Quaternionic fractional Fourier transform for Boehmians

  • R. Roopkumar Central Univ. Tamil Nadu, Thiruvarur, India
Ключові слова: дробове перетворення Фур’є, бьомiани, згортка, функції у кватернiонах

Анотація

УДК 517.9

За допомогою кватернiонної дробової згортки побудовано бьомiанiв простiр функцiй iз значеннями у кватернiонах. Застосовуючи теорему про згортку, ми поширюємо кватернiонне дробове перетворення Фур’є на бьомiанiв простiр та встановлюємо його властивостi.

Посилання

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

Опубліковано
17.06.2020
Як цитувати
RoopkumarR. «Quaternionic Fractional Fourier Transform for Boehmians». Український математичний журнал, вип. 72, вип. 6, Червень 2020, с. 812-21, doi:10.37863/umzh.v72i6.649.
Розділ
Статті