Geodesic
Some Problems of Approximation Theory in the Spaces~$L_p$ on the Line with Power Weight
Matematičeskie zametki, Tome 90 (2011) no. 3, pp. 362-383.

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In the spaces $L_p$ on the line with power weight, we study approximation of functions by entire functions of exponential type. Using the Dunkl difference-differential operator and the Dunkl transform, we define the generalized shift operator, the modulus of smoothness, and the $K$-functional. We prove a direct and an inverse theorem of Jackson–Stechkin type and of Bernstein type. We establish the equivalence between the modulus of smoothness and the $K$-functional.
Keywords: Dunkl difference-differential operator, entire function, Dunkl transform, generalized shift operator, modulus of smoothness, the spaces $L_p$, Jackson–Stechkin theorem. 
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Iong Ping Li; Chun Mei Su; V. I. Ivanov. Some Problems of Approximation Theory in the Spaces~$L_p$ on the Line with Power Weight. Matematičeskie zametki, Tome 90 (2011) no. 3, pp. 362-383. http://geodesic.mathdoc.fr/item/MZM_2011_90_3_a3/

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

[2] S. S. Platonov, “Obobschennye sdvigi Besselya i nekotorye zadachi teorii priblizhenii funktsii na polupryamoi”, Sib. matem. zhurn., 50:1 (2009), 154–174 | MR | Zbl

[3] C. T. Roosenraad, Inequalities with Orthogonal Polynomials, Ph.D. Thesis, Univ. of Wisconsin–Madison, 1969 | MR

[4] M. F. E. de Jeu, “The Dunkl transform”, Invent. Math., 113:1 (1993), 147–162 | DOI | MR | Zbl

[5] N. B. Andersen, M. de Jeu, “Elementary proofs of Paley–Wiener theorems for the Dunkl transform on the real line”, Int. Math. Res. Not., 30 (2005), 1871–1831 | MR | Zbl

[6] S. Thangavelu, Y. Xu, “Convolution operator and maximal function for the Dunkl transform”, J. Anal. Math., 97:1 (2005), 25–55 | DOI | MR | Zbl

[7] M. Rösler, “Generalized Hermite polynomials and the heat equation for Dunkl operators”, Comm. Math. Phys., 192:3 (1998), 519–542 | DOI | MR | Zbl

[8] C. F. Dunkl, “Integral kernels with reflection group invariance”, Canad. J. Math., 43:6 (1991), 1213–1227 | DOI | MR | Zbl

[9] Ó. Ciaurri, J. L. Varona, “A Whittaker–Shannon–Kotel'nikov sampling theorem related to the Dunkl transform”, Proc. Amer. Math. Soc., 135:9 (2007), 2939–2947 | DOI | MR | Zbl

[10] S. S. Platonov, E. S. Belkina, “Ekvivalentnost $K$-funktsionalov i modulei gladkosti, postroennykh po obobschennym sdvigam Danklya”, Izv. vuzov. Matem., 2008, no. 8, 3–15 | MR | Zbl

[11] M. A. Mourou, K. Trimèche, “Calderon's reproducing formula related to the Dunkl operator on the real line”, Monatsh. Math., 136:1 (2002), 47–65 | DOI | MR | Zbl

[12] H. Mejjaoli, N. Sraieb, “Uncertainty principles for the continuous Dunkl Gabor transform and the Dunkl continuous wavelet transform”, Mediterr. J. Math., 5:4 (2008), 443–466 | DOI | MR | Zbl

[13] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1969 | MR | Zbl

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

[15] K. Trimèche, “The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual”, Integral Transform. Spec. Funct., 12:4 (2001), 349–374 | DOI | MR | Zbl