New Type of Gegenbauer–Hermite Monogenic Polynomials and Associated Clifford Wavelets

Abstract

Nowadays, 3D images processing constitutes a challenging topic in many scientific fields such as medicine, computational physics, informatics. Therefore, the construction of suitable functional bases that allow computational aspects to be easily done is a necessity. Wavelets and Clifford algebras are ones of the most import mathematical tools for achieving such necessities. In the present work, new classes of wavelet functions are presented in the framework of Clifford analysis. Firstly, some classes of new monogenic polynomials are provided based on two-parameter weight functions. Such classes englobe the well-known Gegenbauer and Hermite ones. The discovered polynomial sets are next applied to introduce new wavelet functions. Reconstruction formula and Fourier–Plancherel rules have been proved. Computational concrete examples are developed by means of some illustrative examples with graphical representations of the Clifford mother wavelets in some cases. These discovered wavelets have been applied by the next for modeling some biomedical signals such as EEG, ECG and proteins.

Appendix

Proof of Lemma 2

Denote firstly by I the left-hand-side integral. Recall next that for $\lambda \in {\mathbb {R}}$, the Bessel function $J_{\lambda }$ may be written in the integral form

$$\begin{aligned} J_{\lambda }(x)=\dfrac{1}{2\pi }\displaystyle \int _{-\pi }^{\pi }e^{-i(\lambda t-xsint)}\mathrm{d}t. \end{aligned}$$

Next, as the measure $\mathrm{d}\sigma$ is invariant under rotations, we may assume without loss of the generality that $\xi =(1,0,\dots ,0)$. As a result, we obtain

$$\begin{aligned} I=\displaystyle \int _{S^{m-1}}e^{-ir\rho \cos \theta }\mathrm{d}\sigma (\omega ), \end{aligned}$$

where $\theta$ is the angle $(\omega ,e_1)$, with $e_1=(1,0,\dots ,0)$. In spherical coordinates, this means that

$$\begin{aligned} I=\omega _{m-1}\displaystyle \int _{0}^{\pi }e^{-ir\rho \cos \theta } \sin ^{m-2}\theta \,d\theta . \end{aligned}$$

Denoting next $t=\cos \theta$, we get

$$\begin{aligned} I=\omega _{m-1}\displaystyle \int _{-1}^{1}e^{-ir\rho t}(1-t^2)^{(m-3)/2}\mathrm{d}t. \end{aligned}$$

Observing next that the area $\omega _{m-1}$ of the unit sphere $S^{m-1}$ is

$$\begin{aligned} \omega _{m-1}=\dfrac{\pi ^{\frac{m-3}{2}}}{\varGamma (\frac{m-1}{2})} \end{aligned}$$

and in the other hand,

$$\begin{aligned} J_\lambda (x)=\dfrac{|\frac{x}{2}|^{\lambda }}{\pi ^{1/2} \varGamma (\lambda +1/2)}\displaystyle \int _{-1}^{1}e^{-ixt}(1-t^2)^{\lambda -1/2}\mathrm{d}t, \end{aligned}$$

we obtain the desired result. $\square$