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March  2021, 20(3): 975-994. doi: 10.3934/cpaa.2021002

Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials

Ágota P. Horváth

Department of Analysis, Budapest University of Technology and Economics, H-1521 Budapest, Hungary

Received  June 2020 Revised  November 2020 Published  March 2021 Early access  January 2021

Fund Project: The author is supported by the NKFIH-OTKA Grants K128922 and K132097

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.

Keywords: Discrete diffusion semigroup, higher-order parabolic equation, maximal operator, exceptional Jacobi polynomials, Jacobi-Dunkl polynomials.
Mathematics Subject Classification: Primary: 39A12, 39A14, 42C10.
Citation: Ágota P. Horváth. Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials. Communications on Pure & Applied Analysis, 2021, 20 (3) : 975-994. doi: 10.3934/cpaa.2021002
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Durán, Corrigendum to the papers on Exceptional orthogonal polynomials, J. Approx. Theory 253 (2020), 105349. doi: 10.1016/j.jat.2019.105349.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Odake, Recurrence relations of the multi-indexed orthogonal polynomials : II, J Math Phys., 56 (2015), 053506. doi: 10.1063/1.4921230.  Google Scholar

[20]

E. A. Rahmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR-Sb., 46 (1983), 105–l17.  Google Scholar

[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.  Google Scholar

[22] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, Princeton, NJ, 1970.   Google Scholar
[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.  Google Scholar

[24]

G. Szegő, Orthogonal Polynomials, 4$^{th}$ edition, American Mathematical Society, Providence RI, 1975  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Durán, Corrigendum to the papers on Exceptional orthogonal polynomials, J. Approx. Theory 253 (2020), 105349. doi: 10.1016/j.jat.2019.105349.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

S. Odake, Recurrence relations of the multi-indexed orthogonal polynomials : II, J Math Phys., 56 (2015), 053506. doi: 10.1063/1.4921230.  Google Scholar

[20]

E. A. Rahmanov, On the asymptotics of the ratio of orthogonal polynomials, II, Math. USSR-Sb., 46 (1983), 105–l17.  Google Scholar

[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.  Google Scholar

[22] E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, Princeton, NJ, 1970.   Google Scholar
[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.  Google Scholar

[24]

G. Szegő, Orthogonal Polynomials, 4$^{th}$ edition, American Mathematical Society, Providence RI, 1975  Google Scholar

[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.  Google Scholar

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