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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200401.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200402.png" /> with piecewise smooth boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200403.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200404.png" /> be a set of positive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200405.png" />-dimensional Lebesgue measure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200406.png" />.
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The following boundary value problem can then be posed (cf. also [[Boundary value problems of analytic function theory|Boundary value problems of analytic function theory]]): Given a [[Holomorphic function|holomorphic function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200407.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200408.png" /> that is sufficiently well-behaved up to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c1200409.png" /> (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004010.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004012.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004013.png" /> belongs to the Hardy class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004014.png" />) how can it be reconstructed inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004015.png" /> by its values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004016.png" /> by means of an integral formula?
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Let $D$ be a bounded domain in $\mathbf{C} ^ { n }$ with piecewise smooth boundary $\partial D$, and let $M$ be a set of positive $( 2 n - 1 )$-dimensional Lebesgue measure in $\partial D$.
 +
 
 +
The following boundary value problem can then be posed (cf. also [[Boundary value problems of analytic function theory|Boundary value problems of analytic function theory]]): Given a [[Holomorphic function|holomorphic function]] $f$ in $D$ that is sufficiently well-behaved up to the boundary $\partial D$ (for example, $f$ is continuous in $\overline{ D }$, $( f \in H _ { c } ( D ) )$, or $f$ belongs to the Hardy class $H ^ { 1 } ( D )$) how can it be reconstructed inside $D$ by its values on $M$ by means of an integral formula?
  
 
Three methods of solution are known, due to:
 
Three methods of solution are known, due to:
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The following are some very simple solutions:
 
The following are some very simple solutions:
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004017.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004018.png" /> is a smooth arc connecting two points of the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004019.png" /> and lying inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004021.png" /> is the domain bounded by a part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004022.png" /> and the arc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004023.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004024.png" />, then for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004026.png" /> the following Carleman formula holds:
+
a) $n = 1$. If $M = \Gamma$ is a smooth arc connecting two points of the unit circle $\gamma = \{ z _ { 1 } : | z _ { 1 } | = 1 \}$ and lying inside $\gamma$ and $D$ is the domain bounded by a part of $\gamma$ and the arc $\Gamma$, with $0 \notin \overline { D }$, then for $z \in D$ and $f \in H ^ { 1 } ( D )$ the following Carleman formula holds:
  
<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/c/c120/c120040/c12004027.png" /></td> </tr></table>
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\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \frac { 1 } { 2 \pi i } \int _ { \Gamma } f ( \zeta ) \left( \frac { z } { \zeta } \right) ^ { m } \frac { d \zeta } { \zeta - z }. \end{equation*}
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004028.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004029.png" /> be a circular convex bounded domain (a Cartan domain) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004030.png" />-boundary and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004031.png" /> be a piecewise smooth hypersurface intersecting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004032.png" /> and cutting from it the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004033.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004034.png" />. Then there exists a Cauchy–Fantappié formula for the domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004035.png" /> with kernel holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004036.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004038.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004039.png" />. Assume that there exists a vector-valued function (a  "barrier" ) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004042.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004044.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004045.png" /> smoothly extends to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004048.png" />. Then for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004050.png" />, the following Carleman formula with holomorphic kernel is valid (see [[#References|[a2]]]):
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b) $n &gt; 1$. Let $\Omega$ be a circular convex bounded domain (a Cartan domain) with $C ^ { 2 }$-boundary and let $\Gamma$ be a piecewise smooth hypersurface intersecting $\Omega$ and cutting from it the domain $D$, with $0 \notin \overline { D }$. Then there exists a Cauchy–Fantappié formula for the domain $D$ with kernel holomorphic in $z$. Let $\Omega = \{ \zeta : \rho ( \zeta ) &lt; 0 \}$, $\rho \in C ^ { 2 } ( \overline { \Omega } )$, and $\gamma = ( \partial D ) \backslash \Gamma$. Assume that there exists a vector-valued function (a  "barrier" ) $w = w ( z , \zeta )$, $z \in D$, $\zeta \in \Gamma$, such that $\langle w , \zeta - z \rangle \neq 0$, $w \in C _ { \zeta } ^ { 1 } ( \Gamma )$, and $w$ smoothly extends to $\rho ^ { \prime }$ on $\gamma \cap \Gamma$, where $\rho ^ { \prime } = \operatorname { grad } \rho = ( \partial \rho / \partial \zeta _ { 1 } , \dots , \partial \rho / \partial \zeta _ { n } )$. Then for every function $f \in H _ { c } ( D )$ and $z \in D$, the following Carleman formula with holomorphic kernel is valid (see [[#References|[a2]]]):
  
<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/c/c120/c120040/c12004051.png" /></td> </tr></table>
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\begin{equation*} f ( z ) = \end{equation*}
  
