Uncertainty principles for the $q$-Hankel–Stockwell transform

  • Kamel Brahim Department of Mathematics, College of Science, University of Bisha, Saudi Arabia and Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia
  • Hédi Ben Elmonser Department of Mathematics, College of Science Al-Zul, Majmaah University, Al-Majmaah, Saudi Arabia and Department of Mathematics, National Institute of Technologie and Applied Sciences, Tunis, Tunisia
Keywords: q-Harmonic analysis, q-Hankel Stockwell transform, Uncertainty principles

Abstract

UDC 517.3

By using the $q$-Jackson integral and some elements of the $q$-harmonic analysis associated with the $q$-Hankel transform, we introduce and study a $q$-analog of the Hankel–Stockwell transform. We give some harmonic analysis properties (Plancherel formula, inversion formula, reproduicing kernel, etc.).  Furthermore, we establish a version of Heisenberg's uncertainty principles. Finally, we study the $q$-Hankel–Stockwell transform on a subset of finite measure.

References

N. Bettaibi, Uncertainty principles in $q^2$-analogue Fourier analysis, Math. Sci. Res. J., 11, № 11, 590–602 (2007).

B. Nefzi, K. Brahim, Calderуn's reproducing formula and uncertainty principle for the continuous wavelet transform associated with the $q$-Bessel operator, J. Pseudo-Different. Oper. and Appl., 9, 495–522 (2018).

L. Dhaouadi, A. Fitouhi, J. El Kamel, Inequalities in $q$-Fourier analysis, J. Inequal. Pure and Appl. Math., 7, Issue 5, Article 171 (2006).

L. Dhaouadi, On the $q$-Bessel Fourier transform, Bull. Math. Anal. and Appl., 5, Issue 2, 42–60 (2013).

A. Fitouhi, A. Safraoui, Paley–Wiener theorem for the $q^2$-Fourier–Rubin transform, Tamsui Oxf. J. Math. Sci., 26, № 3, 287–304 (2010).

A. Fitouhi, N. Bettaibi, K. Brahim, The mellin transform in quantum calculus, Constr. Approx., 23, 305–323 (2006).

G. Gasper, M. Rahman, Basic hypergeometric series, Encyclopedia Math. and Appl., vol. 35, Cambridge Univ. Press (1990).

W. Heisenberg, Über den anschaulichen inhalt der quantentheoretischen Kinematik und Mechanik, Z. Physik, 43, 172–198 (1927).

F. H. Jackson, On a $q$-definite Integrals, Quart. J. Pure and Appl. Math., 41, 193–203 (1910).

V. G. Kac, P. Cheung, Quantum calculs, Springer-Verlag, New York (2002).

T. H. Koornwinder, R. F. Swarttouw, On $q$-analogues of the Hankel and Fourier transforms, Trans. Amer. Math. Soc., 333, 445–461 (1992).

N. B. Hamadi, Z. Hafirassou, H. Herch, Uncertainty principles for the Hankel–Stockwell transform, J. Pseudo-Different. Oper. and Appl., 1–22 (2020); https: //doi.org/10.1007/s11868-020-00329-z.

S. Saitoh, Theory of reproducing kernels and its applications, Longman Sci. and Technical, Harlow (1988).

R. F. Swarttouw, The Hahn–Exton $q$-Bessel functions, Ph. D. Thesis, Delft Technical Univ. (1992).

Published
25.07.2023
How to Cite
BrahimK., and Ben ElmonserH. “Uncertainty Principles for the $q$-Hankel–Stockwell Transform”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, no. 7, July 2023, pp. 888 -03, doi:10.37863/umzh.v75i7.7166.
Section
Research articles