This paper is devoted to the establishment of two-dimensional sampling theorems for discrete transforms, whose kernels arise from second order partial difference equations. We define a discrete type partial difference operator and investigate its spectral properties. Green’s function is constructed and kernels that generate orthonormal basis of eigenvectors are defined. A discrete Kramer-type lemma is introduced and two sampling theorems of Lagrange interpolation type are proved. Several illustrative examples are depicted. The theory is extendible to higher order settings.

中文翻译：

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多元离散变换的多维采样定理

本文致力于离散变换的二维采样定理的建立，其离散核是由二阶偏差分方程产生的。我们定义了一个离散类型的偏差算子，并研究了它的光谱特性。构造了格林函数，并定义了生成特征向量正交基础的内核。介绍了一个离散的Kramer型引理，并证明了两个Lagrange插值型采样定理。描绘了几个说明性示例。该理论可扩展到更高阶的设置。