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Consider an abstract set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300801.png" /> and a linear set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300802.png" /> of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300803.png" />.
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Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300804.png" /> is equipped with an [[Inner product|inner product]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300806.png" /> is complete with respect to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300807.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300808.png" /> is a [[Hilbert space|Hilbert space]].
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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r1300809.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008010.png" />, is called a reproducing kernel of such a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008011.png" /> if and only if the following two conditions are satisfied:
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Consider an abstract set $E$ and a linear set $F$ of functions $f : E \rightarrow \mathbf{C}$.
  
i) for every fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008012.png" />, the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008013.png" />;
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Assume that $F$ is equipped with an [[Inner product|inner product]] $( f , g )$ and $F$ is complete with respect to the norm $\| f \| = ( f , f ) ^ { 1 / 2 }$. Then $F$ is a [[Hilbert space|Hilbert space]].
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008015.png" />.
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A function $K ( x , y )$, $x , y \in E$, is called a reproducing kernel of such a Hilbert space $H$ if and only if the following two conditions are satisfied:
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i) for every fixed $y \in E$, the function $K ( x , y ) \in H$;
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ii) $( f ( x ) , K ( x , y ) ) = f ( y )$, $\forall f \in H$.
  
 
This definition is given in [[#References|[a1]]]; see also [[#References|[a6]]].
 
This definition is given in [[#References|[a1]]]; see also [[#References|[a6]]].
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Some properties of reproducing kernels are:
 
Some properties of reproducing kernels are:
  
1) If a reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008016.png" /> exists, then it is unique.
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1) If a reproducing kernel $K ( x , y )$ exists, then it is unique.
  
2) A reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008017.png" /> exists if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008020.png" />.
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2) A reproducing kernel $K ( x , y )$ exists if and only if $| f ( y ) | \leq c ( y ) \| f \|$, $\forall f \in H$, where $c ( y ) = \| K ( . , y ) \|$.
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008021.png" /> is a non-negative-definite kernel, that is,
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3) $K ( x , y )$ is a non-negative-definite kernel, that is,
  
<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/r/r130/r130080/r13008022.png" /></td> </tr></table>
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\begin{equation*} \sum _ { i , j = 1 } ^ { n } K ( x _ { i } , x _ { j } ) t _ { j } \overline { t } _ { i } \geq 0 , \forall x _ { i } , y _ { j } \in E , \forall t \in {\bf C} ^ { n }, \end{equation*}
  
 
where the overbar stands for complex conjugation.
 
where the overbar stands for complex conjugation.
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In particular, 3) implies:
 
In particular, 3) implies:
  
<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/r/r130/r130080/r13008023.png" /></td> </tr></table>
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\begin{equation*} K ( x , y ) = \overline { K ( y , x ) } , K ( x , x ) \geq 0, \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/r/r130/r130080/r13008024.png" /></td> </tr></table>
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\begin{equation*} | K ( x , y ) | ^ { 2 } \leq K ( x , x ) K ( y , y ). \end{equation*}
  
Every non-negative-definite kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008025.png" /> generates a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008026.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008027.png" /> is a reproducing kernel (see also [[Reproducing-kernel Hilbert space|Reproducing-kernel Hilbert space]]).
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Every non-negative-definite kernel $K ( x , y )$ generates a Hilbert space $H _ { K }$ for which $K ( x , y )$ is a reproducing kernel (see also [[Reproducing-kernel Hilbert space|Reproducing-kernel Hilbert space]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008028.png" /> is a reproducing kernel, then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008029.png" /> is injective: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008030.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008031.png" />, by reproducing property ii), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008032.png" /> is surjective (cf. also [[Injection|Injection]]; [[Surjection|Surjection]]). Therefore the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008033.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008034.png" />, and since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008035.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008036.png" /> is the identity operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008037.png" />, and so is its inverse.
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If $K ( x , y )$ is a reproducing kernel, then the operator $K f : = ( K f ) ( \cdot ) = ( f , K ( x , ) ) = f ( \cdot )$ is injective: $K f = 0$ implies $f = 0$, by reproducing property ii), and $K : H \rightarrow H$ is surjective (cf. also [[Injection|Injection]]; [[Surjection|Surjection]]). Therefore the inverse operator $K ^ { - 1 }$ is defined on $R ( K ) = H$, and since $K f = f$, the operator $K$ is the identity operator on $H _ { K }$, and so is its inverse.
  
