Abstract
Using the q-harmonic analysis associated with the q-Dunkl operator, we prove an analog of Titchmarsh’s theorem for functions satisfying the q-Dunkl Lipschitz condition, using a generalized q-translation operator in the space \(L_{q,\alpha }^{2}({\mathbb {R}}_{q})\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bettaibi, N., Bettaieb, R.H.: q-Analogue of the Dunkl transform on the real line. Tamsui Oxf. J. Math. Sci. 25(2), 117–205 (2007)
Rubin, R.L.: Duhamel solutions of non-homogeneous \(q^{2}\)-analogue wave equations. Proc. Am. Math. Soc. 135(3), 777–785 (2007)
Titchmarsh, E.C.: Introduction of the Theory of Fourier Integrals. Oxford University Press, Oxford (1937)
Rubin, R.L.: A \(q^{2}\)-analogue operator for \(q^{2}\)-analogue Fourier analysis. J. Math. Anal. Appl. 212, 571–582 (1997)
Gasper, G., Rahman, M.: Basic Hypergeometric Series. Encyclopedia of Mathematics and Its Application, vol. 35. Cambridge University Press, Cambridge (1990)
Jackson, F.H.: On a q-definite integrals. J. Pure Appl. Math. 41, 193–203 (1910)
El Ouadih, S., Daher, R., El Hamma, M.: Moduli of smoothness and K-functional in \(L_{2}({\mathbb{R}}^{+}_{q})\) space with power weight. J. Math. Anal. Appl. 45(3), 491–503 (2019). https://doi.org/10.1007/s10476-019-0830-3
Acknowledgements
The author is grateful to the referees for the useful comments and suggestions in improving the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Daher, R., Tyr, O. An analog of Titchmarsh’s theorem for the q-Dunkl transform in the space \(L_{q,\alpha }^{2}({\mathbb {R}}_{q})\). J. Pseudo-Differ. Oper. Appl. 11, 1933–1949 (2020). https://doi.org/10.1007/s11868-020-00330-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11868-020-00330-6