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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202801.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202802.png" /> and denote by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202803.png" /> the space of all functions holomorphic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202804.png" /> with the topology of uniform convergence on compact subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202805.png" /> (the projective limit topology). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202806.png" /> be a compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202807.png" />. Similarly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202808.png" /> be the space of all functions holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d1202809.png" /> endowed with the following topology: A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028010.png" /> converges to a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028012.png" /> if there exists a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028013.png" /> such that all the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028015.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028016.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028017.png" /> (the inductive limit topology).
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The description of the dual spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028019.png" /> is one of the main problems in the concrete functional analysis of spaces of holomorphic functions.
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Let $D$ be a domain in $\mathbf{C} ^ { n }$ and denote by $A ( D )$ the space of all functions holomorphic in $D$ with the topology of uniform convergence on compact subsets of $D$ (the projective limit topology). Let $K$ be a compact set in $\mathbf{C} ^ { n }$. Similarly, let $A ( K )$ be the space of all functions holomorphic on $K$ endowed with the following topology: A sequence $\{ f_{m} \}$ converges to a function $f$ in $A ( K )$ if there exists a neighbourhood $U \supset K$ such that all the functions $f _ { m } ,\, f \in A ( U )$ and $\{ f_{m} \}$ converges to $f$ in $A ( U )$ (the inductive limit topology).
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The description of the dual spaces $A ( D ) ^ { * }$ and $A ( K ) ^ { * }$ is one of the main problems in the concrete functional analysis of spaces of holomorphic functions.
  
 
==Grothendieck–Köthe–Sebastião e Silva duality.==
 
==Grothendieck–Köthe–Sebastião e Silva duality.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028020.png" /> be a domain in the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028021.png" /> and let
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Let $D$ be a domain in the complex plane ${\bf C} ^ { 1 }$ and let
  
<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/d/d120/d120280/d12028022.png" /></td> </tr></table>
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\begin{equation*} A _ { 0 } ( \overline { \mathbf{C} } \backslash D ) = \{ f : f \in A ( \overline { \mathbf{C} } \backslash D ) , f ( \infty ) = 0 \}. \end{equation*}
  
 
Then one has the isomorphism (see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]])
 
Then one has the isomorphism (see [[#References|[a1]]], [[#References|[a2]]], [[#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/d/d120/d120280/d12028023.png" /></td> </tr></table>
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\begin{equation*} A ( D ) ^ { * } \simeq A _ { 0 } ( \overline { \mathbf{C} } \backslash D ), \end{equation*}
  
 
defined by
 
defined by
  
<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/d/d120/d120280/d12028024.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} F ( f ) = F _ { \phi } ( f ) = \int _ { \Gamma } f ( z ) \phi ( z ) d z, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028025.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028027.png" /> for some domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028028.png" />; and the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028029.png" /> separates the singularities of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028031.png" />. The integral in (a1) does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028032.png" />.
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where $\phi \in A _ { 0 } ( \overline { \mathbf{C} } \backslash D )$. Here, $\phi \in A _ { 0 } ( Q )$, where $\overline { \mathbf{C} } \backslash D \subset Q$ for some domain $Q$; and the curve $\Gamma \subset D \cap Q$ separates the singularities of the functions $f \in A ( D )$ and $\phi$. The integral in (a1) does not depend on the choice of $\Gamma$.
  
 
==Duality and linear convexity.==
 
==Duality and linear convexity.==
When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028033.png" />, the complement of a bounded domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028034.png" /> is not useful for function theory. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028035.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028036.png" />. However, a generalized notion of "exterior" does exist for linearly convex domains and compacta.
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When $n &gt; 1$, the complement of a bounded domain $D \subset \mathbf{C} ^ { x }$ is not useful for function theory. Indeed, if $f \in A _ { 0 } ( \overline { \mathbf{C} } ^ { n } \backslash D )$, then $f \equiv 0$. However, a generalized notion of "exterior" does exist for linearly convex domains and compacta.
  
A domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028037.png" /> is called linearly convex if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028038.png" /> there exists a complex hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028039.png" /> through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028040.png" /> that does not intersect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028041.png" />. A compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028042.png" /> is called linearly convex if it can be approximated from the outside by linearly convex domains. Observe that the topological dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028043.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028044.png" />.
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A domain $D \subset \mathbf{C} ^ { x }$ is called linearly convex if for any $\zeta \in \partial D$ there exists a complex hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028039.png"/> through $\zeta$ that does not intersect $D$. A compact set $K \subset \mathbf{C} ^ { n }$ is called linearly convex if it can be approximated from the outside by linearly convex domains. Observe that the topological dimension of $\alpha$ is $2 n - 2$.
  
 
Some examples:
 
Some examples:
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028045.png" /> be convex; then for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028046.png" /> of the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028047.png" /> there exists a hyperplane of support <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028048.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028049.png" /> that contains the complex hyperplane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028050.png" />.
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1) Let $D$ be convex; then for any point $\zeta$ of the boundary $\partial D$ there exists a hyperplane of support $\beta$ of dimension $2 n - 1$ that contains the complex hyperplane $\alpha$.
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028051.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028053.png" />, are arbitrary plane domains.
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2) Let $D = D _ { 1 } \times \ldots \times D _ { n }$, where $D_{j} \subset {\bf C} ^ { 1 }$, $j = 1 , \ldots , n$, are arbitrary plane domains.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028054.png" /> be approximated from within by the sequence of linearly convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028055.png" /> with smooth boundaries: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028056.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028057.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028058.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028059.png" />. Such an approximation does not always exist, unlike the case of usual convexity. For instance, this approximation is impossible in Example 2) if at least one of the domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028060.png" /> is non-convex.
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Let $D$ be approximated from within by the sequence of linearly convex domains $\{ D _ { m } \}$ with smooth boundaries: $\overline { D } _ { m } \subset D _ { m + 1 } \subset D$, where $D _ { m } = \{ z : \Phi ^ { m } ( z , \bar{z} ) &lt; 0 \}$, $\Phi ^ { m } \in C ^ { 2 } ( \overline { D } _ { m } )$, and $\operatorname { grad } \Phi ^ { m } | _ { \partial D _ { m } } \neq 0$. Such an approximation does not always exist, unlike the case of usual convexity. For instance, this approximation is impossible in Example 2) if at least one of the domains $D _ { j }$ is non-convex.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028061.png" />, one has the isomorphism
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If $0 \in D$, one has the isomorphism
  