 
==Examples of reproducing kernels.==
 
==Examples of reproducing kernels.==
  
  
Consider the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008038.png" /> of analytic functions (cf. [[Analytic function|Analytic function]]) in a bounded [[Simply-connected domain|simply-connected domain]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008039.png" /> of the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008040.png" />-plane. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008041.png" /> is analytic in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008043.png" />, and the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008044.png" />, then
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Consider the Hilbert space $H$ of analytic functions (cf. [[Analytic function|Analytic function]]) in a bounded [[Simply-connected domain|simply-connected domain]] $D$ of the complex $z$-plane. If $f ( z )$ is analytic in $D$, $z _ { 0 } \in D$, and the disc $D _ { z _ { 0 } , r } : = \{ z : | z - z _ { 0 } | \leq r \} \in D$, 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/r/r130/r130080/r13008045.png" /></td> </tr></table>
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\begin{equation*} | f ( z _ { 0 } ) | ^ { 2 } \leq \frac { 1 } { \pi r ^ { 2 } } \int _ { D _ { z _ { 0 } , r } } | f ( \zeta ) | ^ { 2 } d x d y \leq \frac { 1 } { \pi r ^ { 2 } } ( f , f ) _ { L^2(D) }. \end{equation*}
  
Therefore <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008046.png" /> is a reproducing-kernel Hilbert space. Its reproducing kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008047.png" /> is called the Bergman kernel (cf. also [[Bergman kernel function|Bergman kernel function]]).
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Therefore $H$ is a reproducing-kernel Hilbert space. Its reproducing kernel $K _ { D } ( z , \zeta )$ is called the Bergman kernel (cf. also [[Bergman kernel function|Bergman kernel function]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008048.png" /> is an orthonormal basis of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008049.png" /> (cf. also [[Orthogonal system|Orthogonal system]]; [[Basis|Basis]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008050.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008051.png" />.
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If $\{ \phi_j ( z ) \}$ is an orthonormal basis of $H$ (cf. also [[Orthogonal system|Orthogonal system]]; [[Basis|Basis]]), $\phi _ { j } \in H$, then $K _ { D } ( z , \zeta ) = \sum _ { j = 1 } ^ { \infty } \phi _ { j } ( z ) \overline { \phi _ { j } ( \zeta ) }$.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008052.png" /> is the [[Conformal mapping|conformal mapping]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008053.png" /> onto the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008054.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008056.png" />, then [[#References|[a2]]]:
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If $w = f ( z , z_0 )$ is the [[Conformal mapping|conformal mapping]] of $D$ onto the disc $| w | \leq \rho _ { D }$, such that $f ( z , z _ { 0 } ) = 0$, $f ^ { \prime } ( z _ { 0 } , z _ { 0 } ) = 1$, then [[#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/r/r130/r130080/r13008057.png" /></td> </tr></table>
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\begin{equation*} f ( z , z_0 ) = \frac { 1 } { K _ { D } ( z_0 , z _ { 0 } ) } \int _ { z _ { 0 } } ^ { z } K _ { D } ( t , z _ { 0 } ) d t. \end{equation*}
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008058.png" /> be a domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008060.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008061.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008062.png" /> is a finite [[Measure|measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008063.png" />.
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Let $T$ be a domain in ${\bf R} ^ { n }$ and $h ( t , p ) \in L ^ { 2 } ( T , d m )$ for every $p \in E$. Here $m ( t ) > 0$ is a finite [[Measure|measure]] on $T$.
  
