Clifford-Jacobi Polynomials and the Associated Continuous Wavelet Transform in Euclidean Space

Specific wavelet kernel functions for a continuous wavelet transform in Euclidean space are presented within the framework of Clifford analysis. These multi-dimensional wavelets are constructed by taking the Clifford-monogenic extension to Rm+1 of specific functions in R-m generalizing the traditional Jacobi weights. The notion of Clifford-monogenic function is a direct higher dimensional generalization of that of holomorphic function in the complex plane. Moreover, crucial to this construction is the orthogonal decomposition of the space of square integrable functions into the Hardy space H-2(R-m) and its orthogonal complement. In this way a nice relationship is established between the theory of the Clifford Continuous Wavelet Transform on the one hand, and the theory of Hardy spaces on the other hand. Furthermore, also new multi-dimensional polynomials, the so-called Clifford-Jacobi polynomials, are obtained