<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/d/d120/d120280/d12028062.png" /></td> </tr></table>
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\begin{equation*} A ( D ) ^ { * } \simeq A ( \tilde { D } ), \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028063.png" /> is the adjoint set (the generalized complement) defined by
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where $\tilde { D } = \{ w : w _ { 1 } z _ { 1 } + \ldots + w _ { n } z _ { n } \neq 1 , z \in D \}$ is the adjoint set (the generalized complement) defined by
  
<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/d/d120/d120280/d12028064.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} F ( t ) = F _ { \phi } ( f ) = \int _ { \partial D _ { m } } f ( z ) \phi ( w ) \omega ( z , w ). \end{equation}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028066.png" />,
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Here, $f \in A ( D )$, $\phi \in A ( \widetilde { D } )$,
  
<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/d/d120/d120280/d12028067.png" /></td> </tr></table>
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\begin{equation*} w _ { j } = \frac { \Phi ^ { \prime z _ { j } } } { \langle \operatorname { grad } _ { z } \Phi , z \rangle } ,\; j = 1 , \ldots , n, \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/d/d120/d120280/d12028068.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028068.png"/></td> </tr></table>
  
<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/d/d120/d120280/d12028069.png" /></td> </tr></table>
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\begin{equation*} d z = d z _ { 1 } \bigwedge \ldots \bigwedge d z _ { n } , \quad \langle a , b \rangle = a _ { 1 } b _ { 1 } + \ldots + a _ { n } b _ { n }. \end{equation*}
  
The index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028070.png" /> depends on the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028071.png" />, which is holomorphic on the larger compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028072.png" />. The integral in (a4) does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028073.png" />.
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The index $m$ depends on the function $\phi$, which is holomorphic on the larger compact set $\widetilde { D } _ { m } \supset \widetilde { D }$. The integral in (a4) does not depend on the choice of $m$.
  
A similar duality is valid for the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028074.png" /> as well (see [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]]).
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A similar duality is valid for the space $A ( K )$ as well (see [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]]).
  
A. Martineau has defined a strongly linearly convex domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028075.png" /> to be a linearly convex domain for which the above-mentioned isomorphism holds. It is proved in [[#References|[a7]]] that a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028076.png" /> is strongly linearly convex if and only if the intersection of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028077.png" /> with any complex line is connected and simply connected (see also [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]]).
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A. Martineau has defined a strongly linearly convex domain $D$ to be a linearly convex domain for which the above-mentioned isomorphism holds. It is proved in [[#References|[a7]]] that a domain $D$ is strongly linearly convex if and only if the intersection of $D$ with any complex line is connected and simply connected (see also [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]], [[#References|[a11]]]).
  
 
==Duality based on regularized integration over the boundary.==
 
==Duality based on regularized integration over the boundary.==
L. Stout obtained the following result for bounded domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028078.png" /> with the property that, for a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028079.png" />, the Szegö kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028080.png" /> is real-analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028081.png" />. Apparently, this is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028082.png" /> is a strictly pseudo-convex domain with real-analytic boundary. Then the isomorphism
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L. Stout obtained the following result for bounded domains $D \subset \mathbf{C} ^ { x }$ with the property that, for a fixed $z \in D$, the Szegö kernel $K ( z , \zeta )$ is real-analytic in $\zeta \in \partial D$. Apparently, this is true if $D$ is a strictly pseudo-convex domain with real-analytic boundary. Then the isomorphism
  
<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/d/d120/d120280/d12028083.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} A ( D ) ^ { * } \simeq A ( \overline { D } ) \end{equation}
  
 
is defined by the formula
 
is defined by the formula
  
<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/d/d120/d120280/d12028084.png" /></td> </tr></table>
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\begin{equation*} F ( f ) = F _ { \phi } ( f ) = \operatorname { lim } _ { \epsilon \rightarrow 0 } \int _ { \partial D _ { \epsilon } } f ( z ) \overline { \phi ( z ) } d \sigma, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028085.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028086.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028087.png" /> (see [[#References|[a12]]], [[#References|[a13]]]).
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where $D _ { \epsilon } = \{ z : z \in D , \rho ( z , \partial D ) &gt; \epsilon \}$, $f \in A ( D )$, $\phi \in A ( \overline { D } )$ (see [[#References|[a12]]], [[#References|[a13]]]).
  
 
==Nacinovich–Shlapunov–Tarkhanov theorem.==
 
==Nacinovich–Shlapunov–Tarkhanov theorem.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028088.png" /> be a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028089.png" /> with real-analytic boundary and with the property that any neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028090.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028091.png" /> contains a neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028092.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028093.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028094.png" />. This is always the case if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028095.png" /> is a strictly pseudo-convex domain with real-analytic boundary.
+
Let $D$ be a bounded domain in $\mathbf{C} ^ { n }$ with real-analytic boundary and with the property that any neighbourhood $U$ of $\overline{ D }$ contains a neighbourhood $U ^ { \prime } \subset U$ such that $A ( U ^ { \prime } )$ is dense in $A ( D )$. This is always the case if $D$ is a strictly pseudo-convex domain with real-analytic boundary.
  
