiezer, Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR | Zbl

[2] A. F. Timan, Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatlit, M., 1960 | MR

[3] D. V. Gorbachev, “Tochnye neravenstva Bernshteina–Nikolskogo dlya polinomov i tselykh funktsii eksponentsialnogo tipa”, Chebyshevskii sb., 22:5 (2022), 58–110 | DOI

[4] A. I. Kamzolov, “Ob interpolyatsionnoi formule Rissa i neravenstve Bernshteina dlya funktsii na odnorodnykh prostranstvakh”, Matem. zametki, 15:6 (1974), 967–978 | MR | Zbl

[5] V. A. Ivanov, “Tochnye rezultaty v zadache o neravenstve Bernshteina–Nikolskogo na kompaktnykh simmetricheskikh rimanovykh prostranstvakh ranga 1”, Issledovaniya po teorii differentsiruemykh funktsii mnogikh peremennykh i ee prilozheniyam. Chast 14, Tr. MIAN SSSR, 194, Nauka, M., 1992, 111–119 | MR | Zbl

[6] V. A. Ivanov, “O neravenstvakh Bernshteina–Nikolskogo i Favara na kompaktnykh odnorodnykh prostranstvakh ranga 1”, UMN, 38:3 (231) (1983), 179–180 | MR | Zbl

[7] S. S. Platonov, “O nekotorykh zadachakh teorii priblizheniya funktsii na kompaktnykh odnorodnykh mnogoobraziyakh”, Matem. sb., 200:6 (2009), 67–108 | DOI | MR | Zbl

[8] S. S. Platonov, “Garmonicheskii analiz Fure–Yakobi i priblizhenie funktsii”, Izv. RAN. Ser. matem., 78:1 (2014), 117–166 | DOI | MR | Zbl

[9] G. Segë, Ortogonalnye mnogochleny, Fizmatgiz, M., 1962 | MR | Zbl

[10] R. Aski, R. Roi, Dzh. Endryus, Spetsialnye funktsii, MTsNMO, M., 2013

[11] D. V. Chertova, “Teoremy Dzheksona v prostranstvakh $L_p$, $1\le p\le 2$ s periodicheskim vesom Yakobi”, Izv. Tulskogo gos. un-ta. Estestv. nauki, 2009, no. 1, 5–27

[12] O. L. Vinogradov, “O normakh operatorov obobschennogo sdviga, porozhdennykh operatorami Yakobi–Danklya”, Issledovaniya po lineinym operatoram i teorii funktsii. 39, Zap. nauchn. sem. POMI, 389, POMI, SPb., 2011, 34–57

[13] Vo Tkhi Kuk, “Operatory obobschennogo sdviga v prostranstvakh $L_p$ na tore s vesom Yakobi i ikh primenenie”, Izv. Tulskogo gos. un-ta. Estestv. nauki, 2012, no. 1, 17–43

[14] K. Trimèche, Generalized Harmonic Analysis and Wavelet Packets, Gordon and Breach Sci. Publ., Amsterdam, 2001 | MR | Zbl

[15] M. A. Mourou, “Sonine-type integral transforms related to the Dunkl operator on the real line”, Integral Transforms Spec. Funct., 20:12 (2009), 915–924 | DOI | MR | Zbl

[16] G. N. Vatson, Teoriya besselevykh funktsii, Ch. 1, IL, M., 1949 | MR | Zbl

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

[18] R. P. Boas, “Quelques généralisations d'un théorème de S. Bernstein sur la dérivée d'un polynome trigonométrique”, C. R. Acad. Sci. Paris, 227 (1948), 618–619 | MR | Zbl

[19] O. L. Vinogradov, V. V. Zhuk, “Tochnye otsenki pogreshnostei formul tipa chislennogo differentsirovaniya na trigonometricheskikh mnogochlenakh”, Problemy matem. analiza, 21 (2000), 68–109 | Zbl