Discrete function theory in higher-dimensional setting has been in active development since many years. However, available results focus on studying discrete setting for such canonical domains as half-space, while the case of bounded domains generally remained unconsidered. Therefore, this paper presents the extension of the higher-dimensional function theory to the case of arbitrary bounded domains in \({\mathbb {R}}^{n}\). On this way, discrete Stokes’ formula, discrete Borel–Pompeiu formula, as well as discrete Hardy spaces for general bounded domains are constructed. Finally, several discrete Hilbert problems are considered.

中文翻译：

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$$ {\ mathbb {R}} ^ {n} $$ R n中有界域的离散Hardy空间

高维环境中的离散函数理论多年来一直在积极发展。但是，可用的结果集中在研究诸如半空间之类的规范域的离散设置，而有界域的情况通常仍未考虑。因此，本文提出了将高维函数理论扩展到\（{\ mathbb {R}} ^ {n} \）中任意有界域的情况。这样，就构造了离散的Stokes公式，离散的Borel–Pompeiu公式以及用于一般有界域的离散Hardy空间。最后，考虑了几个离散的希尔伯特问题。