Multidimensional quadratic-phase Fourier transform and its uncertainty principles

Year 2025, Volume: 8 Issue: 1, 15 - 34, 17.03.2025

Luís Pinheiro Castro , Rita Guerra

https://doi.org/10.33205/cma.1537893

Abstract

The main aim of this article is to propose a multidimensional quadratic-phase Fourier transform (MQFT) that generalises the well-known and recently introduced quadratic-phase Fourier transform (as well as, of course, the Fourier transform itself) to higher dimensions. In addition to the definition itself, some crucial properties of this new integral transform will be deduced. These include a Riemann-Lebesgue lemma for the MQFT, a Plancherel lemma for the MQFT and a Hausdorff-Young inequality for the MQFT. A second central objective consists of obtaining different uncertainty principles for this MQFT. To this end, using techniques that include obtaining various auxiliary inequalities, the study culminates in the deduction of $L^p$-type Heisenberg-Pauli-Weyl uncertainty principles and $L^p$-type Donoho-Stark uncertainty principles for the MQFT.

Keywords

multidimensional quadratic-phase Fourier transform, Donoho-Stark uncertainty principle, Heisenberg-Pauli-Weyl uncertainty principle, Riemann-Lebesgue lemma, Hausdorff-Young inequality

Project Number

UIDB/04106/2020

References

• E. M. Berkak, E. M. Loualid and R. Daher: Donoho-Stark theorem for the quadratic-phase Fourier integral operators, Methods Funct. Anal. Topology, 27 (4) (2021), 335–339.
• L. P. Castro, L. T. Minh and N. M. Tuan: New convolutions for quadratic-phase Fourier integral operators and their applications, Mediterr. J. Math., 15 (2018), Article ID: 13.
• L. P. Castro, M. R. Haque, M. M. Murshed, S. Saitoh and N. M. Tuan: Quadratic Fourier transforms, Ann. Funct. Anal., 5 (1) (2014), 10–23.
• X. Chen, P. Dang and W. Mai: Lp-theory of linear canonical transforms and related uncertainty principles, Math. Methods Appl. Sci., 47 (2024), 6272–6300.
• S. A. Collins Jr.: Lens-system diffraction integral written in terms of matrix optics, Journal of the Optical Society of America, 60 (1970), 1168–1177.
• M. G. Cowling, J. F. Price: Bandwidth versus time concentration: the Heisenberg–Pauli–Weyl inequality, SIAM J. Math. Anal., 15 (1) (1984), 151–165.
• A. H. Dar, M. Z. M. Abdalla, M. Y. Bhat and A. Asiri: New quadratic phaseWigner distribution and ambiguity function with applications to LFM signals, J. Pseudo-Differ. Oper. Appl., 15 (2) (2024), Article ID: 35.
• G. B. Folland: Real Analysis, Modern Techniques and Applications, 2nd Edition, John Wiley & Sons (1999).
• G. B. Folland, A. Sitaram: The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (3) (1997), 207–238.
• W. Heisenberg: Über den anschaulichen inhalt der quantentheoretischen kinematic und mechanik, Zeit. Für Physik, 43 (3–4) (1927), 172–198.
• M. Kumar, T. Pradhan: Quadratic-phase Fourier transform of tempered distributions and pseudo-differential operators, Integral Transforms Spec. Funct., 33 (6) (2022), 449–465.
• M. Moshinsky, C. Quesne: Linear canonical transformations and their unitary representations, J. Math. Phys., 12 (1971), 1772–1783.
• A. Prasad, T. Kumar and A. Kumar: Convolution for a pair of quadratic-phase Hankel transforms, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 114 (3) (2020), Article ID: 150.
• A. Prasad, P. B. Sharma: The quadratic-phase Fourier wavelet transform, Math. Methods Appl. Sci., 43 (4) (2020), 1953–1969.
• A. Prasad, P. B. Sharma: Pseudo-differential operator associated with quadratic-phase Fourier transform, Bol. Soc. Mat. Mex., 28 (2) (2022), Article ID: 37.
• F. A. Shah, W. Z. Lone: Quadratic-phase wavelet transform with applications to generalized differential equations, Math. Methods Appl. Sci., 45 (3) (2022), 1153–1175.
• P. B. Sharma: Quadratic-phase orthonormal wavelets, Georgian Math. J., 30 (4) (2023), 593–602.
• F. A. Shah, K. S. Nisar,W. Z. Lone and A. Y. Tantary: Uncertainty principles for the quadratic-phase Fourier transforms, Math. Methods Appl. Sci., 44 (13) (2021), 10416–10431.
• L. Tien Minh: Modified ambiguity function and Wigner distribution associated with quadratic-phase Fourier transform, J. Fourier Anal. Appl., 30 (1) (2024), Article ID: 6.
• H. Weyl: Gruppentheorie und Quantenmechanik, Hirzel, Leipzig (1931).
• N. Wiener: The Fourier Integral and Certain of its Applications, Cambridge University Press, Cambridge (1933).