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Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials
Department of Analysis, Budapest University of Technology and Economics, H-1521 Budapest, Hungary |
Some weighted inequalities for the maximal operator with respect to the discrete diffusion semigroups associated with exceptional Jacobi and Jacobi-Dunkl polynomials are given. This setup allows to extend the corresponding results obtained for discrete heat semigroup recently to richer class of differential-difference operators.
References:
[1] |
V. Almeida, J. J. Betancor and L. Rodríguez-Mesa, Discrete Hardy spaces and heat semigroup associated with the discrete Laplacian, Mediterr. J. Math., 16 (2019), 23pp.
doi: 10.1007/s00009-019-1366-2. |
[2] |
A. Arenas, Ó. Ciaurri and E. Labarga, Discrete harmonic analysis associated with Jacobi expansions I: The heat semigroup, J. Math. Anal. Appl., 490 (2020), 123996.
doi: 10.1016/j.jmaa.2020.123996. |
[3] |
F. Astengo and B. Di Blasio,
Dynamics of the heat semigroup in Jacobi analysis, J. Math. Anal. Appl., 391 (2012), 48-56.
doi: 10.1016/j.jmaa.2012.02.033. |
[4] |
J. J. Betancor, A. J. Castro, J. C. Farina and L. Rodríguez-Mesa,
Discrete harmonic analysis associated with ultraspherical expansions, Potential Anal., 53 (2020), 523-563.
doi: 10.1007/s11118-019-09777-9. |
[5] |
F. Chouchene,
Harmonic analysis associated with the Jacobi-Dunkl operator on $\left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$, J. Comput. Appl. Math., 178 (2005), 75-89.
doi: 10.1016/j.cam.2004.02.025. |
[6] |
F. Chouchene, L. Gallardo and M. Mili,
The heat semigroup for the Jacobi-Dunkl operator and the related Markov processes, Potential Anal., 25 (2006), 103-119.
doi: 10.1007/s11118-006-9012-6. |
[7] |
F. Chouchene,
Bounds, asymptotic behavior and recurrence relations for the Jacobi-Dunkl polynomials, Int. J. Open Problems Complex Anal., 6 (2014), 49-77.
doi: 10.12816/0006030. |
[8] |
F. Chouchene and I. Haouala, De La Vallée Poussin Approximations and Jacobi-Dunkl Convolution Structures, Results Math., 75 (2020), b21pp.
doi: 10.1007/s00025-020-1175-8. |
[9] |
O. Ciaurri, T. A. Gillespie, L. Roncal, J. L. Torrea and J. L. Varona,
Harmonic analysis associated with a discrete Laplacian, J. d'Analyse Math., 132 (2017), 109-131.
doi: 10.1007/s11854-017-0015-6. |
[10] |
A. Durán, Corrigendum to the papers on Exceptional orthogonal polynomials, J. Approx. Theory 253 (2020), 105349.
doi: 10.1016/j.jat.2019.105349. |
[11] |
M. Á. García-Ferrero, D. Gómez-Ullate and R. Milson,
A Bochner type classification theorem for exceptional orthogonal polynomials, J. Math. Anal. Appl., 472 (2019), 584-626.
doi: 10.1016/j.jmaa.2018.11.042. |
[12] |
D. Gómez-Ullate, Y. Grandati and R. Milson, Corrigendum on the proof of completeness for exceptional Hermite polynomials, J. Approx. Theory, 253 (2020), 105350.
doi: 10.1016/j.jat.2019.105350. |
[13] |
D. Gómez-Ullate, N. Kamran and R. Milson,
An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl., 359 (2009), 352-367.
doi: 10.1016/j.jmaa.2009.05.052. |
[14] |
D. Gómez-Ullate, F. Marcellán and R. Milson, Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials, J. Math. Anal. Appl., 399 (2013, ) 480–495.
doi: 10.1016/j.jmaa.2012.10.032. |
[15] |
D. V. Gorbachev, V. I. Ivanov and S. Y. Tikhonov,
Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications, Constr. Approx., 49 (2019), 555-605.
doi: 10.1007/s00365-018-9435-5. |
[16] |
Á. P. Horváth,
Asymptotics for Recurrence Coefficients of $X_1$-Jacobi Exceptional Polynomials and Christoffel Function, Integr. Transf. Spec. F., 31 (2020), 87-106.
doi: 10.1080/10652469.2019.1672051. |
[17] |
Á. P. Horváth, Multiplication operator and exceptional Jacobi polynomials, arXiv: 2003.11861. Google Scholar |
[18] |
P. Nevai,
Géza Freud, Orthogonal Polynomials and Christoffel Functions, J. Approx Theory, 48 (1986), 3-167.
doi: 10.1016/0021-9045(86)90016-X. |
[19] |
S. Odake, Recurrence relations of the multi-indexed orthogonal polynomials : II, J Math Phys., 56 (2015), 053506.
doi: 10.1063/1.4921230. |
[20] |
E. A. Rahmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR-Sb., 46 (1983), 105–l17. |
[21] |
M. Rösler, Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Springer, Berlin, 2003.
doi: 10.1007/3-540-44945-0_3. |
[22] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, Princeton, NJ, 1970.
