Dirichlet space

In mathematics, the Dirichlet space on the domain

(named after Peter Gustav Lejeune Dirichlet), is the reproducing kernel Hilbert space of holomorphic functions, contained within the Hardy space

, for which the Dirichlet integral, defined by is finite (here dA denotes the area Lebesgue measure on the complex plane

The latter is the integral occurring in Dirichlet's principle for harmonic functions.

The Dirichlet integral defines a seminorm on

It is not a norm in general, since

whenever f is a constant function.

, we define This is a semi-inner product, and clearly

We may equip

with an inner product given by where

is the usual inner product on

The corresponding norm

is given by Note that this definition is not unique, another common choice is to take

The Dirichlet space is not an algebra, but the space

is a Banach algebra, with respect to the norm

We usually have

(the unit disk of the complex plane

contains all the polynomials and, more generally, all functions

, holomorphic on

The reproducing kernel of

is given by This mathematical analysis–related article is a stub.

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