Abstract
We extend the special affine Fourier transform in the context of quaternion valued functions and study its properties including an uncertainty principle. The same transform is studied on a suitably constructed Boehmian space.
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Abe, S., Sheridan, J.T.: Generalization of the fractional Fourier transformation to an arbitrary linear lossless transformation: an operator approach. J. Phys. 27, 4179–4187 (1994)
Abe, S., Sheridan, J.T.: Optical operations on wave functions as the abelian subgroups of the special affine Fourier transformation. Opt. Lett. 19, 1801–1803 (1994)
Abe, S., Sheridan, J.T.: Almost Fourier and almost Fresnel transformations. Opt. Commun. 113, 385–388 (1995)
Akila, L., Roopkumar, R.: A natural convolution of quaternion valued functions and its applications. Appl. Math. Comput. 242, 633–642 (2014)
Akila, L., Roopkumar, R.: Multidimensional quaternionic Gabor transforms. Adv. Appl. Clifford Algebras 25, 771–1002 (2016)
Atanasiu, D., Mikusiński, P.: On the Fourier transform, Boehmians, and distributions. Colloq. Math. 108, 263–276 (2007)
Arteaga, C., Marrero, I.: The Hankel transform of tempered Boehmians via the exchange property. Appl. Math. Comput. 219, 810–818 (2012)
Bhandari, A., Zayed, A.I.: Convolution and product theorems for the special affine Fourier transform. In: Nashed, M.Z., Li, X. (eds.) Frontiers in Orthogonal Polynomials and q-Series, pp. 119–137. World Scientific, Singapore (2018)
Bhandari, A., Zayed, A.I.: Shift-invariant and sampling spaces associated with the special affine Fourier transform. Appl. Comput. Harmon. Anal. 47, 30–52 (2019)
Cai, L.Z.: Special affine fractional Fourier transformation in frequency domain. Opt. Commun. 185, 271–276 (2000)
El Haoui, Y., Hitzer, E.: Generalized uncertainty principles associated with the quaternionic offset linear canonical transform. Complex Var. Elliptic Equ. (2021). https://doi.org/10.1080/17476933.2021.1916919
Ell, T.A.: Quaternion Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems. In: Proceedings of the 32nd Conference on Decision Control, vol. 1, pp. 1830–1841 (1993)
Ganesan, C., Roopkumar, R.: On generalizations of Boehmians and Hartley transform. Mat. Vesnik. 69, 133–143 (2017)
He, J., Yu, B.: Continuous wavelet transforms on the space \(L^2(R, H, dx)\). Appl. Math. Lett. 17, 111–121 (2004)
Hitzer, E.M.S.: Directional uncertainty principle for quaternion Fourier transform. Adv. Appl. Clifford Algebras 20, 271–284 (2010)
James, D.F.V., Agarwal, G.S.: The generalized Fresnel transform and its applications to optics. Opt. Commun. 126, 207–212 (1996)
Karunakaran, V., Kalpakam, N.V.: Hilbert transform for Boehmians. Integral Transforms Spec. Funct. 9, 19–36 (2000)
Mikusiński, P.: Convergence of Boehmians. Jpn. J. Math. 9, 159–179 (1983)
Mikusiński, P.: On harmonic Boehmians. Proc. Am. Math. Soc. 106, 447–449 (1989)
Mikusiński, P.: On flexibility of Boehmians. Integral Transforms Spec. Funct. 7, 299–312 (1996)
Nemzer, D.: Integrable Boehmians, Fourier transforms, and Poisson’s summation formula. Appl. Anal. Disc. Math. 1, 172–183 (2007)
Nemzer, D.: Quasi-asymptotic behavior of Boehmians. Novi Sad J. Math. 46, 87–102 (2016)
Nemzer, D.: Extending the Stieltjes transform. Sarajevo J. Math. 10, 197–208 (2014)
Moshinsky, M., Quesne, C.: Linear canonical transformations and their unitary representations. J. Math. Phys. 12, 1772–1780 (1971)
Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. IMA J. Appl. Math. 25, 241–265 (1980)
Pei, S.C., Ding, J.J.: Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms. J. Opt. Soc. Am. A 20, 522–532 (2003)
Roopkumar, R.: Quaternionic one-dimensional fractional Fourier transform. Optik 127, 11657–11661 (2016)
Roopkumar, R.: Ripplet transform and its extension to Boehmians. Georgian Math. J. 27, 149–156 (2017)
Roopkumar, R.: Quaternionic fractional Fourier transform for Boehmians. Ukr. J. Math. 72, 942–952 (2020)
Sangwine, S.J.: Fourier transforms of colour images using quaternion or hypercomplex numbers. Electron. Lett. 32, 1979–1980 (1996)
Sangwine, S.J.: Colour image edge detector based on quaternion convolution. Electron. Lett. 34, 969–971 (1998)
Shah, F.A., Tantary, A.Y.: Quaternionic shearlet transform. Optik 175, 115–125 (2018)
Sharma, K.K., Joshi, S.D.: Signal separation using linear canonical and fractional Fourier transforms. Opt. Commun. 265, 454–460 (2006)
Stern, A.: Sampling of compact signals in offset linear canonical transform domains. Signal Image Video Process. 1, 359–367 (2007)
Xiang, Q., Qin, K.: Convolution, correlation, and sampling theorems for the offset linear canonical transform. Signal Image Video Process. 8, 2385–2406 (2013)
Zayed, A.I.: A convolution and product theorem for the fractional Fourier transform. IEEE Signal Process. Lett. 5, 101–103 (1998)
Zemanian, A.H.: Generalized Integral Transformations. Wiley Inc., New York (1968)
Zhi, X., Wei, D., Zhang, W.: A generalized convolution theorem for the special affine Fourier transform and its application to filtering. Optik 127, 2613–2616 (2016)
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This article is part of the Topical Collection on Proceedings ICCA 12, Hefei, 2020, edited by Guangbin Ren, Uwe K\(\ddot{a}\)hler, Rafal Ablamowicz, Fabrizio Colombo, Pierre Dechant, Jacques Helmstetter, G. Stacey Staples, Wei Wang.
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Roopkumar, R. One-Dimensional Quaternionic Special Affine Fourier Transform. Adv. Appl. Clifford Algebras 31, 73 (2021). https://doi.org/10.1007/s00006-021-01174-z
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DOI: https://doi.org/10.1007/s00006-021-01174-z