For any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028096.png" /> there exists a unique solution of the [[Dirichlet problem|Dirichlet problem]]
+
For any function $\phi \in A ( \overline { D } )$ there exists a unique solution of the [[Dirichlet problem|Dirichlet problem]]
  
<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/d/d120/d120280/d12028097.png" /></td> </tr></table>
+
\begin{equation*} \left\{ \begin{array} { l } { \Delta v = 0 } &amp; {\text{in} \ \mathbf{C}^{n} \setminus \overline{D}, }\\ { v = \phi} &amp; { \text { on } \partial D, } \\ { | v | \leq \frac { c } { | z | ^ { 2 n - 2 } }. } \end{array} \right. \end{equation*}
  
 
Here, the isomorphism (a3) can be defined by the formula (see [[#References|[a14]]])
 
Here, the isomorphism (a3) can be defined by the formula (see [[#References|[a14]]])
  
<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/d/d120/d120280/d12028098.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
\begin{equation} \tag{a4} F (\, f ) = F _ { \phi } (\, f ) = \int _ { \partial D _ { m } } f ( z ) \sum ^ { n } _ { k = 1 } ( - 1 ) ^ { k - 1 } \frac { \partial \overline{ v } } { \partial z _ { k } } d \overline{z} [ k ] \bigwedge d z. \end{equation}
  
The integral is well-defined for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d12028099.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280100.png" /> is a sequence of domains with smooth boundaries which approximate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280101.png" /> from within) since the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280102.png" />, which is harmonic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280103.png" />, can be harmonically continued into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280104.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280105.png" /> because of the real analyticity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280107.png" />. The integral in (a4) does not depend on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280108.png" />.
+
The integral is well-defined for some $m$ (where $\{ D _ { m } \}$ is a sequence of domains with smooth boundaries which approximate $D$ from within) since the function $v$, which is harmonic in $\mathbf{C} ^ { n } \backslash \overline { D }$, can be harmonically continued into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280104.png"/> for some $m$ because of the real analyticity of $\partial D$ and $\phi |_{\partial D}$. The integral in (a4) does not depend on the choice of $m$.
  
 
==Duality and cohomology.==
 
==Duality and cohomology.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280109.png" /> be the Dolbeault cohomology space
+
Let $H ^ { n , n - 1 } ( {\bf C} ^ { n } \backslash D )$ be the Dolbeault cohomology space
  
<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/d/d120/d120280/d120280110.png" /></td> </tr></table>
+
\begin{equation*} H ^ { n , n - 1 } = Z ^ { n , n - 1 } / B ^ { n , n - 1 }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280111.png" /> is the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280112.png" />-closed forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280113.png" /> that are in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280114.png" /> on some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280115.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280116.png" /> (the neighbourhood depends on the cocycle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280117.png" />) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280118.png" /> is the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280119.png" /> of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280120.png" />-exact forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280121.png" /> (coboundaries).
+
where $Z ^ { n , n - 1 }$ is the space of all $\overline { \partial }$-closed forms $\alpha$ that are in $C ^ { \infty }$ on some neighbourhood $U$ of $\mathbf{C} ^ { n } \backslash D$ (the neighbourhood depends on the cocycle $\alpha$) and $B ^ { n ,\, n - 1 }$ is the subspace of $Z ^ { n , n - 1 }$ of all $\overline { \partial }$-exact forms $\alpha$ (coboundaries).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280122.png" /> is a bounded pseudo-convex domain, then one has [[#References|[a15]]], [[#References|[a16]]] an isomorphism
+
If $D$ is a bounded pseudo-convex domain, then one has [[#References|[a15]]], [[#References|[a16]]] an isomorphism
  
<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/d/d120/d120280/d120280123.png" /></td> </tr></table>
+
\begin{equation*} A ( D ) ^ { * } \simeq H ^ { n , n - 1 } ( \mathbf{C} ^ { n } \backslash D ), \end{equation*}
  
 
defined by the formula
 
defined by the formula
  
<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/d/d120/d120280/d120280124.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
\begin{equation} \tag{a5} F ( f ) = F _ { g } ( f ) = \int _ { \partial D _ { m } } f g, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280125.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280126.png" />. Here, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280127.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280128.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280129.png" /> is a sequence of domains with smooth boundaries approximating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280130.png" /> from within. Although <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280131.png" /> depends on the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280132.png" />, the integral in (a5) does not depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280133.png" /> (given (a5), the same formula is valid for larger <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280134.png" /> as well).
+
where $f \in A ( D )$, $g \in H ^ { n ,\, n - 1 } ( \mathbf{C} ^ { n } \backslash D )$. Here, for some $U \supset \mathbf{C} ^ { n } \backslash D$ one has $g \in H ^ { n , n - 1 } ( U )$; $\{ D _ { m } \}$ is a sequence of domains with smooth boundaries approximating $D$ from within. Although $m$ depends on the choice of $U$, the integral in (a5) does not depend on $m$ (given (a5), the same formula is valid for larger $m$ as well).
  
A new result [[#References|[a17]]] consists of the following: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280135.png" /> be a bounded pseudo-convex domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280136.png" /> that can be approximated from within by a sequence of strictly pseudo-convex domains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280137.png" />; and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280138.png" /> be the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280139.png" /> consisting of the differential forms of type
+
A new result [[#References|[a17]]] consists of the following: Let $D$ be a bounded pseudo-convex domain in $\mathbf{C} ^ { n }$ that can be approximated from within by a sequence of strictly pseudo-convex domains $\{ D _ { m } \}$; and let $A$ be the subspace of $Z ^ { n , n - 1 }$ consisting of the differential forms of type
  
<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/d/d120/d120280/d120280140.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
\begin{equation} \tag{a6} g _ { u } = \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } \frac { \partial u } { \partial z _ { k } } d \bar{z} [ k ] \wedge d z, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280141.png" /> is a function that is harmonic in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280142.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280143.png" /> (which depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280144.png" />) such that
+
where $u$ is a function that is harmonic in some neighbourhood $U$ of $\mathbf{C} ^ { n } \backslash D$ (which depends on $u$) 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/d/d120/d120280/d120280145.png" /></td> </tr></table>
+
\begin{equation*} | u ( z ) | \leq \frac { C } { | z | ^ { 2 n - 2 } }. \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280146.png" /> be the space of all forms of type (a6) such that the harmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280147.png" /> is representable for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280148.png" /> by the Bochner–Martinelli-type integral (cf. also [[Bochner–Martinelli representation formula|Bochner–Martinelli representation formula]])
+
Let $B$ be the space of all forms of type (a6) such that the harmonic function $\overline { u }$ is representable for some $m$ by the Bochner–Martinelli-type integral (cf. also [[Bochner–Martinelli representation formula|Bochner–Martinelli representation formula]])
  
