Windowed Bessel Fourier transform in quantum calculus and applications
Tóm tắt
This paper deals firstly with some q-harmonic analysis properties for the q-windowed Bessel Fourier transform related to the q-Bessel function of the third kind as Plancherel formula, inversion formula in
$$\mathcal {L}_{q,2,\nu }$$
. Secondly, we give a weak uncertainty principle for it and we show that the portion of the q-windowed Bessel Fourier transform lying outside some set of finite measure cannot be arbitrarily too small. Then, we verify a version of Heisenberg–Pauli–Weyl type uncertainty inequalities for the q-windowed Bessel Fourier transform and its generalization. Finally, using the kernel reproducing theory, given by Saitoh (Theory of reproducing kernels and its applications. Longman Scientific and Technical, Harlow, 1988), we will be able to realize the natural and powerful approximation problems that lead to the q-windowed Bessel Fourier transform inverses.
Tài liệu tham khảo
Bettaibi, N.: Uncertainty principles in \(q^2\)-analogue Fourier analysis. Math. Sci. Res. J. 11(11), 590–602 (2007)
Bonami, A., Demange, B., Jaming, Ph: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev Mat Iberoamericana. 19, 23–55 (2003)
Czaja, W., Gigante, G.: Continuous Gabor transform for strong hypergroups. J. Fourier Anal. Appl. 9, 321–339 (2003)
Dhaouadi, L., Fitouhi, A., El Kamel, J.: Inequalities in \(q\)-Fourier analysis. J. Inequal. Pure Appl. Math. 7(5), 171 (2006)
Dhaouadi, L.: On the \(q\)-Bessel Fourier transform. Bull. Math. Anal. Appl. 5(2), 42–60 (2013)
Dhaoudi, L., Wafa, B., Fitouhi, A.: Paley–Wiener theorem for the \( q\)-Bessel transform and associated \(q\)-sampling formula. Expo. Math. 27, 55–72 (2009)
Dhaouadi, L.: Heisenberg uncertainty principle for the q-Bessel Fourier transform. preprint (2007)
Dhaouadi, L.: On the q-Bessel Fourier transform. Bull. Math. Anal. Appl. 5(2), 42–60 (2013)
Faris, W.G.: Inequalities and uncertainty principles. J. Math. Phys. 19, 461–466 (1978)
Fitouhi, A., Dhaoudi, L.: Positivity of the generalized translation associated with the \(q\)-Hankel transform. Constr. Approx. 34, 435–472 (2011)
Fitouhi, A., Nouri, F., Guesmi, S.: On Heisenberg and uncertainty principles for the q-Dunkl transform. JIPAM J. Inequal. Pure Appl. Math. 10(2), 42 (2009)
Gasper, G., Rahman, M.: Basic hypergeometric series. In: Gasper, G., Rahman, M. (eds.) Encycopedia of Mathematics and Its Applications, vol. 35. Cambridge University Press, Cambridge (1990)
Gröchenig, K.: Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2001)
Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994)
Heisenberg, W.: \(\ddot{U}\)ber den anschaulichen inhalt der quantentheoretischen kinematik und machanik. Z. f. Physik 43, 172–198 (1927)
Hleili, M., Nefzi, B., Bsaissa, A.: A variation on uncertainty principles for the generalized \(q\)-Bessel Fourier transform. J. Math. Anal. Appl. 440, 823–832 (2016)
Jackson, F.H.: On a \(q\)-definite integrals. Q. J. Pure Appl. Math. 41, 193–203 (1910)
Jebbari, E., Soltani, F.: Best approximations for the Laguerre-type Weierstrass transform On \([0,\infty [\times \mathbb{R}\). Int. J. Math. Math. Sci. 17, 2757–2768 (2005)
Kac, V.G., Cheung, P.: Quantum Calculs. Springer, New York (2002)
Koelink, H.T., Swarttouw, R.F.: On the zeros of the Hahn-Exton \(q\)-Bessel function and associated q-Lommel polynomials. J. Math. Anal. Appl. 186, 690–710 (1994)
Koornwinder, T.H., Swarttouw, R.F.: On \(q\)-analogues of the Hankel and Fourier transforms. Trans. AMS 333, 445–461 (1992)
Mejjaoli, H., Sraieb, N.: Gabor transform in quantum calculus and applications. Fract. Calc. Appl. Anal. 12(3), 320–336 (2009)
Price, J.F., Sitaram, A.: Local uncertainty inequalities for locally compact groups. Trans. Amer. Math. Soc. 308, 105–114 (1988)
Rubin, R.L.: A \(q^2\)-analogue operator for \(q^2\)-analogue Fourier analysis. J. Math. Anal. App. 212, 571–582 (1997)
Rsler, M., Voit, M.: Uncertainty principle for Hankel transforms. Proc. Am. Math. Soc. 127, 183–194 (1999)
Omri, S., Rachdi, L.T.: Weierstrass transform associated with the Hankel operator. Bull. Math. Anal. Appl. 1(2), 1–16 (2009)
Saitoh, S.: Applications of Tikhonov regularization to inverse problems using reproducing kernels. J. Phys.: Conf. Ser. 73, 012019 (2007)
Saitoh, S.: Approximate real inversion formulas of the Qaussian convolution. Appl. Anal. 83(7), 727–733 (2004)
Saitoh, S.: Theory of Reproducing Kernels and Its Applications. Longman Scientific and Technical, Harlow (1988)
Saitoh, S., Sawano, Y.: Theory of Reproducing Kernels and Applications Book Series: Developments in Mathematics, vol. 44. Springer, Berlin
Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Mathematical Series, No. 32. Princeton University Press, Princeton (1971)
Swarttouw, R.F.: The Hahn–Exton \(q\)-Bessel functions. Ph. D. Thesis, Delft Technical University (1992)
Wilczok, E.: New uncertainty principles for the continuous Gabor transform and the continuous Wavelet transform. Doc Math. 5, 201–226 (2000)