<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/c/c120/c120040/c12004052.png" /></td> </tr></table>
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\begin{equation*} = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) \left[ \operatorname{CF} ( \zeta - z , w ) - \sum _ { k = 0 } ^ { m } \frac { ( k + n - 1 ) } { k ! } \phi _ { k } \right]; \end{equation*}
  
here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004053.png" /> is the Cauchy–Fantappié differential form (see [[#References|[a3]]])
+
here, $\operatorname {CF}$ is the Cauchy–Fantappié differential form (see [[#References|[a3]]])
  
<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/c/c120/c120040/c12004054.png" /></td> </tr></table>
+
\begin{equation*} \operatorname{CF} ( \zeta - z , w ) = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \frac { \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } w _ { k } d w [ k ] \wedge d \zeta } { \langle w , \zeta - z \rangle ^ { n } }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004057.png" />,
+
where $d w [ k ] = d w _ { 1 } \wedge \ldots \wedge d w _ { k - 1 } \wedge d w _ { k + 1 } \wedge \ldots \wedge d w _ { n }$, $d \zeta = d \zeta _ { 1 } \wedge \ldots \wedge d \zeta _ { n }$, $\langle a , b \rangle = a _ { 1 } b _ { 1 } + \ldots + a _ { n } b _ { n }$,
  
<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/c/c120/c120040/c12004058.png" /></td> </tr></table>
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\begin{equation*} \phi _ { k } = \frac { 1 } { \langle \rho ^ { \prime } , \zeta \rangle ^ { n } } \left\langle \frac { \rho ^ { \prime } ( \zeta ) } { \langle \rho ^ { \prime } ( \zeta ) , \zeta \rangle } , z \right\rangle ^ { k } \sigma, \end{equation*}
  
<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/c/c120/c120040/c12004059.png" /></td> </tr></table>
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\begin{equation*} \sigma = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } \rho ^ { \prime } d \rho ^ { \prime } [ j ] \bigwedge d\zeta . \end{equation*}
  
c) Now, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004060.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004061.png" />-circular domain (a Reinhardt domain); then
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c) Now, let $\Omega$ be an $n$-circular domain (a Reinhardt domain); then
  
<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/c/c120/c120040/c12004062.png" /></td> </tr></table>
+
\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) [ \operatorname{CF} ( \zeta - z , w ) + \end{equation*}
  
<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/c/c120/c120040/c12004063.png" /></td> </tr></table>
+
\begin{equation*} \left. - \frac { 1 } { \langle \rho ^ { \prime } , \zeta \rangle ^ { n } } \sum _ { | \alpha | = 0 } ^ { m } \frac { ( | \alpha | + n - 1 ) ! } { \alpha _ { 1 } ! \ldots \alpha _ { n } ! } \left( \frac { \rho ^ { \prime } ( \zeta ) } { \langle \rho ^ { \prime } , \zeta \rangle } \right) ^ { \alpha } z ^ { \alpha } \sigma \right], \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004064.png" />, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004065.png" /> are non-negative integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004067.png" />.
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where $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $| \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha _ { 1 } } \ldots z _ { n } ^ { \alpha _ { n } }$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004068.png" /> is a ball, then
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If $\Omega = \{ z : | z | &lt; r \}$ is a ball, then
  
<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/c/c120/c120040/c12004069.png" /></td> </tr></table>
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\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) \times \end{equation*}
  
<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/c/c120/c120040/c12004070.png" /></td> </tr></table>
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\begin{equation*} \times \left[ \operatorname {CF} ( \zeta - z , w ) - \frac { ( n - 1 ) ! ( | \zeta | ^ { 2 m } - \langle \overline { \zeta } , z \rangle ^ { m } ) ^ { n } } { [ 2 \pi i | \zeta | ^ { 2 m } \langle \overline { \zeta } , \zeta - z \rangle ] ^ { n } } \sigma _ { 0 } \right], \end{equation*}
  
 
where
 
where
  
<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/c/c120/c120040/c12004071.png" /></td> </tr></table>
+
\begin{equation*} \sigma _ { 0 } = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } \overline { \zeta } _{j} d \overline { \zeta } [ j ] \bigwedge d \zeta . \end{equation*}
  
