Geodesic
On the norms of generalized translation operators generated by Jacobi–Dunkl operators
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 34-57 
O. L. Vinogradov. On the norms of generalized translation operators generated by Jacobi–Dunkl operators. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 39, Tome 389 (2011), pp. 34-57. http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a2/
@article{ZNSL_2011_389_a2,
     author = {O. L. Vinogradov},
     title = {On the norms of generalized translation operators generated by {Jacobi{\textendash}Dunkl} operators},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {34--57},
     year = {2011},
     volume = {389},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a2/}
}
TY  - JOUR
AU  - O. L. Vinogradov
TI  - On the norms of generalized translation operators generated by Jacobi–Dunkl operators
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 34
EP  - 57
VL  - 389
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a2/
LA  - ru
ID  - ZNSL_2011_389_a2
ER  - 
%0 Journal Article
%A O. L. Vinogradov
%T On the norms of generalized translation operators generated by Jacobi–Dunkl operators
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 34-57
%V 389
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_389_a2/
%G ru
%F ZNSL_2011_389_a2

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

We establish an integral representation and improve the norm estimate for the generalized translation operators generated by Jacobi–Dunkl operators $$ \Lambda_{\alpha,\beta}f(x)=f'(x)+\frac{A_{\alpha,\beta}'(x)}{A_{\alpha,\beta}(x)}\,\frac{f(x)-f(-x)}2, $$ where $$ A_{\alpha,\beta}(x)=(1-\cos x)^\alpha(1+\cos x)^\beta|\sin x|, $$ in the spaces $L_p[-\pi,\pi]$ with the weight $A_{\alpha,\beta}$. For $\alpha\ge\beta\ge-\frac12$ we prove that these norms do not exceed $2$.

[1] G. Sege, Ortogonalnye mnogochleny, GIFML, M., 1962

[2] R. Askey, Orthogonal polynomials and special functions, SIAM, Philadelphia, 1975 | MR | Zbl

[3] T. Koornwinder, “Jacobi polynomials. II. An analytic proof of the product formula”, SIAM J. Math. Anal., 5:1 (1974), 125–137 | DOI | MR | Zbl

[4] G. Gasper, W. Trebels, “Multiplier criteria of Marcienkiewicz type for Jacobi expansions”, Trans. Amer. Math. Soc., 231:1 (1977), 117–132 | DOI | MR | Zbl

[5] G. Gasper, “Positivity and the convolution structure for Jacobi series”, Ann. Math., 93:1 (1971), 112–118 | DOI | MR | Zbl

[6] G. Gasper, “Banach algebras for Jacobi series and positivity of a kernel”, Ann. Math., 95:2 (1972), 261–280 | DOI | MR | Zbl

[7] T. Koornwinder, “The addition formula for Jacobi polynomials and spherical harmonics”, SIAM J. Appl. Math., 25:2 (1973), 236–246 | DOI | MR | Zbl

[8] C. F. Dunkl, “Differential-difference operators associated to reflection groups”, Trans. Amer. Math. Soc., 311:1 (1989), 167–183 | DOI | MR | Zbl

[9] M. Rösler, “Dunkl operators: theory and applications”, Lecture notes in math., 1817, Springer-Verlag, Berlin–Heidelberg, 2003, 93–135 | DOI | MR | Zbl

[10] D. V. Chertova, “Teoremy Dzheksona v prostranstvakh $L_p$, $1\le p\le2$ s periodicheskim vesom Yakobi”, Izv. Tulskogo gos. un-ta. Estestvennye nauki, 2009, no. 1, 5–27 | MR

[11] M. Rösler, “Bessel-type signed hypergroups on $\mathbb R$”, Probability measures on groups and related structures, Proc. Conf. (Oberwolfach, 1994), eds. H. Heyer, A. Mukherjea, World Scientific, 1995, 292–304 | MR | Zbl

[12] N. Ben Salem, A. Ould Ahmed Salem, “Convolution structure associated with the Jacobi–Dunkl operator on $\mathbb R$”, Ramanujan Journal, 12 (2006), 359–378 | DOI | MR | Zbl

[13] M. A. Mourou, K. Trimèche, “Transmutation operators and Paley–Wiener theorem associated with a singular differential-difference operator on the real line”, Anal. Appl., 1:1 (2003), 43–70 | DOI | MR | Zbl

[14] M. A. Mourou, “Transmutation operators associated with a Dunkl-type differential-difference operator on the real line and certain of their applications”, Integral Transforms and Special Functions, 12:1 (2001), 77–88 | DOI | MR | Zbl

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

[16] R. Edvards, Ryady Fure v sovremennom izlozhenii, v. 2, Mir, M., 1985

[17] N. P. Korneichuk, Tochnye konstanty v teorii priblizheniya, Nauka, M., 1987 | MR