Hermitian Clifford-Hermite wavelets: an alternative approach

Abstract

Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis, which has proven to be an appropriate framework for developing a higher dimensional continuous wavelet transform theory. In this setting a very specific construction of the wavelets has been established, encompassing all dimensions at once as opposed to the usual tensorial approaches, and being based on generalizations to higher dimension of classical orthogonal polynomials on the real line, such as the radial Clifford--Hermite polynomials, leading to Clifford--Hermite wavelets. More recently, Hermitian Clifford analysis has emerged as a new and successful branch of Clifford analysis, offering yet a refinement of the orthogonal case. In this new setting a Clifford--Hermite continuous wavelet transform has already been introduced in earlier work, its norm preserving character however being expressed in terms of suitably adapted scalar valued inner products on the respective $L_2$--spaces of signals and of transforms involved. In this contribution we present an alternative Hermitian Clifford--Hermite wavelet theory with Clifford algebra valued inner products, based on an orthogonal decomposition of the space of square integral functions, which is obtained by introducing a new Hilbert transform in the Hermitian setting.