The two-dimensional Clifford-Fourier transform. (English) Zbl 1122.42016
As well known the classical uni-dimensional Fourier transform is a very powerful tool which is very useful in Analysis as well as in many other areas of mathematics. It is also used in many practical applications like information storage procedures, for instance.
There have been several generalizations of the classical Fourier transform to higher dimensions like the hypercomplex Fourier or the Quaternionic Fourier transform which were introduced by Bülow and Sommer. One type of these generalizations is formed by the Clifford Fourier transforms, which are based on Clifford algebra on \(\mathbb R^m\) and Clifford analysis, whose basic notions are briefly and neatly introduced in the paper under review. One of the main difficulties of the study of these generalizations is the non-commutativity of Clifford Algebra and finding a kernel that allow one to work with the transform to extend the results of the uni-dimensional classical Fourier Transform.
The authors of the present paper in [“The Clifford-Fourier transform”, J. Fourier Anal. Appl. 11, No. 6, 669–681 (2005; Zbl 1122.42015)] introduced one of these types of Clifford transforms. This transform can be defined by \[ (\mathcal F f)(y)={1\over 2\pi} \int_{\mathbb R^2} e^{x \wedge y} f(x)\,dA(x), \] where \(f\) is an integrable complex function on \(\mathbb R^2\). Originally, it was defined by decomposing the uni-dimensional Fourier transform a an exponential of a scalar operator involving the Laplacian. All these facts are clearly presented in the paper under review. Indeed, the concepts are motivated and kindly introduced so that even someone who is not familiar with the subject can read the paper. Furthermore, a deep study of their Clifford Fourier transform is done. In particular, thanks to the knowledge of the kernel of the transform, they are able to show that the classical results, like the inversion and the convolution theorem, extend to their Clifford Fourier transform. Also, they study other properties like relating their transform to the standard tensorial Fourier transform. The authors thus propose their generalization for higher dimensions. They compute Clifford Fourier transform of the characteristic function and give some application to the vector field analysis signals.
It is quite sure that the results will be very useful in future works on different areas.
There have been several generalizations of the classical Fourier transform to higher dimensions like the hypercomplex Fourier or the Quaternionic Fourier transform which were introduced by Bülow and Sommer. One type of these generalizations is formed by the Clifford Fourier transforms, which are based on Clifford algebra on \(\mathbb R^m\) and Clifford analysis, whose basic notions are briefly and neatly introduced in the paper under review. One of the main difficulties of the study of these generalizations is the non-commutativity of Clifford Algebra and finding a kernel that allow one to work with the transform to extend the results of the uni-dimensional classical Fourier Transform.
The authors of the present paper in [“The Clifford-Fourier transform”, J. Fourier Anal. Appl. 11, No. 6, 669–681 (2005; Zbl 1122.42015)] introduced one of these types of Clifford transforms. This transform can be defined by \[ (\mathcal F f)(y)={1\over 2\pi} \int_{\mathbb R^2} e^{x \wedge y} f(x)\,dA(x), \] where \(f\) is an integrable complex function on \(\mathbb R^2\). Originally, it was defined by decomposing the uni-dimensional Fourier transform a an exponential of a scalar operator involving the Laplacian. All these facts are clearly presented in the paper under review. Indeed, the concepts are motivated and kindly introduced so that even someone who is not familiar with the subject can read the paper. Furthermore, a deep study of their Clifford Fourier transform is done. In particular, thanks to the knowledge of the kernel of the transform, they are able to show that the classical results, like the inversion and the convolution theorem, extend to their Clifford Fourier transform. Also, they study other properties like relating their transform to the standard tensorial Fourier transform. The authors thus propose their generalization for higher dimensions. They compute Clifford Fourier transform of the characteristic function and give some application to the vector field analysis signals.
It is quite sure that the results will be very useful in future works on different areas.
Reviewer: Alfonso Montes-Rodriguez (Sevilla)
MSC:
| 42C15 | General harmonic expansions, frames |
| 30G35 | Functions of hypercomplex variables and generalized variables |
Citations:
Zbl 1122.42015References:
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