Herglotz's theorem for Jacobi-Dunkl positive definite sequences
-
Frej Chouchene
and
Iness Haouala
Abstract
In this work, we introduce the notion of Jacobi-Dunkl positive definite sequences using the properties of the discrete Jacobi-Dunkl translation. Then, we establish the analogous of Herglotz’s theorem to characterize these sequences.
(Communicated by Tomasz Natkaniec)
Acknowledgement
Thanks to the referees for careful readings and helpful comments and suggestions.
References
[1] Bloom, W. R.—Heyer, H.: Harmonic Analysis of Probability Measures on Hypergroups, de Gruyter, Berlin, 1995.10.1515/9783110877595Search in Google Scholar
[2] Chouchene, F.: Harmonic analysis associated with the Jacobi-Dunkl operator on ]−
[3] Chouchene, F.: Bounds, asymptotic behavior and recurrence relations for the Jacobi-Dunkl polynomials, International Journal of Open Problems Complex Analysis 6(1) (2014), 49–77.10.12816/0006030Search in Google Scholar
[4] Chouchene, F.: Recurrence and Christoffel-Darboux formulas for the Jacobi-Dunkl polynomials and applications, Commun. Math. Anal. 16(1) (2014), 123–142.Search in Google Scholar
[5] Einsiedler, M.—Ward, T.: Functional analysis, spectral theory, and applications. Grad. Texts in Math. 276, Springer International Publishing AG, 2017.10.1007/978-3-319-58540-6Search in Google Scholar
[6] Grenander, U.—Szegö, G.: Toeplitz Forms and their Applications. California Monographs in Mathematical Sciences, University of California Press, Berkeley, 1958.Search in Google Scholar
[7] Helson, H.: Harmonic Analysis, The Wadsworth & Brooks/ Cole Advanced Books & Software, Addision-Wesley, 1983, reprinted by Wadsworth, 1991.Search in Google Scholar
[8] Li, Zh.-k.: Hardy spaces for Jacobi expansions. I. The basic theory, Analysis 16(1) (1996), 27–49.10.1524/anly.1996.16.1.27Search in Google Scholar
[9] Li, Zh.-k.: Conjugate Jacobi series and conjugate functions, J. Approx. Theory 86(2) (1996), 179–196.10.1006/jath.1996.0061Search in Google Scholar
[10] Li, Zh.-k.—LIU, L.-M.: Uncertainty principles for Jacobi expansions, J. Math. Anal. Appl. 286(2) (2003), 652–663.10.1016/S0022-247X(03)00507-9Search in Google Scholar
[11] Li, Zh.-k.—ZHU, Zh.-h.: Characterizations of smoothness of functions in terms of the basis conjugate to Jacobi polynomials, J. Approx. Theory 160(1–2) (2009), 113–134.10.1016/j.jat.2008.03.006Search in Google Scholar
[12] Li, Zh.-k.—Liao, J.-Q.: Hardy spaces for Dunkl-Gegenbauer expansions, J. Funct. Anal. 265(5) (2013), 687–742.10.1016/j.jfa.2013.05.024Search in Google Scholar
[13] Rösler, M.: Trigonometric convolution structures on ℤ derived from Jacobi polynomials, J. Comp. Appl. Math. 65 (1995), 357–368.10.1016/0377-0427(95)00122-0Search in Google Scholar
[14] Rudin, W.: Real and Complex Analysis, 3rd ed., McGraw-Hill Higher Education, 1986.Search in Google Scholar
[15] Sunder, V. S.: The Riesz representation theorem, Indian J. Pure Appl. Math. 39(6) (2008), 467–481.Search in Google Scholar
[16] Szegö, G.: Orthogonal Polynomials. Amer. Math. Soc. Colloq. Pub., Vol. 23, Amer. Math. Soc. Providence, R.I., 1975.Search in Google Scholar
© 2021 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Regular papers
- Properties of implication in effect algebras
- On a nonlinear relation for computing the overpartition function
- Six-cycle systems
- Representation of bifinite domains by BF-closure spaces
- L-fuzzy cosets in universal algebras
- Density of sets with missing differences and applications
- Degree of independence of numbers
- A generalization of a result on the sum of element orders of a finite group
- A New fuzzy McShane integrability
- Hankel determinants of second and third order for the class 𝓢 of univalent functions
- Herglotz's theorem for Jacobi-Dunkl positive definite sequences
- Functional inequalities for Gaussian hypergeometric function and generalized elliptic integral of the first kind
- Existence on solutions of a class of casual differential equations on a time scale
- On a general system of difference equations defined by homogeneous functions
- Jordan amenability of banach algebras
- A new notion of orthogonality involving area and length
- Reeb flow invariant ∗-Ricci operators on trans-Sasakian three-manifolds
- A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function
- On first countable quasitopological homotopy groups
Articles in the same Issue
- Regular papers
- Properties of implication in effect algebras
- On a nonlinear relation for computing the overpartition function
- Six-cycle systems
- Representation of bifinite domains by BF-closure spaces
- L-fuzzy cosets in universal algebras
- Density of sets with missing differences and applications
- Degree of independence of numbers
- A generalization of a result on the sum of element orders of a finite group
- A New fuzzy McShane integrability
- Hankel determinants of second and third order for the class 𝓢 of univalent functions
- Herglotz's theorem for Jacobi-Dunkl positive definite sequences
- Functional inequalities for Gaussian hypergeometric function and generalized elliptic integral of the first kind
- Existence on solutions of a class of casual differential equations on a time scale
- On a general system of difference equations defined by homogeneous functions
- Jordan amenability of banach algebras
- A new notion of orthogonality involving area and length
- Reeb flow invariant ∗-Ricci operators on trans-Sasakian three-manifolds
- A finite graph is homeomorphic to the Reeb graph of a Morse–Bott function
- On first countable quasitopological homotopy groups
