上半格上离散全纯函数的Hardy空间
Hardy spaces of discrete holomorphic functions on the upper half-lattice
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中文总结 AI 辅助
该研究在正方形格点离散全纯性框架下，发展上半格离散全纯函数的Hardy空间\(H^p\)理论，证明相关公式与恒等式，描述再生核等，比较离散与经典理论，还证明对偶性等结果。
中文摘要 AI 辅助
我们在正方形格点上离散全纯性的经典框架内，发展了上半格上离散全纯函数的Hardy空间\(H^p\)理论。我们证明了柯西和泊松再生公式，建立了边界范数恒等式，并得到了这些空间的帕利-维纳型特征。在希尔伯特空间情形下，我们描述了相关的再生核和塞戈投影，并通过经典\(H^2\)函数的一族离散全纯逼近，将离散理论与上半平面上的经典Hardy空间进行比较。我们还证明了\(1<p<\infty\)时\(H^p\)的对偶性结果，建立了水平线上的唯一性和采样结果，并引入了伯格曼型空间，比较了两种自然加权尺度。
英文摘要
We develop a theory of Hardy spaces $H^p$ of discrete holomorphic functions on the upper half-lattice, within the classical framework of discrete holomorphicity on the square lattice. We prove Cauchy and Poisson reproducing formulas, establish a boundary norm identity, and obtain Paley--Wiener type characterizations for these spaces. In the Hilbert space case, we describe the associated reproducing kernel and Szegő projection, and we compare the discrete theory with the classical Hardy space on the upper half-plane through a family of discrete holomorphic approximants of classical $H^2$-functions. We also prove duality results for $H^p$, $1<p<\infty$, establish uniqueness and sampling results on horizontal lines, and introduce Bergman-type spaces, comparing two natural weighted scales.