![]() |
[23] |
M. H. Stone, Linear Transformations in Hilbert Space and their Applications to Analysis, American Mathematical Society, New York, 1932.
doi: 10.1090/coll/015. |
[24] |
G. Szegő, Orthogonal Polynomials, 4$^{th}$ edition, American Mathematical Society, Providence RI, 1975 |
[25] |
O. L. Vinogradov,
On the norms of generalized translation operators generated by the Jacobi-Dunkl operators, J. of Math. Sci., 182 (2012), 603-616.
doi: 10.1007/s10958-012-0765-8. |
show all references
References:
[1] |
V. Almeida, J. J. Betancor and L. Rodríguez-Mesa, Discrete Hardy spaces and heat semigroup associated with the discrete Laplacian, Mediterr. J. Math., 16 (2019), 23pp.
doi: 10.1007/s00009-019-1366-2. |
[2] |
A. Arenas, Ó. Ciaurri and E. Labarga, Discrete harmonic analysis associated with Jacobi expansions I: The heat semigroup, J. Math. Anal. Appl., 490 (2020), 123996.
doi: 10.1016/j.jmaa.2020.123996. |
[3] |
F. Astengo and B. Di Blasio,
Dynamics of the heat semigroup in Jacobi analysis, J. Math. Anal. Appl., 391 (2012), 48-56.
doi: 10.1016/j.jmaa.2012.02.033. |
[4] |
J. J. Betancor, A. J. Castro, J. C. Farina and L. Rodríguez-Mesa,
Discrete harmonic analysis associated with ultraspherical expansions, Potential Anal., 53 (2020), 523-563.
doi: 10.1007/s11118-019-09777-9. |
[5] |
F. Chouchene,
Harmonic analysis associated with the Jacobi-Dunkl operator on $\left]-\frac{\pi}{2}, \frac{\pi}{2}\right[$, J. Comput. Appl. Math., 178 (2005), 75-89.
doi: 10.1016/j.cam.2004.02.025. |
[6] |
F. Chouchene, L. Gallardo and M. Mili,
The heat semigroup for the Jacobi-Dunkl operator and the related Markov processes, Potential Anal., 25 (2006), 103-119.
doi: 10.1007/s11118-006-9012-6. |
[7] |
F. Chouchene,
Bounds, asymptotic behavior and recurrence relations for the Jacobi-Dunkl polynomials, Int. J. Open Problems Complex Anal., 6 (2014), 49-77.
doi: 10.12816/0006030. |
[8] |
F. Chouchene and I. Haouala, De La Vallée Poussin Approximations and Jacobi-Dunkl Convolution Structures, Results Math., 75 (2020), b21pp.
doi: 10.1007/s00025-020-1175-8. |
[9] |
O. Ciaurri, T. A. Gillespie, L. Roncal, J. L. Torrea and J. L. Varona,
Harmonic analysis associated with a discrete Laplacian, J. d'Analyse Math., 132 (2017), 109-131.
doi: 10.1007/s11854-017-0015-6. |
[10] |
A. Durán, Corrigendum to the papers on Exceptional orthogonal polynomials, J. Approx. Theory 253 (2020), 105349.
doi: 10.1016/j.jat.2019.105349. |
[11] |
M. Á. García-Ferrero, D. Gómez-Ullate and R. Milson,
A Bochner type classification theorem for exceptional orthogonal polynomials, J. Math. Anal. Appl., 472 (2019), 584-626.
doi: 10.1016/j.jmaa.2018.11.042. |
[12] |
D. Gómez-Ullate, Y. Grandati and R. Milson, Corrigendum on the proof of completeness for exceptional Hermite polynomials, J. Approx. Theory, 253 (2020), 105350.
doi: 10.1016/j.jat.2019.105350. |
[13] |
D. Gómez-Ullate, N. Kamran and R. Milson,
An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl., 359 (2009), 352-367.
doi: 10.1016/j.jmaa.2009.05.052. |
[14] |
D. Gómez-Ullate, F. Marcellán and R. Milson, Asymptotic and interlacing properties of zeros of exceptional Jacobi and Laguerre polynomials, J. Math. Anal. Appl., 399 (2013, ) 480–495.
doi: 10.1016/j.jmaa.2012.10.032. |
[15] |
D. V. Gorbachev, V. I. Ivanov and S. Y. Tikhonov,
Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications, Constr. Approx., 49 (2019), 555-605.
doi: 10.1007/s00365-018-9435-5. |
[16] |
Á. P. Horváth,
Asymptotics for Recurrence Coefficients of $X_1$-Jacobi Exceptional Polynomials and Christoffel Function, Integr. Transf. Spec. F., 31 (2020), 87-106.
doi: 10.1080/10652469.2019.1672051. |
[17] |
Á. P. Horváth, Multiplication operator and exceptional Jacobi polynomials, arXiv: 2003.11861. Google Scholar |
[18] |
P. Nevai,
Géza Freud, Orthogonal Polynomials and Christoffel Functions, J. Approx Theory, 48 (1986), 3-167.
doi: 10.1016/0021-9045(86)90016-X. |
[19] |
S. Odake, Recurrence relations of the multi-indexed orthogonal polynomials : II, J Math Phys., 56 (2015), 053506.
doi: 10.1063/1.4921230. |
[20] |
E. A. Rahmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR-Sb., 46 (1983), 105–l17. |
[21] |
M. Rösler, Dunkl operators: theory and applications, in Orthogonal Polynomials and Special Functions (Leuven, 2002), Springer, Berlin, 2003.
doi: 10.1007/3-540-44945-0_3. |
[22] |
E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, Princeton, NJ, 1970.
![]() |
[23] |
M. H. Stone, Linear Transformations in Hilbert Space and their Applications to Analysis, American Mathematical Society, New York, 1932.
doi: 10.1090/coll/015. |
[24] |
G. Szegő, Orthogonal Polynomials, 4$^{th}$ edition, American Mathematical Society, Providence RI, 1975 |
[25] |
O. L. Vinogradov,
On the norms of generalized translation operators generated by the Jacobi-Dunkl operators, J. of Math. Sci., 182 (2012), 603-616.
doi: 10.1007/s10958-012-0765-8. |
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