In all the above Carleman formulas the limits are understood in the sense of uniform convergence on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c120/c120040/c12004072.png" />. A description of the class of holomorphic functions representable by Carleman formulas is given in [[#References|[a4]]]. In [[#References|[a1]]] applications of Carleman formulas in analysis and in mathematical physics can be found as well.
+
In all the above Carleman formulas the limits are understood in the sense of uniform convergence on compact subsets of $D$. A description of the class of holomorphic functions representable by Carleman formulas is given in [[#References|[a4]]]. In [[#References|[a1]]] applications of Carleman formulas in analysis and in mathematical physics can be found as well.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. Aizenberg,  "Carleman's formulas in complex analysis" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L. Aizenberg,  "Carleman's formulas and conditions of analytic extendability" , ''Topics in Complex Analysis'' , ''Banach Centre Publ.'' , '''31''' , Banach Centre  (1995)  pp. 27–34</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Aizenberg,  A.P. Yuzhakov,  "Integral representation and residues in multidimensional complex analysis" , Amer. Math. Soc.  (1983)  (In Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  L. Aizenberg,  A. Tumanov,  A. Vidras,  "The class of holomorphic functions representable by Carleman formula"  ''Ann. Scuola Norm. Pisa'' , '''27''' :  1  (1998)  pp. 93–105</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  L. Aizenberg,  "Carleman's formulas in complex analysis" , Kluwer Acad. Publ.  (1993)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  L. Aizenberg,  "Carleman's formulas and conditions of analytic extendability" , ''Topics in Complex Analysis'' , ''Banach Centre Publ.'' , '''31''' , Banach Centre  (1995)  pp. 27–34</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  L. Aizenberg,  A.P. Yuzhakov,  "Integral representation and residues in multidimensional complex analysis" , Amer. Math. Soc.  (1983)  (In Russian)</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  L. Aizenberg,  A. Tumanov,  A. Vidras,  "The class of holomorphic functions representable by Carleman formula"  ''Ann. Scuola Norm. Pisa'' , '''27''' :  1  (1998)  pp. 93–105</td></tr></table>

Revision as of 16:58, 1 July 2020

Let $D$ be a bounded domain in $\mathbf{C} ^ { n }$ with piecewise smooth boundary $\partial D$, and let $M$ be a set of positive $( 2 n - 1 )$-dimensional Lebesgue measure in $\partial D$.

The following boundary value problem can then be posed (cf. also Boundary value problems of analytic function theory): Given a holomorphic function $f$ in $D$ that is sufficiently well-behaved up to the boundary $\partial D$ (for example, $f$ is continuous in $\overline{ D }$, $( f \in H _ { c } ( D ) )$, or $f$ belongs to the Hardy class $H ^ { 1 } ( D )$) how can it be reconstructed inside $D$ by its values on $M$ by means of an integral formula?

Three methods of solution are known, due to:

1) Carleman–Goluzin–Krylov;

2) M.M. Lavrent'ev; and

3) A.M. Kytmanov. See [a1].

The following are some very simple solutions:

a) $n = 1$. If $M = \Gamma$ is a smooth arc connecting two points of the unit circle $\gamma = \{ z _ { 1 } : | z _ { 1 } | = 1 \}$ and lying inside $\gamma$ and $D$ is the domain bounded by a part of $\gamma$ and the arc $\Gamma$, with $0 \notin \overline { D }$, then for $z \in D$ and $f \in H ^ { 1 } ( D )$ the following Carleman formula holds:

\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \frac { 1 } { 2 \pi i } \int _ { \Gamma } f ( \zeta ) \left( \frac { z } { \zeta } \right) ^ { m } \frac { d \zeta } { \zeta - z }. \end{equation*}

b) $n > 1$. Let $\Omega$ be a circular convex bounded domain (a Cartan domain) with $C ^ { 2 }$-boundary and let $\Gamma$ be a piecewise smooth hypersurface intersecting $\Omega$ and cutting from it the domain $D$, with $0 \notin \overline { D }$. Then there exists a Cauchy–Fantappié formula for the domain $D$ with kernel holomorphic in $z$. Let $\Omega = \{ \zeta : \rho ( \zeta ) < 0 \}$, $\rho \in C ^ { 2 } ( \overline { \Omega } )$, and $\gamma = ( \partial D ) \backslash \Gamma$. Assume that there exists a vector-valued function (a "barrier" ) $w = w ( z , \zeta )$, $z \in D$, $\zeta \in \Gamma$, such that $\langle w , \zeta - z \rangle \neq 0$, $w \in C _ { \zeta } ^ { 1 } ( \Gamma )$, and $w$ smoothly extends to $\rho ^ { \prime }$ on $\gamma \cap \Gamma$, where $\rho ^ { \prime } = \operatorname { grad } \rho = ( \partial \rho / \partial \zeta _ { 1 } , \dots , \partial \rho / \partial \zeta _ { n } )$. Then for every function $f \in H _ { c } ( D )$ and $z \in D$, the following Carleman formula with holomorphic kernel is valid (see [a2]):

