Discrete weierstrass transform in discrete hermitian clifford analysis

• Published in: Applied mathematics and computation 2021-02, Vol.391, p.125641, Article 125641
• Main Authors: Massé, A., Sommen, F., De Ridder, H., Raeymaekers, T.
• Format: Article
• Language: English
• Subjects: Delta functions; Discrete clifford analysis; Hermite polynomials; Weierstrass space; Weierstrass transform
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2020.125641

The classical Weierstrass transform is an isometric operator mapping elements of the weighted L2−space L2(R,exp(−x2/2)) to the Fock space. It has numereous applications in physics and applied mathematics. In this paper, we define an analogue version of this transform in discrete Hermitian Clifford analysis, where functions are defined on a grid rather than the continuous space. This new transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. Furthermore, a discrete Weierstrass space with appropriate inner product is constructed, for which the discrete Hermite polynomials form a basis. In this setting, we also investigate the behaviour of the discrete delta functions and check if they are elements of this newly defined Weierstrass space.