<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/d/d120/d120280/d120280149.png" /></td> </tr></table>
+
\begin{equation*} \overline { u } ( z ) = \int _ { \partial D _ { m } } w ( \zeta ) \frac { \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } ( \overline { \zeta } _ { k } - \overline{z} _ { k } ) d \overline { \zeta } [ k ] \wedge d \zeta } { | \zeta - z | ^ { 2 n } }, \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280150.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280151.png" />. Then one has an isomorphism
+
where $z \in \mathbf{C} ^ { n } \backslash \overline { D } _ { m }$ and $\omega ( \zeta ) \in C ( \partial D _ { m } )$. Then one has an isomorphism
  
<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/d/d120/d120280/d120280152.png" /></td> </tr></table>
+
\begin{equation*} A ( D ) ^ { * } \simeq A / B; \end{equation*}
  
 
it is defined by the formula
 
it is defined by the formula
  
<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/d/d120/d120280/d120280153.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
+
<table class="eq" style="width:100%;"> <tr><td style="width:94%;text-align:center;" valign="top"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280153.png"/></td> <td style="width:5%;text-align:right;" valign="top">(a7)</td></tr></table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280154.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280155.png" />. Note that (a7) gives a more concrete description of the duality than does (a5). The integral in (a7) is also independent of the choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280156.png" />.
+
where $f \in A ( D )$ and $g _ { u } \in A / B$. Note that (a7) gives a more concrete description of the duality than does (a5). The integral in (a7) is also independent of the choice of $m$.
  
 
Other descriptions of the spaces dual to spaces of holomorphic functions for special classes of domains can be found in [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]], [[#References|[a10]]].
 