Define a linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008064.png" /> by
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Define a linear mapping $L : L ^ { 2 } ( T , d m ) \rightarrow F$ 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/r/r130/r130080/r13008065.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} f ( p ) = L g : = \int _ { T } g ( t ) \overline { h ( t , p ) } d m ( t ). \end{equation}
  
 
Define the kernel
 
Define the kernel
  
<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/r/r130/r130080/r13008066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} K ( p , q ) : = \int _ { T } h ( t , q ) \overline { h ( t , p ) } d m ( t ) , p , q \in E. \end{equation}
  
 
This kernel is non-negative-definite:
 
This kernel is non-negative-definite:
  
<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/r/r130/r130080/r13008067.png" /></td> </tr></table>
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\begin{equation*} \sum _ { i , j + 1 } ^ { n } K ( p _ { i } , p _ { j } ) \xi _ { j } \overline { \xi _ { i } } = \int _ { T } | \sum _ { j = 1 } ^ { n } \xi _ { j } h ( t , p _ { j } ) | ^ { 2 } d m ( t ) > 0 \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/r/r130/r130080/r13008068.png" /></td> </tr></table>
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\begin{equation*} \xi \neq 0, \end{equation*}
  
provided that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008069.png" /> the set of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008070.png" /> is linearly independent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008071.png" /> (cf. [[Linear independence|Linear independence]]).
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provided that for any set $\{ p _ { 1 } , \dots , p _ { n } \} \in E$ the set of functions $\{ h ( t , p _ { j } ) \} _ { 1 \leq j \leq n}$ is linearly independent in $L ^ { 2 } ( T , d m )$ (cf. [[Linear independence|Linear independence]]).
  
In this case the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008072.png" /> generates a uniquely determined reproducing-kernel Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008073.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008074.png" /> is the reproducing kernel.
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In this case the kernel $K ( p , q )$ generates a uniquely determined reproducing-kernel Hilbert space $H _ { K }$ for which $K ( p , q )$ is the reproducing kernel.
  
In [[#References|[a6]]] it is claimed that a convenient characterization of the range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008075.png" /> of the linear transformation (a1) is given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008076.png" />. In [[#References|[a4]]] it is shown by examples that such a characterization is often useless in practice: in general the norm in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008077.png" /> can not be described in terms of the standard Sobolev or Hölder norms, and the assumption in [[#References|[a6]]] that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008078.png" /> can be realized as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008079.png" /> is not justified and is not correct, in general.
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In [[#References|[a6]]] it is claimed that a convenient characterization of the range $R ( L )$ of the linear transformation (a1) is given by the formula $R ( L ) = H _ { K }$. In [[#References|[a4]]] it is shown by examples that such a characterization is often useless in practice: in general the norm in $H _ { K }$ can not be described in terms of the standard Sobolev or Hölder norms, and the assumption in [[#References|[a6]]] that $H _ { K }$ can be realized as $L ^ { 2 } ( E , d \mu )$ is not justified and is not correct, in general.
  
However, in [[#References|[a6]]] there are some examples of characterizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008080.png" /> for some special operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008081.png" /> and in [[#References|[a5]]] a characterization of the range of a wide class of multi-dimensional linear transforms, whose kernels are kernels of positive rational functions of self-adjoint elliptic operators, is given.
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However, in [[#References|[a6]]] there are some examples of characterizations of $H _ { K }$ for some special operators $L$ and in [[#References|[a5]]] a characterization of the range of a wide class of multi-dimensional linear transforms, whose kernels are kernels of positive rational functions of self-adjoint elliptic operators, is given.
  
Reproducing kernels are discussed in [[#References|[a5]]] for rigged triples of Hilbert spaces (cf. also [[Rigged Hilbert space|Rigged Hilbert space]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008082.png" /> is a Hilbert space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008083.png" /> is a linear [[Compact operator|compact operator]] defined on all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008084.png" />, then the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008085.png" /> in the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008086.png" /> is a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008087.png" />. The space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008088.png" />, with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008089.png" />, is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008090.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008091.png" />. The inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008092.png" /> is given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008093.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008094.png" />, equipped with this inner product, is a Hilbert space.
+
Reproducing kernels are discussed in [[#References|[a5]]] for rigged triples of Hilbert spaces (cf. also [[Rigged Hilbert space|Rigged Hilbert space]]). If $H _ { 0 }$ is a Hilbert space and $A > 0$ is a linear [[Compact operator|compact operator]] defined on all of $H$, then the closure of $H _ { 0 }$ in the norm $( A u , u ) ^ { 1 / 2 } = \| A ^ { 1 / 2 } u \|$ is a Hilbert space $H _ { - } \supset H _ { 0 }$. The space dual to $H_-$, with respect to $H _ { 0 }$, is denoted by $H _ { + }$, $H _ { + } \subset H _ { 0 } \subset H _ { - }$. The inner product in $H _ { + }$ is given by the formula $( u , v ) _ { + } = ( A ^ { - 1 / 2 } u , A ^ { - 1 / 2 } v ) _ { 0 }$. The space $H _ { + } = R ( A ^ { 1 / 2 } )$, equipped with this inner product, is a Hilbert space.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008095.png" />, where the eigenvalues <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008096.png" /> are counted according to their multiplicities and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008097.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008098.png" /> is the Kronecker delta.
+
Let $A \varphi _ { j } = \lambda _ { j } \varphi _ { j }$, where the eigenvalues $\lambda_j$ are counted according to their multiplicities and $( \varphi_j , \varphi _ { m } ) _ { 0 } = \delta _ { j m }$, where $\delta _ { j m }$ is the Kronecker delta.
  
Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r13008099.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080100.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080101.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080102.png" />.
+
Assume that $| \varphi_j ( x ) | < c$ for all $j$ and all $x$, and $\Lambda ^ { 2 } : = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } < \infty$.
  
Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080103.png" /> is a reproducing-kernel Hilbert space and its reproducing kernel is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080104.png" />.
+
Then $H _ { + }$ is a reproducing-kernel Hilbert space and its reproducing kernel is $K ( x , y ) = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } \varphi _ { j } ( y ) \overline { \varphi _ { j } ( x ) }$.
  
To check that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080105.png" /> is indeed the reproducing kernel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080106.png" />, one calculates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080107.png" />. Indeed, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080108.png" /> is the identity operator because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080109.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080110.png" /> is the kernel of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080111.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080112.png" />.
+
To check that $K ( x , y )$ is indeed the reproducing kernel of $H _ { + }$, one calculates $( A ^ { - 1 / 2 } u , A ^ { - 1 / 2 } K ) _ { 0 } = ( u , A ^ { - 1 } K ) _ { 0 } = u ( y )$. Indeed, $A ^ { - 1 } K = I$ is the identity operator because $A u = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ( u , \varphi _ { j } ) \varphi _ { j } ( x )$, so that $K ( x , y )$ is the kernel of the operator $A$ in $H _ { 0 }$.
  
The value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080113.png" /> is a linear functional in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080114.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080115.png" /> is a reproducing-kernel Hilbert space. Indeed, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080116.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080117.png" />. Therefore, denoting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080118.png" /> and using the [[Cauchy inequality|Cauchy inequality]] and [[Parseval equality|Parseval equality]] one gets:
+
The value $u ( y )$ is a linear functional in $H _ { + }$, so that $H _ { + }$ is a reproducing-kernel Hilbert space. Indeed, if $u \in H _ { + }$, then $v : = A ^ { - 1 / 2 } u \in H _ { 0 }$. Therefore, denoting $v _ { j } : = ( v , \varphi _ { j } ) _ { 0 }$ and using the [[Cauchy inequality|Cauchy inequality]] and [[Parseval equality|Parseval equality]] one gets:
  
<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/r/r130/r130080/r130080119.png" /></td> </tr></table>
+
\begin{equation*} | u ( y ) | = \left| \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { 1 / 2 } v _ { j } \varphi _ { j } ( x ) \right| < c \Lambda \| v \| _ { 0 } = c \Lambda \| u \| _ { + }, \end{equation*}
  
 
as claimed.
 
as claimed.
  
From the representation of the inner product in the reproducing-kernel Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080120.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080121.png" /> it is clear that, in general, the inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080122.png" /> is not an inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080123.png" />.
+
From the representation of the inner product in the reproducing-kernel Hilbert space $H _ { + }$ by the formula $( u , v ) _ { + } = ( A ^ { - 1 / 2 } u , A ^ { - 1 / 2 } v ) _ { 0 }$ it is clear that, in general, the inner product in $H _ { + }$ is not an inner product in $L ^ { 2 } ( E , d \mu )$.
  
