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Difference between revisions of "Analytic functional"

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An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122601.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122602.png" />, the dual of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122603.png" /> of analytic functions defined on an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122604.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122605.png" />, i.e. a functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122606.png" />. Thus, a distribution with compact support is an analytic functional. There exists a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122607.png" />, said to be the support of the analytic functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122608.png" />, on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a0122609.png" /> is concentrated: For any open set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226010.png" /> the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226011.png" /> can be extended to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226012.png" /> so that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226013.png" /> the following inequality is valid:
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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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226014.png" /></td> </tr></table>
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$$|f(u)|\leq C_\omega\sup|u|,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226015.png" /> is a constant depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226016.png" />. There exists a measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226017.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226018.png" /> such that
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where $C_\omega$ is a constant depending on $\omega$. There exists a measure $\mu$ with support in $K$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012260/a01226019.png" /></td> </tr></table>
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$$f(u)=\int\limits_\omega ud\mu.$$
  
 
An analytic functional is defined in a similar manner on a space of real-valued functions.
 
An analytic functional is defined in a similar manner on a space of real-valued functions.

Latest revision as of 17:01, 16 August 2014

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

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

References

[a1] L. Ehrenpreis, "Fourier analysis in several complex variables" , Wiley (Interscience) (1970)
[a2] L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Sect. 4.5
How to Cite This Entry:
Analytic functional. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_functional&oldid=13936
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article