Research PaperNo Access

Paley–Wiener-type theorems for the canonical Fourier–Bessel transform

Minggang Fei

https://orcid.org/0000-0002-2205-5112

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China

E-mail Address: fei@uestc.edu.cn

Corresponding author.

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and

Yuanyuan Si

School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China

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https://doi.org/10.1142/S0219691324500449Cited by:0 (Source: Crossref)

Abstract

The aim of this paper is to establish several versions of Paley–Wiener-type theorems for the canonical Fourier–Bessel transform, which is a Bessel type of linear canonical transform indexed by a matrix parameter m $\in S L (2, ℝ)$ .

Keywords:

AMSC: 33C52, 42B10

References

1. N. B. Andersen , Real Paley–Wiener theorems, Bull. London Math. Soc. 36(4) (2004) 504–508. Crossref, Google Scholar
2. N. B. Andersen , Real Paley–Wiener theorems for the Dunkl transform on $ℝ$ , Integral Transforms Spec. Funct. 17 (2006) 543–547. Crossref, Web of Science, Google Scholar
3. N. B. Andersen and M. de Jeu , Real Paley–Wiener theorems and local spectral radius formulas, Trans. Amer. Math. Soc. 362 (2010) 3613–3640. Crossref, Web of Science, Google Scholar
4. H. H. Bang , A property of infinitely differentiable functions, Proc. Amer. Math. Soc. 108 (1990) 73–76. Crossref, Web of Science, Google Scholar
5. H. H. Bang , Functions with bounded spectrum, Trans. Amer. Math. Soc. 347 (1995) 1067–1080. Crossref, Web of Science, Google Scholar
6. B. Barshan, M. Kutay and H. Ozaktas , Optimal filtering with linear canonical transformations, Opt. Commun. 135 (1997) 32–36. Crossref, Web of Science, Google Scholar
7. A. Bultheel and H. Martinez-Sulbaran , Recent developments in the theory of the fractional Fourier and linear canonical transforms, Bull. Belg. Math. Soc. Simon Stevin 13 (2007) 971–1005. Crossref, Web of Science, Google Scholar
8. Q. H. Chen, L. Q. Li and G. B. Ren , Generalized Paley–Wiener theorems, Int. J. Wavelets Multiresolut. Inf. Process. 10(2) (2012) 1250020. Link, Web of Science, Google Scholar
9. S. A. Collins , Lens-system diffraction integral written in terms of matrix optics, J. Opt. Soc. Amer. 60 (1970) 1168–1177. Crossref, Web of Science, Google Scholar
10. L. Dhaouadi, J. Sahbani and A. Fitouhi , Harmonic analysis associated to the canonical Fourier–Bessel transform, Integral Transforms Spec. Funct. 32(4) (2021) 290–315. Crossref, Web of Science, Google Scholar
11. M. Fei and Q. Li , Real Paley–Wiener theorems for the special relativistic space-time Fourier transform, Int. J. Wavelets Multiresolut. Inf. Process. 20(6) (2022) 2250022. Link, Google Scholar
12. M. Fei, Y. Xu and J. Yan , Real Paley–Wiener theorem for the quaternion Fourier transform, Complex Var. Elliptic Equ. 62(8) (2017) 1072–1080. Crossref, Web of Science, Google Scholar
13. Y. X. Fu and L. Q. Li , Paley–Wiener and Boas theorems for the quaternion Fourier transform, Adv. Appl. Clifford Algebras 23 (2013) 837–848. Crossref, Web of Science, Google Scholar
14. Y. X. Fu and L. Q. Li , Real Paley–Wiener theorems for the Clifford Fourier transform, Sci. China Math. 57(11) (2014) 2381–2392. Crossref, Web of Science, Google Scholar
15. S. Ghazouani and J. Sahbani , Canonical Fourier–Bessel transform and their applications, J. Pseudo-Differ. Oper. Appl. 14(1) (2023) Article 3. Crossref, Web of Science, Google Scholar
16. S. Ghazouani, E. A. Soltani and A. Fitouhi , A unified class of integral transforms related to the Dunkl transform, J. Math. Anal. Appl. 449 (2017) 1797–1849. Crossref, Web of Science, Google Scholar
17. S. Li, J. Leng and M. Fei , Paley–Wiener-type theorems for the Clifford–Fourier transform, Math. Methods Appl. Sci. 42(18) (2019) 6101–6113. Crossref, Web of Science, Google Scholar
18. S. Li, J. Leng and M. Fei , Real Paley–Wiener theorems for the $(k, a)$ -generalized Fourier transform, Math. Methods Appl. Sci. 43(11) (2020) 6985–6994. Crossref, Google Scholar
19. Y. Li and G. Ren , Real Paley–Wiener theorem for the octonion Fourier transform, Math. Methods Appl. Sci. 47(10) (2024) 7902–7917. Crossref, Web of Science, Google Scholar
20. H. Mejjaoli and K. Trimèche , Spectrum of functions for the Dunkl transform on $ℝ^{d}$ , Fract. Calc. Appl. Anal. 10 (2007) 19–38. Google Scholar
21. M. Moshinsky and C. Quesne , Linear canonical transformations and their unitary representations, J. Math. Phys. 12 (1971) 1772–1780. Crossref, Web of Science, Google Scholar
22. Y. Othmani and K. Trimèche , Real Paley–Wiener theorems associated with the Weinstein operator, Mediterr. J. Math. 3 (2006) 105–118. Crossref, Web of Science, Google Scholar
23. H. Ozaktas, Z. Zalevsky and M. Kutay , The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001). Google Scholar
24. R. Paley and N. Wiener , The Fourier Transforms in the Complex Domain, Colloquium Publications Series, Vol. 19 (American Mathematical Society, Providence, RI, 1934). Google Scholar
25. S. Pei and J. Ding , Eigenfunctions of linear canonical transform, IEEE Trans. Signal Process. 50 (2002) 11–26. Crossref, Web of Science, Google Scholar
26. J. Sahbani , Quantitative uncertainty principles for the canonical Fourier–Bessel transform, Acta Math. Sin. (Engl. Ser.) 38(2) (2022) 331–346. Crossref, Google Scholar
27. J. Sahbani and T. Guettiti , Prolate spheroidal wave functions associated with the canonical Fourier–Bessel transform and uncertainty principles, Math. Methods Appl. Sci. 46(14) (2023) 15506–15525. Crossref, Web of Science, Google Scholar
28. E. M. Stein and G. Weiss , Introduction to Fourier Analysis on Euclidean Spaces (Princeton Univ. Press, Princeton, 1971). Google Scholar
29. V. K. Tuan , On the Paley–Wiener theorem, in Theory of Functions and Applications, Collection of Works dedicated to the Memory of Mkhitar M. Djrbashian (Louys Publishing House, Yerevan, 1995), pp. 193–196. Google Scholar
30. V. K. Tuan , Paley–Wiener-type theorems, Fract. Calc. Appl. Anal. 2(2) (1999) 135–144. Google Scholar
31. V. K. Tuan , On the supports of functions, Numer. Funct. Anal. Optim. 20(3–4) (1999) 387–394. Crossref, Web of Science, Google Scholar
32. K. B. Wolf , Canonical transform. I. Complex linear transforms, J. Math. Phys. 15 (1974) 1295–1301. Crossref, Google Scholar