The inner product in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080124.png" /> is of the form
+
The inner product in $H _ { + }$ is of the form
  
<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/r/r130/r130080/r130080125.png" /></td> </tr></table>
+
\begin{equation*} ( u , v )_ + = \int _ { D } \int _ { D } B ( x , y ) u ( y ) \overline { v ( x ) } d y d x \;\text { if } H _ { 0 } = L ^ { 2 } ( D ), \end{equation*}
  
where the distributional kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080126.png" /> acts on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080127.png" /> by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080128.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080129.png" /> is the Fourier coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080130.png" /> (cf. also [[Fourier coefficients|Fourier coefficients]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080131.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080132.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080133.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080134.png" />. Thus, the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080135.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130080/r130080136.png" />.
+
where the distributional kernel $B ( x , y ) = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { - 1 } \varphi _ { j } ( x ) \overline { \varphi _ { j } ( y ) }$ acts on $u \in R ( A )$ by the formula $\int _ { D } B ( x , y ) u ( y ) d y = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { - 1 } ( u , \varphi _ { j } ) _ { 0 } \varphi _ { j } ( x )$, where $( u , \varphi _ { j } ) _ { 0 } : = \int _ { D } u ( y ) \overline { \varphi _ { j } ( y ) } d y$ is the Fourier coefficient of $u$ (cf. also [[Fourier coefficients|Fourier coefficients]]). If $u \in R ( A )$, then $u = A w$ for some $w \in H _ { 0 }$, and $( u , \varphi_j ) = \lambda _ { j } w _ { j }$. Thus, the series $\sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { - 1 } ( u , \varphi_j ) _ { 0 } \varphi _ { j } ( x ) = \sum _ { j = 1 } ^ { \infty } w _ { j } \varphi _ { j } ( x ) = w ( x )$ converges in $H _ { 0 } = L ^ { 2 } ( D )$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N. Aronszajn,  "Theory of reproducing kernels"  ''Trans. Amer. Math. Soc.'' , '''68'''  (1950)  pp. 337–404</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Bergman,  "The kernel function and conformal mapping" , Amer. Math. Soc.  (1950)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.G. Ramm,  "On the theory of reproducing kernel Hilbert spaces"  ''J. Inverse Ill-Posed Probl.'' , '''6''' :  5  (1998)  pp. 515–520</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.G. Ramm,  "On Saitoh's characterization of the range of linear transforms"  A.G. Ramm (ed.) , ''Inverse Problems, Tomography and Image Processing'' , Plenum  (1998)  pp. 125–128</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.G. Ramm,  "Random fields estimation theory" , Longman/Wiley  (1990)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S. Saitoh,  "Integral transforms, reproducing kernels and their applications" , ''Pitman Res. Notes'' , Longman  (1997)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Schwartz,  "Sous-espaces hilbertiens d'espaces vectoriels topologique et noyaux associes"  ''Anal. Math.'' , '''13'''  (1964)  pp. 115–256</TD></TR></table>
+
<table>
 +
<tr><td valign="top">[a1]</td> <td valign="top">  N. Aronszajn,  "Theory of reproducing kernels"  ''Trans. Amer. Math. Soc.'' , '''68'''  (1950)  pp. 337–404</td></tr>
 +
<tr><td valign="top">[a2]</td> <td valign="top">  S. Bergman,  "The kernel function and conformal mapping" , Amer. Math. Soc.  (1950)</td></tr>
 +
<tr><td valign="top">[a3]</td> <td valign="top">  A.G. Ramm,  "On the theory of reproducing kernel Hilbert spaces"  ''J. Inverse Ill-Posed Probl.'' , '''6''' :  5  (1998)  pp. 515–520</td></tr>
 +
<tr><td valign="top">[a4]</td> <td valign="top">  A.G. Ramm,  "On Saitoh's characterization of the range of linear transforms"  A.G. Ramm (ed.) , ''Inverse Problems, Tomography and Image Processing'' , Plenum  (1998)  pp. 125–128</td></tr>
 +
<tr><td valign="top">[a5]</td> <td valign="top">  A.G. Ramm,  "Random fields estimation theory" , Longman/Wiley  (1990)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  S. Saitoh,  "Integral transforms, reproducing kernels and their applications" , ''Pitman Res. Notes'' , Longman  (1997)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  L. Schwartz,  "Sous-espaces hilbertiens d'espaces vectoriels topologique et noyaux associés"  ''Anal. Math.'' , '''13'''  (1964)  pp. 115–256</td></tr>
 +
</table>

Latest revision as of 17:31, 4 February 2024

Consider an abstract set $E$ and a linear set $F$ of functions $f : E \rightarrow \mathbf{C}$.