Other descriptions of the spaces dual to spaces of holomorphic functions for special classes of domains can be found in [[#References|[a18]]], [[#References|[a19]]], [[#References|[a20]]], [[#References|[a21]]], [[#References|[a22]]], [[#References|[a23]]], [[#References|[a24]]], [[#References|[a10]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Grothendieck,   "Sur certain espaces de fonctions holomorphes" ''J. Reine Angew. Math.'' , '''192''' (1953) pp. 35–64; 77–95</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G. Köthe,   "Dualität in der Funktionentheorie" ''J. Reine Angew. Math.'' , '''191''' (1953) pp. 30–39</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Sebastião e Silva,   "Analytic functions in functional analysis" ''Portug. Math.'' , '''9''' (1950) pp. 1–130</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> A. Martineau,   "Sur la topologies des espaces de fonctions holomorphes" ''Math. Ann.'' , '''163''' (1966) pp. 62–88</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> L. Aizenberg,   "The general form of a linear continuous functional in spaces of functions holomorphic in convex domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280157.png" />" ''Soviet Math. Dokl.'' , '''7''' (1966) pp. 198–202</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> L. Aizenberg,   "Linear convexity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280158.png" /> and the distributions of the singularities of holomorphic functions" ''Bull. Acad. Polon. Sci. Ser. Math. Astr. Phiz.'' , '''15''' (1967) pp. 487–495 (In Russian)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> S.V. Zhamenskij,   "A geometric criterion of strong linear convexity" ''Funct. Anal. Appl.'' , '''13''' (1979) pp. 224–225</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> M. Andersson,   "Cauchy–Fantappié–Leray formulas with local sections and the inverse Fantappié transform" ''Bull. Soc. Math. France'' , '''120''' (1992) pp. 113–128</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> S.G. Gindikin,   G.M. Henkin,   "Integral geometry for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280159.png" />-cohomologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280160.png" />-linearly concave domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280161.png" />"  ''Funct. Anal. Appl.'' , '''12''' (1978) pp. 6–23</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> S.V. Znamenskij,   "Strong linear convexity. I: Duality of spaces of holomorphic functions" ''Sib. Math. J.'' , '''26''' (1985) pp. 331–341</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> S.V. Znamenskij,   "Strong linear convexity. II: Existence of holomorphic solutions of linear systems of equations" ''Sib. Math. J.'' , '''29''' (1988) pp. 911–925</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> L. Aizenberg,   S.G. Gindikin,   "The general form of a linear continuous functional in spaces of holomorphic functions"  ''Moskov. Oblast. Ped. Just. Uchen. Zap.'' , '''87''' (1964) pp. 7–15  (In Russian)</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E.L. Stout,   "Harmonic duality, hyperfunctions and removable singularities" ''Izv. Akad. Nauk Ser. Mat.'' , '''59''' (1995) pp. 133–170</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> M. Nacinovich,  A. Shlapunov,   N. Tarkhanov,  "Duality in the spaces of solutions of elliptic systems"  ''Ann. Scuola Norm. Sup. Pisa'' , '''26''' (1998) pp. 207–232</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> J.P. Serre,   "Une théorème de dualité" ''Comment. Math. Helvetici'' , '''29''' (1955) pp. 9–26</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> A. Martineau,   "Sur les fonctionelles analytiques et la transformation de Fourier–Borel" ''J. Anal. Math.'' , '''9''' (1963) pp. 1–164</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> L. Aizenberg,   "Duality in complex analysis" , ''Israel Math. Conf. Proc.'' , '''11''' (1997) pp. 27–35</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> H.G. Tillman,   "Randverteilungen analytischer funktionen und distributionen" ''Math. Z.'' , '''59''' (1953) pp. 61–83</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> S. Rolewicz,   "On spaces of holomorphic function" ''Studia Math.'' , '''21''' (1962) pp. 135–160</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> L. Aizenberg,   B.S. Mityagin,  "The spaces of functions analytic in multi-circular domains"  ''Sib. Mat. Zh.'' , '''1'''  (1960) pp. 153–170  (In Russian)</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top">  L. Aizenberg,  "The spaces of functions analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120280/d120280162.png" />-circular domains" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 75–82</TD></TR><TR><TD valign="top">[a22]</TD> <TD valign="top"> L.J. Ronkin,   "On general form of functionals in space of functions, analytic in semicircular domain" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 673–686</TD></TR><TR><TD valign="top">[a23]</TD> <TD valign="top"> S.G. Gindikin,   "Analytic functions in tubular domains" ''Soviet Math. Dokl.'' , '''3''' (1962)</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top"> S.D. Simonzhenkov,   "Description of the conjugate space of functions that are holomorphic in the domain of a special type" ''Sib. Math. J.'' , '''22''' (1981)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> A. Grothendieck, "Sur certain espaces de fonctions holomorphes" ''J. Reine Angew. Math.'' , '''192''' (1953) pp. 35–64; 77–95 {{MR|0062335}} {{MR|0058865}} {{ZBL|}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> G. Köthe, "Dualität in der Funktionentheorie" ''J. Reine Angew. Math.'' , '''191''' (1953) pp. 30–39 {{MR|0056824}} {{ZBL|0050.33502}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> J. Sebastião e Silva, "Analytic functions in functional analysis" ''Portug. Math.'' , '''9''' (1950) pp. 1–130</td></tr><tr><td valign="top">[a4]</td> <td valign="top"> A. Martineau, "Sur la topologies des espaces de fonctions holomorphes" ''Math. Ann.'' , '''163''' (1966) pp. 62–88 {{MR|190697}} {{ZBL|}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> L. Aizenberg, "The general form of a linear continuous functional in spaces of functions holomorphic in convex domains in $\mathbf{C} ^ { n }$" ''Soviet Math. Dokl.'' , '''7''' (1966) pp. 198–202</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> L. Aizenberg, "Linear convexity in $\mathbf{C} ^ { n }$ and the distributions of the singularities of holomorphic functions" ''Bull. Acad. Polon. Sci. Ser. Math. Astr. Phiz.'' , '''15''' (1967) pp. 487–495 (In Russian) {{MR|0222346}} {{ZBL|}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> S.V. Zhamenskij, "A geometric criterion of strong linear convexity" ''Funct. Anal. Appl.'' , '''13''' (1979) pp. 224–225</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> M. Andersson, "Cauchy–Fantappié–Leray formulas with local sections and the inverse Fantappié transform" ''Bull. Soc. Math. France'' , '''120''' (1992) pp. 113–128 {{MR|}} {{ZBL|0757.32008}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> S.G. Gindikin, G.M. Henkin, "Integral geometry for $\overline { \partial }$-cohomologies in $q$-linearly concave domains in $\mathbf{CP} ^ { n }$" ''Funct. Anal. Appl.'' , '''12''' (1978) pp. 6–23</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> S.V. Znamenskij, "Strong linear convexity. I: Duality of spaces of holomorphic functions" ''Sib. Math. J.'' , '''26''' (1985) pp. 331–341 {{MR|}} {{ZBL|0596.32017}} </td></tr><tr><td valign="top">[a11]</td> <td valign="top"> S.V. Znamenskij, "Strong linear convexity. II: Existence of holomorphic solutions of linear systems of equations" ''Sib. Math. J.'' , '''29''' (1988) pp. 911–925 {{MR|}} {{ZBL|0689.47004}} {{ZBL|0672.47009}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> L. Aizenberg, S.G. Gindikin, "The general form of a linear continuous functional in spaces of holomorphic functions" ''Moskov. Oblast. Ped. Just. Uchen. Zap.'' , '''87''' (1964) pp. 7–15 (In Russian) {{MR|0180699}} {{ZBL|}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> E.L. Stout, "Harmonic duality, hyperfunctions and removable singularities" ''Izv. Akad. Nauk Ser. Mat.'' , '''59''' (1995) pp. 133–170 {{MR|1481618}} {{ZBL|0876.32003}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> M. Nacinovich, A. Shlapunov, N. Tarkhanov, "Duality in the spaces of solutions of elliptic systems" ''Ann. Scuola Norm. Sup. Pisa'' , '''26''' (1998) pp. 207–232 {{MR|1631573}} {{ZBL|0919.35040}} </td></tr><tr><td valign="top">[a15]</td> <td valign="top"> J.P. Serre, "Une théorème de dualité" ''Comment. Math. Helvetici'' , '''29''' (1955) pp. 9–26</td></tr><tr><td valign="top">[a16]</td> <td valign="top"> A. Martineau, "Sur les fonctionelles analytiques et la transformation de Fourier–Borel" ''J. Anal. Math.'' , '''9''' (1963) pp. 1–164</td></tr><tr><td valign="top">[a17]</td> <td valign="top"> L. Aizenberg, "Duality in complex analysis" , ''Israel Math. Conf. Proc.'' , '''11''' (1997) pp. 27–35 {{MR|1476701}} {{ZBL|0907.46018}} </td></tr><tr><td valign="top">[a18]</td> <td valign="top"> H.G. Tillman, "Randverteilungen analytischer funktionen und distributionen" ''Math. Z.'' , '''59''' (1953) pp. 61–83 {{MR|}} {{ZBL|0051.08901}} </td></tr><tr><td valign="top">[a19]</td> <td valign="top"> S. Rolewicz, "On spaces of holomorphic function" ''Studia Math.'' , '''21''' (1962) pp. 135–160 {{MR|0154146}} {{ZBL|}} </td></tr><tr><td valign="top">[a20]</td> <td valign="top"> L. Aizenberg, B.S. Mityagin, "The spaces of functions analytic in multi-circular domains" ''Sib. Mat. Zh.'' , '''1''' (1960) pp. 153–170 (In Russian) {{MR|124526}} {{ZBL|}} </td></tr><tr><td valign="top">[a21]</td> <td valign="top"> L. Aizenberg, "The spaces of functions analytic in $( p , q )$-circular domains" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 75–82</td></tr><tr><td valign="top">[a22]</td> <td valign="top"> L.J. Ronkin, "On general form of functionals in space of functions, analytic in semicircular domain" ''Soviet Math. Dokl.'' , '''2''' (1961) pp. 673–686 {{MR|131577}} {{ZBL|}} </td></tr><tr><td valign="top">[a23]</td> <td valign="top"> S.G. Gindikin, "Analytic functions in tubular domains" ''Soviet Math. Dokl.'' , '''3''' (1962)</td></tr><tr><td valign="top">[a24]</td> <td valign="top"> S.D. Simonzhenkov, "Description of the conjugate space of functions that are holomorphic in the domain of a special type" ''Sib. Math. J.'' , '''22''' (1981) {{MR|610784}} {{ZBL|}} </td></tr></table>

