Abstract The classical Weierstrass transform is an isometric operator mapping elements of the weighted L 2 − space L 2 ( R , exp ( − x 2 / 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.

中文翻译：

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离散厄米克崖分析中的离散魏尔斯特拉斯变换

摘要 经典的 Weierstrass 变换是将加权 L 2 − 空间 L 2 ( R , exp ( − x 2 / 2 ) ) 的元素映射到 Fock 空间的等距算子。它在物理学和应用数学中有许多应用。在本文中，我们在离散 Hermitian Clifford 分析中定义了这种变换的模拟版本，其中函数在网格上而不是在连续空间上定义。这种新变换基于经典定义，结合高斯函数的离散版本和经典 Hermite 多项式的离散对应项。此外，还构建了具有适当内积的离散 Weierstrass 空间，离散 Hermite 多项式构成了该空间的基。在这个设定中，