Assume that $F$ is equipped with an inner product $( f , g )$ and $F$ is complete with respect to the norm $\| f \| = ( f , f ) ^ { 1 / 2 }$. Then $F$ is a Hilbert space.

A function $K ( x , y )$, $x , y \in E$, is called a reproducing kernel of such a Hilbert space $H$ if and only if the following two conditions are satisfied:

i) for every fixed $y \in E$, the function $K ( x , y ) \in H$;

ii) $( f ( x ) , K ( x , y ) ) = f ( y )$, $\forall f \in H$.

This definition is given in [a1]; see also [a6].

Some properties of reproducing kernels are:

1) If a reproducing kernel $K ( x , y )$ exists, then it is unique.

2) A reproducing kernel $K ( x , y )$ exists if and only if $| f ( y ) | \leq c ( y ) \| f \|$, $\forall f \in H$, where $c ( y ) = \| K ( . , y ) \|$.

3) $K ( x , y )$ is a non-negative-definite kernel, that is,

\begin{equation*} \sum _ { i , j = 1 } ^ { n } K ( x _ { i } , x _ { j } ) t _ { j } \overline { t } _ { i } \geq 0 , \forall x _ { i } , y _ { j } \in E , \forall t \in {\bf C} ^ { n }, \end{equation*}

where the overbar stands for complex conjugation.

In particular, 3) implies:

\begin{equation*} K ( x , y ) = \overline { K ( y , x ) } , K ( x , x ) \geq 0, \end{equation*}

\begin{equation*} | K ( x , y ) | ^ { 2 } \leq K ( x , x ) K ( y , y ). \end{equation*}

Every non-negative-definite kernel $K ( x , y )$ generates a Hilbert space $H _ { K }$ for which $K ( x , y )$ is a reproducing kernel (see also Reproducing-kernel Hilbert space).

If $K ( x , y )$ is a reproducing kernel, then the operator $K f : = ( K f ) ( \cdot ) = ( f , K ( x , ) ) = f ( \cdot )$ is injective: $K f = 0$ implies $f = 0$, by reproducing property ii), and $K : H \rightarrow H$ is surjective (cf. also Injection; Surjection). Therefore the inverse operator $K ^ { - 1 }$ is defined on $R ( K ) = H$, and since $K f = f$, the operator $K$ is the identity operator on $H _ { K }$, and so is its inverse.

Examples of reproducing kernels.

Consider the Hilbert space $H$ of analytic functions (cf. Analytic function) in a bounded simply-connected domain $D$ of the complex $z$-plane. If $f ( z )$ is analytic in $D$, $z _ { 0 } \in D$, and the disc $D _ { z _ { 0 } , r } : = \{ z : | z - z _ { 0 } | \leq r \} \in D$, then

\begin{equation*} | f ( z _ { 0 } ) | ^ { 2 } \leq \frac { 1 } { \pi r ^ { 2 } } \int _ { D _ { z _ { 0 } , r } } | f ( \zeta ) | ^ { 2 } d x d y \leq \frac { 1 } { \pi r ^ { 2 } } ( f , f ) _ { L^2(D) }. \end{equation*}

Therefore $H$ is a reproducing-kernel Hilbert space. Its reproducing kernel $K _ { D } ( z , \zeta )$ is called the Bergman kernel (cf. also Bergman kernel function).

If $\{ \phi_j ( z ) \}$ is an orthonormal basis of $H$ (cf. also Orthogonal system; Basis), $\phi _ { j } \in H$, then $K _ { D } ( z , \zeta ) = \sum _ { j = 1 } ^ { \infty } \phi _ { j } ( z ) \overline { \phi _ { j } ( \zeta ) }$.