Latest revision as of 17:44, 1 July 2020

Let $D$ be a domain in $\mathbf{C} ^ { n }$ and denote by $A ( D )$ the space of all functions holomorphic in $D$ with the topology of uniform convergence on compact subsets of $D$ (the projective limit topology). Let $K$ be a compact set in $\mathbf{C} ^ { n }$. Similarly, let $A ( K )$ be the space of all functions holomorphic on $K$ endowed with the following topology: A sequence $\{ f_{m} \}$ converges to a function $f$ in $A ( K )$ if there exists a neighbourhood $U \supset K$ such that all the functions $f _ { m } ,\, f \in A ( U )$ and $\{ f_{m} \}$ converges to $f$ in $A ( U )$ (the inductive limit topology).

The description of the dual spaces $A ( D ) ^ { * }$ and $A ( K ) ^ { * }$ is one of the main problems in the concrete functional analysis of spaces of holomorphic functions.

Grothendieck–Köthe–Sebastião e Silva duality.

Let $D$ be a domain in the complex plane ${\bf C} ^ { 1 }$ and let

\begin{equation*} A _ { 0 } ( \overline { \mathbf{C} } \backslash D ) = \{ f : f \in A ( \overline { \mathbf{C} } \backslash D ) , f ( \infty ) = 0 \}. \end{equation*}

Then one has the isomorphism (see [a1], [a2], [a3])

\begin{equation*} A ( D ) ^ { * } \simeq A _ { 0 } ( \overline { \mathbf{C} } \backslash D ), \end{equation*}

defined by

\begin{equation} \tag{a1} F ( f ) = F _ { \phi } ( f ) = \int _ { \Gamma } f ( z ) \phi ( z ) d z, \end{equation}

where $\phi \in A _ { 0 } ( \overline { \mathbf{C} } \backslash D )$. Here, $\phi \in A _ { 0 } ( Q )$, where $\overline { \mathbf{C} } \backslash D \subset Q$ for some domain $Q$; and the curve $\Gamma \subset D \cap Q$ separates the singularities of the functions $f \in A ( D )$ and $\phi$. The integral in (a1) does not depend on the choice of $\Gamma$.

Duality and linear convexity.

When $n > 1$, the complement of a bounded domain $D \subset \mathbf{C} ^ { x }$ is not useful for function theory. Indeed, if $f \in A _ { 0 } ( \overline { \mathbf{C} } ^ { n } \backslash D )$, then $f \equiv 0$. However, a generalized notion of "exterior" does exist for linearly convex domains and compacta.

A domain $D \subset \mathbf{C} ^ { x }$ is called linearly convex if for any $\zeta \in \partial D$ there exists a complex hyperplane through $\zeta$ that does not intersect $D$. A compact set $K \subset \mathbf{C} ^ { n }$ is called linearly convex if it can be approximated from the outside by linearly convex domains. Observe that the topological dimension of $\alpha$ is $2 n - 2$.

Some examples:

1) Let $D$ be convex; then for any point $\zeta$ of the boundary $\partial D$ there exists a hyperplane of support $\beta$ of dimension $2 n - 1$ that contains the complex hyperplane $\alpha$.

2) Let $D = D _ { 1 } \times \ldots \times D _ { n }$, where $D_{j} \subset {\bf C} ^ { 1 }$, $j = 1 , \ldots , n$, are arbitrary plane domains.

Let $D$ be approximated from within by the sequence of linearly convex domains $\{ D _ { m } \}$ with smooth boundaries: $\overline { D } _ { m } \subset D _ { m + 1 } \subset D$, where $D _ { m } = \{ z : \Phi ^ { m } ( z , \bar{z} ) < 0 \}$, $\Phi ^ { m } \in C ^ { 2 } ( \overline { D } _ { m } )$, and $\operatorname { grad } \Phi ^ { m } | _ { \partial D _ { m } } \neq 0$. Such an approximation does not always exist, unlike the case of usual convexity. For instance, this approximation is impossible in Example 2) if at least one of the domains $D _ { j }$ is non-convex.

If $0 \in D$, one has the isomorphism

\begin{equation*} A ( D ) ^ { * } \simeq A ( \tilde { D } ), \end{equation*}

where $\tilde { D } = \{ w : w _ { 1 } z _ { 1 } + \ldots + w _ { n } z _ { n } \neq 1 , z \in D \}$ is the adjoint set (the generalized complement) defined by

\begin{equation} \tag{a2} F ( t ) = F _ { \phi } ( f ) = \int _ { \partial D _ { m } } f ( z ) \phi ( w ) \omega ( z , w ). \end{equation}

Here, $f \in A ( D )$, $\phi \in A ( \widetilde { D } )$,

\begin{equation*} w _ { j } = \frac { \Phi ^ { \prime z _ { j } } } { \langle \operatorname { grad } _ { z } \Phi , z \rangle } ,\; j = 1 , \ldots , n, \end{equation*}

\begin{equation*} d z = d z _ { 1 } \bigwedge \ldots \bigwedge d z _ { n } , \quad \langle a , b \rangle = a _ { 1 } b _ { 1 } + \ldots + a _ { n } b _ { n }. \end{equation*}

The index $m$ depends on the function $\phi$, which is holomorphic on the larger compact set $\widetilde { D } _ { m } \supset \widetilde { D }$. The integral in (a4) does not depend on the choice of $m$.

