CONVOLUTION THEOREMS FOR CLIFFORD FOURIER TRANSFORM AND PROPERTIES

Mawardi Bahri, Ryuichi Ashino, Rémi Vaillancourt


Abstract


The non-commutativity of the Clifford multiplication gives different aspects from the classical Fourier analysis.We establish main properties of convolution theorems for the Clifford Fourier transform. Some properties of these generalized convolutionsare extensions of the corresponding convolution theorems of the classical Fourier transform.

DOI : http://dx.doi.org/10.22342/jims.20.2.143.125-140


Keywords


Clifford convolution, Clifford algebra, Clifford Fourier transform

References


Mawardi, B. and Hitzer, E.,

Clifford Fourier transformation and uncertainty principle for

the Clifford geometric algebra Cl_{3,0},

Adv. Appl. Clifford Algebr., 16(1) (2006), 41--61.

Mawardi, B., Adji, S. and Zhao, J.,

Clifford algebra-valued wavelet transform on multivector fields,

Adv. Appl. Clifford Algebr., 21(1) (2011), 13--30.

Hitzer, E.,

Quaternion Fourier transform on quaternion fields and generalizations,

Adv. Appl. Clifford Algebr., 17(3) (2007), 497--517.

Hitzer, E. and Mawardi, B.,

Clifford Fourier transform on multivector fields and uncertainty principle

for dimensions n = 2 (mod 4) and n = 3 (mod 4),

Adv. Appl. Clifford Algebr., 18}(3-4) (2008), 715--736.

Brackx, F., Delanghe, R. and Sommen, F.,

Clifford Analysis,

Research Notes in Mathematics, 76.

Pitman, Boston, MA, 1982.

Sommen, F.,

A product and an exponential function in hypercomplex function theory,

Applicable Anal., 12(1) (1981), 13--26.

De Bie, H., De Schepper, N. and Sommen, F.,

The class of Clifford-Fourier transforms,

J. Fourier Anal. Appl., 17(6) (2011), 1198--1231.

De Bie, H. and De Schepper, N.,

The fractional Clifford-Fourier transforms,

Complex Anal. Oper. Theory, 6(5) (2012), 1047--1067.

Ebling, J. and Scheuermann, G.,

Clifford Fourier transform on vector fields,

IEEE Transactions on Visualization and Computer Graphics, 11(4) (2005), 469--479.

Bahri, M.,

Clifford windowed Fourier transform applied to linear time-varying systems,

Appl. Math. Sci. (Ruse), 6(58) (2012), 2857--2864.

Bracewell, R. N.,

The Fourier Transform and its Applications},

McGraw-Hill International Editions, Singapore, 2000.

Bujack, R., Scheuermann, G. and Hitzer, E.,

A general geometric Fourier transform convolution theorem,

Adv. Appl. Clifford Algebr., 23(1) (2013), 15--38.

Sangwine, S. J.,

Color images edge detector based on quaternion convolution,

Electronics Letters, 34(10) (1998), 969--971.

Batard, T., Berthier, M. and Saint-Jean, C.,

Clifford-Fourier transform for color image processing,

In: Geometric Algebra Computing in Engineering and Computer Science,

pp. 135--162, Springer London, 2010.

Fu, Y., Kahler, U. and Cerejeiras, P.,

The Balian-Low theorem for the windowed quaternionic Fourier transform,

Adv. Appl. Clifford Algebr., 22(4) (2012), 1025--1040.


Full Text: PDF

Refbacks

  • There are currently no refbacks.


Journal of the Indonesian Mathematical Society ( p-ISSN:2086-8952 | e-ISSN:2460-0245) published by the Indonesian Mathematical Society (IndoMS).

Indexed by:

logo DOAJLogo SintaThe Indonesian Publication Index-Portal Garuda Google Scholar logo zbMath Logo AMSLogo CrossrefLogo Thomson Reuters

Visitor Number : web statistics View My Stats


Creative Commons License
Journal of the Indonesian Mathematical Society by http://jims-a.org/index.php/jimsa is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

p-ISSN:2086-8952e-ISSN:2460-0245