Dirichlet Space

In mathematics, the Dirichlet space on the domain $\Omega \subseteq \mathbb, \, \mathcal(\Omega)$ (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

:$\mathcal(f) := \iint_\Omega , f^\prime(z), ^2 \, dA = \iint_\Omega , \partial_x f, ^2 + , \partial_y f, ^2 \, dx \, dy$

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

For $f,\, g \in \mathcal(\Omega)$, we define

:$\mathcal(f, \, g) : = \iint_\Omega f'(z) \overline \, dA(z).$

This is a semi-inner product, and clearly $\mathcal(f, \, f) = \mathcal(f)$. We may equip $\mathcal(\Omega)$ with an inner product given by

:$\langle f, g \rangle_ := \langle f, \, g \rangle_ + \mathcal(f, \, g) \; \; \; \; \; (f, \, g \in \mathcal(\Omega)),$

where $\langle \cdot, \, \cdot \rangle_$ is the usual inner product on $H^2 (\Omega).$ The corresponding norm $\, \cdot \, _$ is given by

:$\, f\, ^2_ := \, f\, ^2_ + \mathcal(f) \; \; \; \; \; (f \in \mathcal (\Omega)).$

Note that this definition is not unique, another common choice is to take $\, f\, ^2 = , f(c), ^2 + \mathcal(f)$, for some fixed $c \in \Omega$.

The Dirichlet space is not an algebra, but the space $\mathcal(\Omega) \cap H^\infty(\Omega)$ is a Banach algebra, with respect to the norm

:$\, f\, _ := \, f\, _ + \mathcal(f)^ \; \; \; \; \; (f \in \mathcal(\Omega) \cap H^\infty(\Omega)).$

We usually have $\Omega = \mathbb$ (the unit disk of the complex plane $\mathbb$), in that case $\mathcal(\mathbb):=\mathcal$, and if

:$f(z) = \sum_ a_n z^n \; \; \; \; \; (f \in \mathcal),$

then

:$D(f) =\sum_ n , a_n, ^2,$

and

:$\, f \, ^2_\mathcal = \sum_ (n+1) , a_n, ^2.$

Clearly, $\mathcal$ contains all the polynomials and, more generally, all functions $f$, holomorphic on $\mathbb$ such that $f'$ is bounded on $\mathbb$.

The reproducing kernel of $\mathcal$ at $w \in \mathbb \setminus \$ is given by

:$k_w(z) = \frac \log \left( \frac \right) \; \; \; \; \; (z \in \mathbb \setminus \).$

See also

*Banach space
* Bergman space
*Hardy space
*Hilbert space

References

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Complex analysis
Functional analysis
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