Quaternionic fractional Fourier transform for Boehmians

  • R. Roopkumar Central Univ. Tamil Nadu, Thiruvarur, India
Keywords: Fractional Fourier transform, quaternion valued functions, convolution, Boehmians

Abstract

UDC 517.9

We construct a Boehmian space of quaternion valued functions using the quaternionic fractional convolution. Applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the context of Boehmians and its properties are established.

References

L. Akila, R. Roopkumar, A natural convolution of quaternion valued functions and its applications, Appl. Math. and Comput., 242, No 1, 633 – 642 (2014) https://doi.org/10.1016/j.amc.2014.06.007 DOI: https://doi.org/10.1016/j.amc.2014.06.007

L. Akila, R. Roopkumar, Ridgelet transform on quaternion valued functions, Int. J. Wavelets Multiresolut. Inf. Process., 14, No 1 (2016), 18 p. https://doi.org/10.1142/S0219691316500065 DOI: https://doi.org/10.1142/S0219691316500065

L. Akila, R. Roopkumar, Multidimensional quaternionic Gabor transforms, Adv. Appl. Clifford Algebras, 25 , 771 – 1002 (2016) https://doi.org/10.1007/s00006-015-0634-x DOI: https://doi.org/10.1007/s00006-015-0634-x

L. Akila, R. Roopkumar, Quaternionic Stockwell transform, Integral Transforms and Spec. Funct., 27 , No 6, 484 – 504 (2016) https://doi.org/10.1080/10652469.2016.1155570 DOI: https://doi.org/10.1080/10652469.2016.1155570

L. Akila, R. Roopkumar, Quaternionic curvelet transform, Optik, 131, 255 – 266 (2017). DOI: https://doi.org/10.1016/j.ijleo.2016.11.011

L. B. Almeida, The fractional order Fourier transform and time-frequency representations, IEEE Trans. Signal Process., 42 , No 11, 3084 – 3091 (1994).

L. B. Almeida, Product and convolution theorems for the fractional Fourier transform, IEEE Signal Process. Lett., 4 , No 1, 15 – 17 (1997).

C. Arteaga, I. Marrero, The Hankel transform of tempered Boehmians via the exchange property, Appl. Math. and Comput., 219, 810 – 818 (2012) https://doi.org/10.1016/j.amc.2012.06.043 DOI: https://doi.org/10.1016/j.amc.2012.06.043

F. Brackx, E. Hitzer, S. Sangwine, History of quaternion and Clifford – Fourier transforms and wavelets, Quaternion and Clifford Fourier Transforms and Wavelets, Trends Math., 27 , 11 – 27 (2013) DOI: https://doi.org/10.1007/978-3-0348-0603-9

T. Bulo ̈w, Hypercomplex spectral signal representations for the processing and analysis of images, Ph. D. thesis, Christian-Albrechts-Univ. zu Kiel (1999)

C. Ganesan, R. Roopkumar, Convolution theorems for fractional Fourier cosine and sine transforms and their extensions to Boehmians>, Commun. Korean Math. Soc., 31 , No 4, 791 – 809 (2016) https://doi.org/10.4134/CKMS.c150244 DOI: https://doi.org/10.4134/CKMS.c150244

C. Ganesan, R. Roopkumar, On generalizations of Boehmian space and Hartley transform, Mat. Vesnik, 69 , 133 – 143 (2017) https://doi.org/10.1111/jere.12167 DOI: https://doi.org/10.1111/jere.12167

X. Guanlei, W. Xiaotong, X. Xiaogang, Fractional quaternion Fourier transform, Signal Processing, 88 , No 10, 2511 – 2517 (2008)

J. He, B. Yu, Continuous wavelet transforms on the space L2(R; H; dx), Appl. Math. Lett., 17 , 111 – 121 (2004) https://doi.org/10.1016/S0893-9659(04)90021-3 DOI: https://doi.org/10.1016/S0893-9659(04)90021-3

E. M. S. Hitzer, Quaternion Fourier transform on quaternion elds and generalizations, Adv. Appl. Clifford Algebras, 17 , No 3, 497 – 517 (2007) https://doi.org/10.1007/s00006-007-0037-8 DOI: https://doi.org/10.1007/s00006-007-0037-8

E. Hitzer, S. Sangwine, The orthogonal 2D planes split of quaternions and steerable quaternion Fourier transformations, Quaternion and Clifford Fourier Transforms and Wavelets, Trends Math., Birkha ̈user, Basel (2013) https://doi.org/10.1007/978-3-0348-0603-9_2 DOI: https://doi.org/10.1007/978-3-0348-0603-9_2

X.-X. Hu, K. I. Kou, Quaternion Fourier and linear canonical inversion theorems, Math. Methods Appl. Sci., 40 , No 7, 2421 – 2440 (2017) https://doi.org/10.1002/mma.4148 DOI: https://doi.org/10.1002/mma.4148

V. Karunakarn, R. Roopkumar, Ultra Boehmians and their Fourier transforms, Fract. Calc. and Appl. Anal., 5 , No 2, 181 – 194 (2002).