If $w = f ( z , z_0 )$ is the conformal mapping of $D$ onto the disc $| w | \leq \rho _ { D }$, such that $f ( z , z _ { 0 } ) = 0$, $f ^ { \prime } ( z _ { 0 } , z _ { 0 } ) = 1$, then [a2]:

\begin{equation*} f ( z , z_0 ) = \frac { 1 } { K _ { D } ( z_0 , z _ { 0 } ) } \int _ { z _ { 0 } } ^ { z } K _ { D } ( t , z _ { 0 } ) d t. \end{equation*}

Let $T$ be a domain in ${\bf R} ^ { n }$ and $h ( t , p ) \in L ^ { 2 } ( T , d m )$ for every $p \in E$. Here $m ( t ) > 0$ is a finite measure on $T$.

Define a linear mapping $L : L ^ { 2 } ( T , d m ) \rightarrow F$ by

\begin{equation} \tag{a1} f ( p ) = L g : = \int _ { T } g ( t ) \overline { h ( t , p ) } d m ( t ). \end{equation}

Define the kernel

\begin{equation} \tag{a2} K ( p , q ) : = \int _ { T } h ( t , q ) \overline { h ( t , p ) } d m ( t ) , p , q \in E. \end{equation}

This kernel is non-negative-definite:

\begin{equation*} \sum _ { i , j + 1 } ^ { n } K ( p _ { i } , p _ { j } ) \xi _ { j } \overline { \xi _ { i } } = \int _ { T } | \sum _ { j = 1 } ^ { n } \xi _ { j } h ( t , p _ { j } ) | ^ { 2 } d m ( t ) > 0 \end{equation*}

\begin{equation*} \xi \neq 0, \end{equation*}

provided that for any set $\{ p _ { 1 } , \dots , p _ { n } \} \in E$ the set of functions $\{ h ( t , p _ { j } ) \} _ { 1 \leq j \leq n}$ is linearly independent in $L ^ { 2 } ( T , d m )$ (cf. Linear independence).

In this case the kernel $K ( p , q )$ generates a uniquely determined reproducing-kernel Hilbert space $H _ { K }$ for which $K ( p , q )$ is the reproducing kernel.

In [a6] it is claimed that a convenient characterization of the range $R ( L )$ of the linear transformation (a1) is given by the formula $R ( L ) = H _ { K }$. In [a4] it is shown by examples that such a characterization is often useless in practice: in general the norm in $H _ { K }$ can not be described in terms of the standard Sobolev or Hölder norms, and the assumption in [a6] that $H _ { K }$ can be realized as $L ^ { 2 } ( E , d \mu )$ is not justified and is not correct, in general.

However, in [a6] there are some examples of characterizations of $H _ { K }$ for some special operators $L$ and in [a5] a characterization of the range of a wide class of multi-dimensional linear transforms, whose kernels are kernels of positive rational functions of self-adjoint elliptic operators, is given.

Reproducing kernels are discussed in [a5] for rigged triples of Hilbert spaces (cf. also Rigged Hilbert space). If $H _ { 0 }$ is a Hilbert space and $A > 0$ is a linear compact operator defined on all of $H$, then the closure of $H _ { 0 }$ in the norm $( A u , u ) ^ { 1 / 2 } = \| A ^ { 1 / 2 } u \|$ is a Hilbert space $H _ { - } \supset H _ { 0 }$. The space dual to $H_-$, with respect to $H _ { 0 }$, is denoted by $H _ { + }$, $H _ { + } \subset H _ { 0 } \subset H _ { - }$. The inner product in $H _ { + }$ is given by the formula $( u , v ) _ { + } = ( A ^ { - 1 / 2 } u , A ^ { - 1 / 2 } v ) _ { 0 }$. The space $H _ { + } = R ( A ^ { 1 / 2 } )$, equipped with this inner product, is a Hilbert space.

Let $A \varphi _ { j } = \lambda _ { j } \varphi _ { j }$, where the eigenvalues $\lambda_j$ are counted according to their multiplicities and $( \varphi_j , \varphi _ { m } ) _ { 0 } = \delta _ { j m }$, where $\delta _ { j m }$ is the Kronecker delta.

Assume that $| \varphi_j ( x ) | < c$ for all $j$ and all $x$, and $\Lambda ^ { 2 } : = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } < \infty$.

Then $H _ { + }$ is a reproducing-kernel Hilbert space and its reproducing kernel is $K ( x , y ) = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } \varphi _ { j } ( y ) \overline { \varphi _ { j } ( x ) }$.