A similar duality is valid for the space $A ( K )$ as well (see [a4], [a5], [a6]).

A. Martineau has defined a strongly linearly convex domain $D$ to be a linearly convex domain for which the above-mentioned isomorphism holds. It is proved in [a7] that a domain $D$ is strongly linearly convex if and only if the intersection of $D$ with any complex line is connected and simply connected (see also [a8], [a9], [a10], [a11]).

Duality based on regularized integration over the boundary.

L. Stout obtained the following result for bounded domains $D \subset \mathbf{C} ^ { x }$ with the property that, for a fixed $z \in D$, the Szegö kernel $K ( z , \zeta )$ is real-analytic in $\zeta \in \partial D$. Apparently, this is true if $D$ is a strictly pseudo-convex domain with real-analytic boundary. Then the isomorphism

\begin{equation} \tag{a3} A ( D ) ^ { * } \simeq A ( \overline { D } ) \end{equation}

is defined by the formula

\begin{equation*} F ( f ) = F _ { \phi } ( f ) = \operatorname { lim } _ { \epsilon \rightarrow 0 } \int _ { \partial D _ { \epsilon } } f ( z ) \overline { \phi ( z ) } d \sigma, \end{equation*}

where $D _ { \epsilon } = \{ z : z \in D , \rho ( z , \partial D ) > \epsilon \}$, $f \in A ( D )$, $\phi \in A ( \overline { D } )$ (see [a12], [a13]).

Nacinovich–Shlapunov–Tarkhanov theorem.

Let $D$ be a bounded domain in $\mathbf{C} ^ { n }$ with real-analytic boundary and with the property that any neighbourhood $U$ of $\overline{ D }$ contains a neighbourhood $U ^ { \prime } \subset U$ such that $A ( U ^ { \prime } )$ is dense in $A ( D )$. This is always the case if $D$ is a strictly pseudo-convex domain with real-analytic boundary.

For any function $\phi \in A ( \overline { D } )$ there exists a unique solution of the Dirichlet problem

\begin{equation*} \left\{ \begin{array} { l } { \Delta v = 0 } & {\text{in} \ \mathbf{C}^{n} \setminus \overline{D}, }\\ { v = \phi} & { \text { on } \partial D, } \\ { | v | \leq \frac { c } { | z | ^ { 2 n - 2 } }. } \end{array} \right. \end{equation*}

Here, the isomorphism (a3) can be defined by the formula (see [a14])

\begin{equation} \tag{a4} F (\, f ) = F _ { \phi } (\, f ) = \int _ { \partial D _ { m } } f ( z ) \sum ^ { n } _ { k = 1 } ( - 1 ) ^ { k - 1 } \frac { \partial \overline{ v } } { \partial z _ { k } } d \overline{z} [ k ] \bigwedge d z. \end{equation}

The integral is well-defined for some $m$ (where $\{ D _ { m } \}$ is a sequence of domains with smooth boundaries which approximate $D$ from within) since the function $v$, which is harmonic in $\mathbf{C} ^ { n } \backslash \overline { D }$, can be harmonically continued into for some $m$ because of the real analyticity of $\partial D$ and $\phi |_{\partial D}$. The integral in (a4) does not depend on the choice of $m$.

Duality and cohomology.

Let $H ^ { n , n - 1 } ( {\bf C} ^ { n } \backslash D )$ be the Dolbeault cohomology space

\begin{equation*} H ^ { n , n - 1 } = Z ^ { n , n - 1 } / B ^ { n , n - 1 }, \end{equation*}

where $Z ^ { n , n - 1 }$ is the space of all $\overline { \partial }$-closed forms $\alpha$ that are in $C ^ { \infty }$ on some neighbourhood $U$ of $\mathbf{C} ^ { n } \backslash D$ (the neighbourhood depends on the cocycle $\alpha$) and $B ^ { n ,\, n - 1 }$ is the subspace of $Z ^ { n , n - 1 }$ of all $\overline { \partial }$-exact forms $\alpha$ (coboundaries).

If $D$ is a bounded pseudo-convex domain, then one has [a15], [a16] an isomorphism

\begin{equation*} A ( D ) ^ { * } \simeq H ^ { n , n - 1 } ( \mathbf{C} ^ { n } \backslash D ), \end{equation*}

defined by the formula

\begin{equation} \tag{a5} F ( f ) = F _ { g } ( f ) = \int _ { \partial D _ { m } } f g, \end{equation}

where $f \in A ( D )$, $g \in H ^ { n ,\, n - 1 } ( \mathbf{C} ^ { n } \backslash D )$. Here, for some $U \supset \mathbf{C} ^ { n } \backslash D$ one has $g \in H ^ { n , n - 1 } ( U )$; $\{ D _ { m } \}$ is a sequence of domains with smooth boundaries approximating $D$ from within. Although $m$ depends on the choice of $U$, the integral in (a5) does not depend on $m$ (given (a5), the same formula is valid for larger $m$ as well).

