Analytic functional

An element $f$ of the space $H'(\Omega)$, the dual of the space $H(\Omega)$ of analytic functions defined on an open subset $\Omega$ of $\mathbf C^n$, i.e. a functional on $H(\Omega)$. Thus, a distribution with compact support is an analytic functional. There exists a compact set $K\subset\Omega$, said to be the support of the analytic functional $f$, on which $f$ is concentrated: For any open set $\omega\supset K$ the functional $f$ can be extended to $H(\omega)$ so that for all $u\in H(\Omega)$ the following inequality is valid:

$$|f(u)|\leq C_\omega\sup|u|,$$

where $C_\omega$ is a constant depending on $\omega$. There exists a measure $\mu$ with support in $K$ such that

$$f(u)=\int\limits_\omega ud\mu.$$

An analytic functional is defined in a similar manner on a space of real-valued functions.

#### Comments[edit]

For applications to partial differential equations, see [a1].

#### References[edit]

[a1] |  L. Ehrenpreis, "Fourier analysis in several complex variables" , Wiley (Interscience) (1970)

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[a2] |  L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Sect. 4.5