V. Karunakaran, C. Prasanna Devi, The Laplace transform on a Boehmian space, Ann. Polon. Math., 97 , 151 – 157 (2010) https://doi.org/10.4064/ap97-2-4 DOI: https://doi.org/10.4064/ap97-2-4

Y. F. Luchko, H. Mart ́inez, J. J. Trujillo, Fractional Fourier transform and some of its applications, Fract. Calc. and Appl. Anal., 11 , No 4, 457 – 470 (2008).

A. C. McBride, Fractional calculus and integral transforms of generalised functions, Pitman Publ., London (1979) iv+179 pp. ISBN: 0-273-08415-1

A. C. McBride, F. H. Kerr, On Namias’s fractional Fourier transforms, IMA J. Appl. Math., 3 9 , No 2, 159 – 175 (1987) https://doi.org/10.1093/imamat/39.2.159 DOI: https://doi.org/10.1093/imamat/39.2.159

P. Mikusinski, Convergence of Boehmians, Japan. J. Math., 9 , 159–179 (1983) https://doi.org/10.4099/math1924.9.159 DOI: https://doi.org/10.4099/math1924.9.159

P. Mikusin ́ski, On exibility of Boehmians, Integral Transforms Spec. Funct., 7 , 299–312 (1996)

D. Mustard, The fractional Fourier transform and the Wigner distribution, J. Aust. Math. Soc., Ser. B, 38 , 209 – 219 (1996) https://doi.org/10.1017/S0334270000000606 DOI: https://doi.org/10.1017/S0334270000000606

V. Namias, The fractional order Fourier transform and its application to quantum mechanics, IMA J. Appl. Math., 25 , No 3, 241 – 265 (1980)

D. Nemzer, Extending the Stieltjes transform, Sarajevo J. Math., 10 , 197 – 208 (2014) https://doi.org/10.5644/SJM.10.2.06 DOI: https://doi.org/10.5644/SJM.10.2.06

D. Nemzer, Extending the Stieltjes transform II, Fract. Calc. and Appl. Anal., 17 , 1060 – 1074 (2014) https://doi.org/10.2478/s13540-014-0214-0 DOI: https://doi.org/10.2478/s13540-014-0214-0

H. M. Ozaktas, D. Mendlovic, Fourier transforms of fractional order and their optical interpretation, Opt. Commun., 101 , 163 – 169 (1993) DOI: https://doi.org/10.1016/0030-4018(93)90359-D

H. M. Ozaktas, D. Mendlovic, Fractional Fourier optics, J. Opt. Soc. Amer. A, 12 , 743 – 751 (1995) DOI: https://doi.org/10.1364/JOSAA.12.000743

R. Roopkumar, On extension of Gabor transform to Boehmians, Mat. Vesnik, 65 , 431 – 444 (2013)

R. Roopkumar, Quaternionic one-dimensional fractional Fourier transform, Optik, 127 , 11657 – 11661 (2016) DOI: https://doi.org/10.1016/j.ijleo.2016.09.069

R. Roopkumar, E. R. Negrin, Poisson transform on Boehmians, Appl. Math. and Comput., 216 , 2740 – 2748 (2010) https://doi.org/10.1016/j.amc.2010.03.122 DOI: https://doi.org/10.1016/j.amc.2010.03.122

R. Roopkumar, E. R. Negrin, A uni ed extension of Stieltjes and Poisson transforms to Boehmians, Integral Transforms Spec. Funct., 22 , No 3, 195 – 206 (2011) https://doi.org/10.1080/10652469.2010.511208 DOI: https://doi.org/10.1080/10652469.2010.511208

E. Sejdic ́, I. Djurovic ́, L. Stankovic ́, Fractional Fourier transform as a signal processing tool: an overview of recent developments, Signal Processing, 91 , No 6, 1351 – 1369 (2011).

R. Subash Moorthy, R. Roopkumar, Curvelet transform for Boehmians, Arab J. Math. Sci., 20 , 264 – 279 (2014) https://doi.org/10.1016/j.ajmsc.2013.10.001 DOI: https://doi.org/10.1016/j.ajmsc.2013.10.001

K. Viswanath, Normal operations on quaternionic Hilbert spaces, Trans. Amer. Math. Soc., 162 , 337 – 350 (1971) https://doi.org/10.2307/1995758 DOI: https://doi.org/10.2307/1995758

D. Wei, Y. Li, Different forms of Plancherel theorem for fractional quaternion Fourier transform, Optik, 124 , No 24, 6999 – 7002 (2013) https://doi.org/10.1007/978-3-0348-0603-9_1 DOI: https://doi.org/10.1007/978-3-0348-0603-9_1

A. I. Zayed, A convolution and product theorem for the fractional Fourier transform, IEEE Signal Proc. Lett., 5 , No 4, 101 – 103 (1998) https://doi.org/10.1109/78.984750 DOI: https://doi.org/10.1109/78.984750

A. I. Zayed, Fractional Fourier transforms of generalized functions, Integral Transforms Spec. Funct., 7 , 299 – 312 (1998) https://doi.org/10.1080/10652469808819206 DOI: https://doi.org/10.1080/10652469808819206

Published
17.06.2020
How to Cite
Roopkumar, R. “Quaternionic Fractional Fourier Transform for Boehmians”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 6, June 2020, pp. 812-21, doi:10.37863/umzh.v72i6.649.
Section
Research articles