Two-index Clifford-Hermite polynomials with applications in wavelet analysis |
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Authors: | F Brackx N De Schepper F Sommen |
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Institution: | Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, B-9000 Gent, Belgium |
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Abstract: | Clifford analysis may be regarded as a higher-dimensional analogue of the theory of holomorphic functions in the complex plane. It has proven to be an appropriate framework for higher-dimensional continuous wavelet transforms, based on specific types of multi-dimensional orthogonal polynomials, such as the Clifford-Hermite polynomials, which form the building blocks for so-called Clifford-Hermite wavelets, offering a refinement of the traditional Marr wavelets. In this paper, a generalization of the Clifford-Hermite polynomials to a two-parameter family is obtained by taking the double monogenic extension of a modulated Gaussian, i.e. the classical Morlet wavelet. The eventual goal being the construction of new Clifford wavelets refining the Morlet wavelet, we first investigate the properties of the underlying polynomials. |
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Keywords: | Hermite polynomials Clifford analysis Wavelet analysis |
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