A new result [a17] consists of the following: Let $D$ be a bounded pseudo-convex domain in $\mathbf{C} ^ { n }$ that can be approximated from within by a sequence of strictly pseudo-convex domains $\{ D _ { m } \}$; and let $A$ be the subspace of $Z ^ { n , n - 1 }$ consisting of the differential forms of type

\begin{equation} \tag{a6} g _ { u } = \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } \frac { \partial u } { \partial z _ { k } } d \bar{z} [ k ] \wedge d z, \end{equation}

where $u$ is a function that is harmonic in some neighbourhood $U$ of $\mathbf{C} ^ { n } \backslash D$ (which depends on $u$) such that

\begin{equation*} | u ( z ) | \leq \frac { C } { | z | ^ { 2 n - 2 } }. \end{equation*}

Let $B$ be the space of all forms of type (a6) such that the harmonic function $\overline { u }$ is representable for some $m$ by the Bochner–Martinelli-type integral (cf. also Bochner–Martinelli representation formula)

\begin{equation*} \overline { u } ( z ) = \int _ { \partial D _ { m } } w ( \zeta ) \frac { \sum _ { k = 1 } ^ { n } ( - 1 ) ^ { k - 1 } ( \overline { \zeta } _ { k } - \overline{z} _ { k } ) d \overline { \zeta } [ k ] \wedge d \zeta } { | \zeta - z | ^ { 2 n } }, \end{equation*}

where $z \in \mathbf{C} ^ { n } \backslash \overline { D } _ { m }$ and $\omega ( \zeta ) \in C ( \partial D _ { m } )$. Then one has an isomorphism

\begin{equation*} A ( D ) ^ { * } \simeq A / B; \end{equation*}

it is defined by the formula

(a7)

where $f \in A ( D )$ and $g _ { u } \in A / B$. Note that (a7) gives a more concrete description of the duality than does (a5). The integral in (a7) is also independent of the choice of $m$.

Other descriptions of the spaces dual to spaces of holomorphic functions for special classes of domains can be found in [a18], [a19], [a20], [a21], [a22], [a23], [a24], [a10].

References

[a1] A. Grothendieck, "Sur certain espaces de fonctions holomorphes" J. Reine Angew. Math. , 192 (1953) pp. 35–64; 77–95 MR0062335 MR0058865
[a2] G. Köthe, "Dualität in der Funktionentheorie" J. Reine Angew. Math. , 191 (1953) pp. 30–39 MR0056824 Zbl 0050.33502
[a3] J. Sebastião e Silva, "Analytic functions in functional analysis" Portug. Math. , 9 (1950) pp. 1–130
[a4] A. Martineau, "Sur la topologies des espaces de fonctions holomorphes" Math. Ann. , 163 (1966) pp. 62–88 MR190697
[a5] L. Aizenberg, "The general form of a linear continuous functional in spaces of functions holomorphic in convex domains in $\mathbf{C} ^ { n }$" Soviet Math. Dokl. , 7 (1966) pp. 198–202
[a6] L. Aizenberg, "Linear convexity in $\mathbf{C} ^ { n }$ and the distributions of the singularities of holomorphic functions" Bull. Acad. Polon. Sci. Ser. Math. Astr. Phiz. , 15 (1967) pp. 487–495 (In Russian) MR0222346
[a7] S.V. Zhamenskij, "A geometric criterion of strong linear convexity" Funct. Anal. Appl. , 13 (1979) pp. 224–225
[a8] M. Andersson, "Cauchy–Fantappié–Leray formulas with local sections and the inverse Fantappié transform" Bull. Soc. Math. France , 120 (1992) pp. 113–128 Zbl 0757.32008
[a9] S.G. Gindikin, G.M. Henkin, "Integral geometry for $\overline { \partial }$-cohomologies in $q$-linearly concave domains in $\mathbf{CP} ^ { n }$" Funct. Anal. Appl. , 12 (1978) pp. 6–23
[a10] S.V. Znamenskij, "Strong linear convexity. I: Duality of spaces of holomorphic functions" Sib. Math. J. , 26 (1985) pp. 331–341 Zbl 0596.32017
[a11] S.V. Znamenskij, "Strong linear convexity. II: Existence of holomorphic solutions of linear systems of equations" Sib. Math. J. , 29 (1988) pp. 911–925 Zbl 0689.47004 Zbl 0672.47009
[a12] L. Aizenberg, S.G. Gindikin, "The general form of a linear continuous functional in spaces of holomorphic functions" Moskov. Oblast. Ped. Just. Uchen. Zap. , 87 (1964) pp. 7–15 (In Russian) MR0180699
[a13] E.L. Stout, "Harmonic duality, hyperfunctions and removable singularities" Izv. Akad. Nauk Ser. Mat. , 59 (1995) pp. 133–170 MR1481618 Zbl 0876.32003
[a14] M. Nacinovich, A. Shlapunov, N. Tarkhanov, "Duality in the spaces of solutions of elliptic systems" Ann. Scuola Norm. Sup. Pisa , 26 (1998) pp. 207–232 MR1631573 Zbl 0919.35040
[a15] J.P. Serre, "Une théorème de dualité" Comment. Math. Helvetici , 29 (1955) pp. 9–26
[a16] A. Martineau, "Sur les fonctionelles analytiques et la transformation de Fourier–Borel" J. Anal. Math. , 9 (1963) pp. 1–164
[a17] L. Aizenberg, "Duality in complex analysis" , Israel Math. Conf. Proc. , 11 (1997) pp. 27–35 MR1476701 Zbl 0907.46018
[a18] H.G. Tillman, "Randverteilungen analytischer funktionen und distributionen" Math. Z. , 59 (1953) pp. 61–83 Zbl 0051.08901
[a19] S. Rolewicz, "On spaces of holomorphic function" Studia Math. , 21 (1962) pp. 135–160 MR0154146
[a20] L. Aizenberg, B.S. Mityagin, "The spaces of functions analytic in multi-circular domains" Sib. Mat. Zh. , 1 (1960) pp. 153–170 (In Russian) MR124526
[a21] L. Aizenberg, "The spaces of functions analytic in $( p , q )$-circular domains" Soviet Math. Dokl. , 2 (1961) pp. 75–82
[a22] L.J. Ronkin, "On general form of functionals in space of functions, analytic in semicircular domain" Soviet Math. Dokl. , 2 (1961) pp. 673–686 MR131577
[a23] S.G. Gindikin, "Analytic functions in tubular domains" Soviet Math. Dokl. , 3 (1962)
[a24] S.D. Simonzhenkov, "Description of the conjugate space of functions that are holomorphic in the domain of a special type" Sib. Math. J. , 22 (1981) MR610784
How to Cite This Entry:
Duality in complex analysis. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Duality_in_complex_analysis&oldid=18633
This article was adapted from an original article by L. Aizenberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article