Optimal Arguments in the Jackson--Stechkin Inequality in~$L_2(\mathbb{R}^d)$ with Dunkl Weight

V. I. Ivanov ; A. V. Ivanov

Matematičeskie zametki, Tome 96 (2014) no. 5, pp. 674-686.

Résumé

The paper is devoted to the determination of the optimal arguments in the sharp Jackson–Stechkin inequality with modulus of continuity of order $r$ in the space $L_2(\mathbb{R}^d)$ with Dunkl weight defined by the root system $R$ and a nonnegative function of multiplicity $k$. If $$ \lambda_k=\frac d2-1+\sum_{\alpha\in R_+}k(\alpha)=\frac12, $$ where $R_+$ is the positive subsystem of the root system, then the optimal arguments for all $r$ coincide. If $\lambda_k\ne 1/2$, then the optimal argument for the modulus of continuity of second order is greater than for the first order. Such patterns are related to the arithmetic properties of zeros of Bessel functions.

Détail

Keywords: Jackson–Stechkin inequality, the space $L_2(\mathbb{R}^d)$ with Dunkl weight, modulus of continuity, Logan problem, Dunkl transform, Bessel function, Hankel transform, Borel probability measure.

Bibliographie

[1] M. Rösler, “Dunkl operators. Theory and applications”, Orthogonal Polynomials and Special Functions, Lecture Notes in Math., 1817, Springer-Verlag, Berlin, 2002, 93–135 | DOI | MR | Zbl

[2] M. Rösler, “A positive radial product formula for the Dunkl kernel”, Trans. Amer. Math. Soc., 355:6 (2003), 2413–2438 | DOI | MR | Zbl

[3] Kha Tkhi Min Khue, “O svyazi mnogomernykh i odnomernykh konstant Dzheksona v prostranstvakh $L_2$ so stepennymi vesami”, Izv. TulGU. Estestvennye nauki, 2012, no. 6, 114–123

[4] A. V. Ivanov, V. I. Ivanov, Kha Tkhi Min Khue, “Obobschennaya konstanta Dzheksona v prostranstve $L_2(\mathbb{R}^d)$ s vesom Danklya”, Izv. TulGU. Estestvennye nauki, 2013, no. 3, 74–90

[5] V. I. Ivanov, Kha Tkhi Min Khue, “Obobschennoe neravenstvo Dzheksona v prostranstve $L_2(\mathbb R^d)$ s vesom Danklya”, Tr. IMM UrO RAN, 20, no. 1, 2014, 109–118

[6] S. N. Vasilev, “Neravenstvo Dzheksona v $L_2(\mathbb R^N)$ s obobschennym modulem nepreryvnosti”, Tr. IMM UrO RAN, 16, no. 4, 2010, 93–99

[7] D. V. Gorbachev, “Otsenka optimalnogo argumenta v tochnom mnogomernom $L_2$-neravenstve Dzheksona–Stechkina”, Tr. IMM UrO RAN, 20, no. 1, 2014, 83–91

[8] N. I. Chernykh, “O neravenstve Dzheksona v $L_2$”, Priblizhenie funktsii v srednem, Tr. MIAN SSSR, 88, 1967, 71–74 | MR | Zbl

[9] V. V. Arestov, N. I. Chernykh, “On the $L_2$-approximation of periodic functions by trigonometric polynomials”, Approximation and Functions Spaces, North-Holland, Amsterdam, 1981, 25–43 | MR | Zbl

[10] D. V. Gorbachev, “Ekstremalnye zadachi dlya tselykh funktsii eksponentsialnogo sfericheskogo tipa”, Matem. zametki, 68:2 (2000), 179–187 | DOI | MR | Zbl

[11] E. E. Berdysheva, “Dve vzaimosvyazannye ekstremalnye zadachi dlya tselykh funktsii mnogikh peremennykh”, Matem. zametki, 66:3 (1999), 336–350 | DOI | MR | Zbl

[12] A. V. Ivanov, “Nekotorye ekstremalnye zadachi dlya tselykh funktsii v vesovykh prostranstvakh”, Izv. TulGU. Estestvennye nauki, 2010, no. 1, 26–44

[13] A. V. Ivanov, “Zadacha Logana dlya tselykh funktsii mnogikh peremennykh i konstanty Dzheksona v vesovykh prostranstvakh”, Izv. TulGU. Estestvennye nauki, 2011, no. 2, 29–58

[14] A. V. Ivanov, V. I. Ivanov, “Optimalnye argumenty v neravenstve Dzheksona v prostranstve $L_2(\mathbb{R}^d)$ so stepennym vesom”, Matem. zametki, 94:3 (2013), 338–348 | DOI | Zbl

[15] D. V. Gorbachev, S. A. Strankovskii, “Odna ekstremalnaya zadacha dlya chetnykh polozhitelno opredelennykh tselykh funktsii eksponentsialnogo tipa”, Matem. zametki, 80:5 (2006), 712–717 | DOI | MR | Zbl

[16] V. V. Arestov, A. G. Babenko, “On the optimal point in Jackson's inequality in $L_2(-\infty,\infty)$ with the second modulus of continuity”, East J. Approx., 10:1-2 (2004), 201–214 | MR | Zbl

[17] B. M. Levitan, I. S. Sargsyan, Vvedenie v spektralnuyu teoriyu. Samosopryazhennye obyknovennye differentsialnye operatory, Nauka, M., 1970 | MR | Zbl

[18] Kha Tkhi Min Khue, “Obobschennoe neravenstvo Dzheksona–Stechkina v prostranstve $L_2(\mathbb{R}^d)$ s vesom Danklya”, Izv. TulGU. Estestvennye nauki, 2014, no. 1, 63–82

[19] B. F. Logan, “Extremal problems for positive-definite bandlimited functions. I. Eventually positive functions with zero integral”, SIAM J. Math. Anal., 14:2 (1983), 249–253 | DOI | MR

[20] A. V. Ivanov, V. I. Ivanov, “Optimalnyi argument v obobschennom neravenstve Dzheksona v prostranstve $L_2(\mathbb{R}^d)$ s vesom Danklya i obobschennaya zadacha Logana”, Izv. TulGU. Estestvennye nauki, 2014, no. 1, 22–36

[21] M. Plancherel, G. Pólya, “Fonctions entiéres et intégrales de Fourier multiples”, Comment. Math. Helv., 9 (1937), 224–248 | DOI | Zbl

[22] C. Frappier, P. Oliver, “A quadrature formula involving zeros of Bessel functions”, Math. Comp., 60:201 (1993), 303–316 | DOI | MR | Zbl

[23] G. R. Grozev, Q. I. Rahman, “A quadrature formulae with zeros of Bessel functions as nodes”, Math. Comp., 64:210 (1995), 715–725 | DOI | MR | Zbl

[24] M. de Jeu, “Paley–Wiener theorems for the Dunkl transform”, Trans. Amer. Math. Soc., 358:10 (2006), 4225–4250 | DOI | MR | Zbl

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

[26] D. V. Gorbachev, Ekstremalnye zadachi teorii funktsii i teorii priblizhenii i ikh prilozheniya, Dis. $\dots$ dokt. fiz.-matem. nauk, Ekaterinburg, 2006