\begin{equation*} f ( z ) = \end{equation*}

\begin{equation*} = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) \left[ \operatorname{CF} ( \zeta - z , w ) - \sum _ { k = 0 } ^ { m } \frac { ( k + n - 1 ) } { k ! } \phi _ { k } \right]; \end{equation*}

here, $\operatorname {CF}$ is the Cauchy–Fantappié differential form (see [a3])

\begin{equation*} \operatorname{CF} ( \zeta - z , w ) = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \frac { \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } w _ { k } d w [ k ] \wedge d \zeta } { \langle w , \zeta - z \rangle ^ { n } }, \end{equation*}

where $d w [ k ] = d w _ { 1 } \wedge \ldots \wedge d w _ { k - 1 } \wedge d w _ { k + 1 } \wedge \ldots \wedge d w _ { n }$, $d \zeta = d \zeta _ { 1 } \wedge \ldots \wedge d \zeta _ { n }$, $\langle a , b \rangle = a _ { 1 } b _ { 1 } + \ldots + a _ { n } b _ { n }$,

\begin{equation*} \phi _ { k } = \frac { 1 } { \langle \rho ^ { \prime } , \zeta \rangle ^ { n } } \left\langle \frac { \rho ^ { \prime } ( \zeta ) } { \langle \rho ^ { \prime } ( \zeta ) , \zeta \rangle } , z \right\rangle ^ { k } \sigma, \end{equation*}

\begin{equation*} \sigma = \frac { ( n - 1 ) ! } { ( 2 \pi i ) ^ { n } } \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } \rho ^ { \prime } d \rho ^ { \prime } [ j ] \bigwedge d\zeta . \end{equation*}

c) Now, let $\Omega$ be an $n$-circular domain (a Reinhardt domain); then

\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) [ \operatorname{CF} ( \zeta - z , w ) + \end{equation*}

\begin{equation*} \left. - \frac { 1 } { \langle \rho ^ { \prime } , \zeta \rangle ^ { n } } \sum _ { | \alpha | = 0 } ^ { m } \frac { ( | \alpha | + n - 1 ) ! } { \alpha _ { 1 } ! \ldots \alpha _ { n } ! } \left( \frac { \rho ^ { \prime } ( \zeta ) } { \langle \rho ^ { \prime } , \zeta \rangle } \right) ^ { \alpha } z ^ { \alpha } \sigma \right], \end{equation*}

where $\alpha = ( \alpha _ { 1 } , \ldots , \alpha _ { n } )$, all $\alpha_j$ are non-negative integers, $| \alpha | = \alpha _ { 1 } + \ldots + \alpha _ { n }$, $z ^ { \alpha } = z _ { 1 } ^ { \alpha _ { 1 } } \ldots z _ { n } ^ { \alpha _ { n } }$.

If $\Omega = \{ z : | z | < r \}$ is a ball, then

\begin{equation*} f ( z ) = \operatorname { lim } _ { m \rightarrow \infty } \int _ { \Gamma } f ( \zeta ) \times \end{equation*}

\begin{equation*} \times \left[ \operatorname {CF} ( \zeta - z , w ) - \frac { ( n - 1 ) ! ( | \zeta | ^ { 2 m } - \langle \overline { \zeta } , z \rangle ^ { m } ) ^ { n } } { [ 2 \pi i | \zeta | ^ { 2 m } \langle \overline { \zeta } , \zeta - z \rangle ] ^ { n } } \sigma _ { 0 } \right], \end{equation*}

where

\begin{equation*} \sigma _ { 0 } = \sum _ { j = 1 } ^ { n } ( - 1 ) ^ { j - 1 } \overline { \zeta } _{j} d \overline { \zeta } [ j ] \bigwedge d \zeta . \end{equation*}

In all the above Carleman formulas the limits are understood in the sense of uniform convergence on compact subsets of $D$. A description of the class of holomorphic functions representable by Carleman formulas is given in [a4]. In [a1] applications of Carleman formulas in analysis and in mathematical physics can be found as well.

References

[a1] L. Aizenberg, "Carleman's formulas in complex analysis" , Kluwer Acad. Publ. (1993)
[a2] L. Aizenberg, "Carleman's formulas and conditions of analytic extendability" , Topics in Complex Analysis , Banach Centre Publ. , 31 , Banach Centre (1995) pp. 27–34
[a3] L. Aizenberg, A.P. Yuzhakov, "Integral representation and residues in multidimensional complex analysis" , Amer. Math. Soc. (1983) (In Russian)
[a4] L. Aizenberg, A. Tumanov, A. Vidras, "The class of holomorphic functions representable by Carleman formula" Ann. Scuola Norm. Pisa , 27 : 1 (1998) pp. 93–105
How to Cite This Entry:
Carleman formulas. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Carleman_formulas&oldid=16201
This article was adapted from an original article by L. Aizenberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article