Dirichlet space

In mathematics, the Dirichlet space on the domain $\Omega \subseteq \mathbb{C}, \, \mathcal{D}(\Omega)$ (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space $H^2(\Omega)$ , for which the Dirichlet integral, defined by

is finite (here dA denotes the area Lebesgue measure on the complex plane $\mathbb{C}$ ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on $\mathcal{D}(\Omega)$ . It is not a norm in general, since $\mathcal{D}(f) = 0$ whenever f is a constant function.

单词	Dirichlet space
释义	Dirichlet space 英语百科 Dirichlet space In mathematics, the Dirichlet space on the domain $\Omega \subseteq \mathbb{C}, \, \mathcal{D}(\Omega)$ (named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space $H^2(\Omega)$ , for which the Dirichlet integral, defined by is finite (here dA denotes the area Lebesgue measure on the complex plane $\mathbb{C}$ ). The latter is the integral occurring in Dirichlet's principle for harmonic functions. The Dirichlet integral defines a seminorm on $\mathcal{D}(\Omega)$ . It is not a norm in general, since $\mathcal{D}(f) = 0$ whenever f is a constant function.
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