To check that $K ( x , y )$ is indeed the reproducing kernel of $H _ { + }$, one calculates $( A ^ { - 1 / 2 } u , A ^ { - 1 / 2 } K ) _ { 0 } = ( u , A ^ { - 1 } K ) _ { 0 } = u ( y )$. Indeed, $A ^ { - 1 } K = I$ is the identity operator because $A u = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ( u , \varphi _ { j } ) \varphi _ { j } ( x )$, so that $K ( x , y )$ is the kernel of the operator $A$ in $H _ { 0 }$.

The value $u ( y )$ is a linear functional in $H _ { + }$, so that $H _ { + }$ is a reproducing-kernel Hilbert space. Indeed, if $u \in H _ { + }$, then $v : = A ^ { - 1 / 2 } u \in H _ { 0 }$. Therefore, denoting $v _ { j } : = ( v , \varphi _ { j } ) _ { 0 }$ and using the Cauchy inequality and Parseval equality one gets:

\begin{equation*} | u ( y ) | = \left| \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { 1 / 2 } v _ { j } \varphi _ { j } ( x ) \right| < c \Lambda \| v \| _ { 0 } = c \Lambda \| u \| _ { + }, \end{equation*}

as claimed.

From the representation of the inner product in the reproducing-kernel Hilbert space $H _ { + }$ by the formula $( u , v ) _ { + } = ( A ^ { - 1 / 2 } u , A ^ { - 1 / 2 } v ) _ { 0 }$ it is clear that, in general, the inner product in $H _ { + }$ is not an inner product in $L ^ { 2 } ( E , d \mu )$.

The inner product in $H _ { + }$ is of the form

\begin{equation*} ( u , v )_ + = \int _ { D } \int _ { D } B ( x , y ) u ( y ) \overline { v ( x ) } d y d x \;\text { if } H _ { 0 } = L ^ { 2 } ( D ), \end{equation*}

where the distributional kernel $B ( x , y ) = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { - 1 } \varphi _ { j } ( x ) \overline { \varphi _ { j } ( y ) }$ acts on $u \in R ( A )$ by the formula $\int _ { D } B ( x , y ) u ( y ) d y = \sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { - 1 } ( u , \varphi _ { j } ) _ { 0 } \varphi _ { j } ( x )$, where $( u , \varphi _ { j } ) _ { 0 } : = \int _ { D } u ( y ) \overline { \varphi _ { j } ( y ) } d y$ is the Fourier coefficient of $u$ (cf. also Fourier coefficients). If $u \in R ( A )$, then $u = A w$ for some $w \in H _ { 0 }$, and $( u , \varphi_j ) = \lambda _ { j } w _ { j }$. Thus, the series $\sum _ { j = 1 } ^ { \infty } \lambda _ { j } ^ { - 1 } ( u , \varphi_j ) _ { 0 } \varphi _ { j } ( x ) = \sum _ { j = 1 } ^ { \infty } w _ { j } \varphi _ { j } ( x ) = w ( x )$ converges in $H _ { 0 } = L ^ { 2 } ( D )$.

References

[a1] N. Aronszajn, "Theory of reproducing kernels" Trans. Amer. Math. Soc. , 68 (1950) pp. 337–404
[a2] S. Bergman, "The kernel function and conformal mapping" , Amer. Math. Soc. (1950)
[a3] A.G. Ramm, "On the theory of reproducing kernel Hilbert spaces" J. Inverse Ill-Posed Probl. , 6 : 5 (1998) pp. 515–520
[a4] A.G. Ramm, "On Saitoh's characterization of the range of linear transforms" A.G. Ramm (ed.) , Inverse Problems, Tomography and Image Processing , Plenum (1998) pp. 125–128
[a5] A.G. Ramm, "Random fields estimation theory" , Longman/Wiley (1990)
[a6] S. Saitoh, "Integral transforms, reproducing kernels and their applications" , Pitman Res. Notes , Longman (1997)
[a7] L. Schwartz, "Sous-espaces hilbertiens d'espaces vectoriels topologique et noyaux associés" Anal. Math. , 13 (1964) pp. 115–256
How to Cite This Entry:
Reproducing kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reproducing_kernel&oldid=16915
This article was adapted from an original article by A.G. Ramm (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article