Two-index Clifford–Hermite polynomials with applications in wavelet analysis

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.