Best approximation of Hartley-Bessel multiplier operators on weighted Sobolev spaces
DOI:
https://doi.org/10.33993/jnaat541-1512Keywords:
Hartley transform, Bessel functions, Calder´on’s reproducing formulas, Approximation theoryAbstract
The main goal of this paper is to introduce the Hartley-Bessel \(L^2_\alpha\)-multiplier operators and to give for them some new results as Plancherel’s, Calderon’s reproducing formulas and Heisenberg’s, Donoho-Stark’s uncertainty principles. Next, using the theory of reproducing kernels we give best approximation and an integral representation of the extremal functions related to these operators on weighted Sobolev spaces.
Downloads
References
[1] Aronszajn, N. (1950). Theory of reproducing kernels. Transactions of the American mathematical society, 68(3), 337-404. DOI: https://doi.org/10.1090/S0002-9947-1950-0051437-7
[2] Ba˜nuelos, R., & Bogdan, K. (2007). L´evy processes and Fourier multipliers. Journal of Functional Analysis, 250(1), 197-213.https://doi.org/10.1016/j.jfa.2007.05.013 DOI: https://doi.org/10.1016/j.jfa.2007.05.013
[3] Bouzeffour, F. (2024). The Hartley–Bessel function: product formula and convolution structure. Journal of Pseudo-Differential Operators and Applications, 15(2), 42.https://doi.org/10.1007/s11868-024-00610-5 DOI: https://doi.org/10.1007/s11868-024-00610-5
[4] Bouzeffour, F. (2014). The generalized Hartley transform. Integral Transforms and Special Functions, 25(3), 230-
239.https://doi.org/10.1080/10652469.2013.838760 DOI: https://doi.org/10.1080/10652469.2013.838760
[5] Bouzeffour, F., & Garayev, M. (2021). The Hartley transform via SUSY quantum mechanics. Memoirs on Differential Equations and Mathematical Physics, 83, 31-41.
[6] Bracewell, R. N. (1989). Physical aspects of the Hartley transform. Journal of atmospheric and terrestrial physics, 51(9-10), 791-795. DOI: https://doi.org/10.1016/0021-9169(89)90036-6
[7] Chana, A., & Akhlidj, A. (2024). Calder´on’s formulas and uncertainty principles for the Laguerre–Bessel L 2-multiplier operators. Integral Transforms and Special Functions, 1-18.https://doi.org/10.1080/10652469.2024.2382787 DOI: https://doi.org/10.1080/10652469.2024.2382787
[8] Donoho, D. L., & Stark, P. B. (1989). Uncertainty principles and signal recovery. SIAM Journal on Applied Mathematics, 49(3), 906-931. DOI: https://doi.org/10.1137/0149053
[9] Dunkl, C. F. (1989). Differential-difference operators associated to reflection groups. Transactions of the American Mathematical Society, 311(1), 167-183. DOI: https://doi.org/10.1090/S0002-9947-1989-0951883-8
[10] H¨ormander, L. (1960). Estimates for translation invariant operators in L p spaces.
[11] Hartley, R. V. (1942). A more symmetrical Fourier analysis applied to transmission problems. Proceedings of the IRE, 30(3), 144-150. DOI: https://doi.org/10.1109/JRPROC.1942.234333
[12] Matsuura, T., Saitoh, S., & Trong, D. D. (2005). Approximate and analytical inversion formulas in heat conduction on multidimensional spaces. DOI: https://doi.org/10.1163/156939405775297452
[13] McConnell, T. R. (1984). On Fourier multiplier transformations of Banach-valued functions. Transactions of the American Mathematical Society, 285(2), 739-757. DOI: https://doi.org/10.1090/S0002-9947-1984-0752501-X
[14] Mejjaoli, H. (2017). Harmonic analysis associated with the generalized differential-difference operator on the real line and quantitative uncertainty principles for its Hartley transform. Applicable Analysis, 96(7), 1146-1169. DOI: https://doi.org/10.1080/00036811.2016.1178723
[15] Mikhlin, S.G. On the multipliers of Fourier integrals. Dokl. Akad. Nauk SSSR 109 (1956),701-703
[16] ROSLER, MARGIT. Bessel-type signed hypergroups on R. Probability measures on groups and related structures XI. Proceedings, Oberwolfach, 1994, p. 292-304.
[17] Saitoh, S. (1983). Hilbert spaces induced by Hilbert space valued functions. Proceedings of the American Mathematical Society, 89(1), 74-78. DOI: https://doi.org/10.1090/S0002-9939-1983-0706514-9
[18] Saitoh, S.,Matsuura, T. & (2006). Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley–Wiener spaces. Applicable Analysis, 85(8), 901-915. DOI: https://doi.org/10.1080/00036810600643662
[19] Soltani, F. (2004). Lp-Fourier multipliers for the Dunkl operator on the real line. Journal of Functional Analysis, 209(1), 16-35. DOI: https://doi.org/10.1016/j.jfa.2003.11.009
[20] Soltani, F., & Maktouf, I. (2024). Dunkl–Weinstein Multiplier Operators and Applications to Reproducing Kernel Theory. Mediterranean Journal of Mathematics, 21(3), 80. DOI: https://doi.org/10.1007/s00009-024-02623-2
[21] Yakubovich, S. (2014). On the half-Hartley transform, its iteration and compositions with Fourier transforms. DOI: https://doi.org/10.1216/JIE-2014-26-4-581
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Ahmed Chana, Abdellatif Akhlidj
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
