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A complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470601.png" /> that is square-integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470602.png" /> and satisfies the condition (of Hermitian symmetry)
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<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/h/h047/h047060/h0470603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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{{TEX|done}}
  
for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470604.png" />. The bar in (1) denotes transition to the complex-conjugate value. If a Hermitian kernel vanishes almost nowhere, then it has at least one characteristic (eigen) value, that is, there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470605.png" /> such that the equation
+
A complex-valued function  $  K ( x , y ) $
 +
that is square-integrable on  $  [ a , b ] \times [ a , b ] $
 +
and satisfies the condition (of Hermitian symmetry)
  
<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/h/h047/h047060/h0470606.png" /></td> </tr></table>
+
$$ \tag{1 }
 +
\overline{ {K ( x , y ) }}\; = K ( y , x )
 +
$$
  
has a non-zero solution (an eigen function corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470607.png" />). All eigen values are real and on any interval there are only finitely many. The eigen functions corresponding to distinct eigen values are orthogonal to each other.
+
for almost-all  $  ( x , y ) \in [ a , b ] \times [ a , b ] $.  
 +
The bar in (1) denotes transition to the complex-conjugate value. If a Hermitian kernel vanishes almost nowhere, then it has at least one characteristic (eigen) value, that is, there exists a number  $  \lambda $
 +
such that the equation
  
There is an orthonormal (finite or infinite) sequence of eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470608.png" /> corresponding to the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470609.png" />. The series
+
$$
 +
\phi ( x) - \lambda \int\limits _ { a } ^ { b }  K ( x , y ) \phi ( y)  d y  = 0
 +
$$
  
<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/h/h047/h047060/h04706010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
has a non-zero solution (an eigen function corresponding to  $  \lambda $).  
 +
All eigen values are real and on any interval there are only finitely many. The eigen functions corresponding to distinct eigen values are orthogonal to each other.
  
converges in the mean on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706011.png" /> to the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706012.png" />. If the kernel is continuous and the series (2) converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706013.png" />, then its sum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706014.png" />. The system of eigen values and eigen functions of a Hermitian kernel is finite if and only if the kernel is degenerate (cf. [[Degenerate kernel|Degenerate kernel]]).
+
There is an orthonormal (finite or infinite) sequence of eigen functions  $  \phi _ {1} , \phi _ {2} \dots $
 +
corresponding to the eigen values  $  | \lambda _ {1} | \leq  | \lambda _ {2} | \leq  \dots $.
 +
The series
 +
 
 +
$$ \tag{2 }
 +
\sum _ { k= } 1 ^  \infty 
 +
 
 +
\frac{\phi _ {k} ( x) \overline{ {\phi _ {k} ( y) }}\; }{\lambda _ {k} }
 +
 
 +
$$
 +
 
 +
converges in the mean on the square $  [ a , b ] \times [ a , b ] $
 +
to the kernel $  K $.  
 +
If the kernel is continuous and the series (2) converges uniformly on $  [ a , b ] \times [ a , b ] $,  
 +
then its sum is $  K $.  
 +
The system of eigen values and eigen functions of a Hermitian kernel is finite if and only if the kernel is degenerate (cf. [[Degenerate kernel|Degenerate kernel]]).
  
 
All iterated kernels (cf. [[Iterated kernel|Iterated kernel]]) of a Hermitian kernel are also Hermitian.
 
All iterated kernels (cf. [[Iterated kernel|Iterated kernel]]) of a Hermitian kernel are also Hermitian.
  
The linear integral operator generated by a Hermitian kernel is self-adjoint. A Hermitian kernel is called complete (or closed) if the system of its eigen functions is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706015.png" />; otherwise it is called incomplete. A Hermitian kernel is called positive (negative) if all its eigen values are positive (negative). A complete positive (negative) kernel is called positive (negative) definite.
+
The linear integral operator generated by a Hermitian kernel is self-adjoint. A Hermitian kernel is called complete (or closed) if the system of its eigen functions is complete in $  L _ {2} [ a , b ] $;  
 +
otherwise it is called incomplete. A Hermitian kernel is called positive (negative) if all its eigen values are positive (negative). A complete positive (negative) kernel is called positive (negative) definite.
  
The interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706016.png" /> can be replaced by a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706017.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706018.png" />-dimensional Euclidean space.
+
The interval $  [ a , b ] $
 +
can be replaced by a domain $  \Omega $
 +
in an $  n $-
 +
dimensional Euclidean space.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  pp. Chapt. 1  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Gleichungen der mathematischen Physik" , MIR  (1984)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  pp. Chapt. 1  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.S. Vladimirov,  "Gleichungen der mathematischen Physik" , MIR  (1984)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
Another result connected with the series (2) is Mercer's theorem: Let the kernel be continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706019.png" /> and suppose that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706021.png" />,
+
Another result connected with the series (2) is Mercer's theorem: Let the kernel be continuous on $  [ a , b ] \times [ a , b ] $
 +
and suppose that for all $  f $
 +
in $  L _ {2} [ a , b ] $,
  
<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/h/h047/h047060/h04706022.png" /></td> </tr></table>
+
$$
 +
\int\limits _ { a } ^ { b }  \int\limits _ { a } ^ { b }
 +
K ( x , y ) f ( y)
 +
\overline{ {f ( x) }}\; d y  d x  \geq  0 .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706023.png" /> is an orthonormal sequence of (continuous) eigen functions corresponding to the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706024.png" /> then the series (2) converges absolutely and uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706025.png" /> and the sum of (2) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706026.png" />.
+
If $  \phi _ {1} , \phi _ {2} ,\dots $
 +
is an orthonormal sequence of (continuous) eigen functions corresponding to the eigen values $  \lambda _ {1} , \lambda _ {2} \dots $
 +
then the series (2) converges absolutely and uniformly on $  [ a , b ] \times [ a , b ] $
 +
and the sum of (2) is $  K ( x , y ) $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.G. Tricomi,  "Integral equations" , Interscience  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F.G. Tricomi,  "Integral equations" , Interscience  (1957)</TD></TR></table>

Revision as of 22:10, 5 June 2020


A complex-valued function $ K ( x , y ) $ that is square-integrable on $ [ a , b ] \times [ a , b ] $ and satisfies the condition (of Hermitian symmetry)

$$ \tag{1 } \overline{ {K ( x , y ) }}\; = K ( y , x ) $$

for almost-all $ ( x , y ) \in [ a , b ] \times [ a , b ] $. The bar in (1) denotes transition to the complex-conjugate value. If a Hermitian kernel vanishes almost nowhere, then it has at least one characteristic (eigen) value, that is, there exists a number $ \lambda $ such that the equation

$$ \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x , y ) \phi ( y) d y = 0 $$

has a non-zero solution (an eigen function corresponding to $ \lambda $). All eigen values are real and on any interval there are only finitely many. The eigen functions corresponding to distinct eigen values are orthogonal to each other.

There is an orthonormal (finite or infinite) sequence of eigen functions $ \phi _ {1} , \phi _ {2} \dots $ corresponding to the eigen values $ | \lambda _ {1} | \leq | \lambda _ {2} | \leq \dots $. The series

$$ \tag{2 } \sum _ { k= } 1 ^ \infty \frac{\phi _ {k} ( x) \overline{ {\phi _ {k} ( y) }}\; }{\lambda _ {k} } $$

converges in the mean on the square $ [ a , b ] \times [ a , b ] $ to the kernel $ K $. If the kernel is continuous and the series (2) converges uniformly on $ [ a , b ] \times [ a , b ] $, then its sum is $ K $. The system of eigen values and eigen functions of a Hermitian kernel is finite if and only if the kernel is degenerate (cf. Degenerate kernel).

All iterated kernels (cf. Iterated kernel) of a Hermitian kernel are also Hermitian.

The linear integral operator generated by a Hermitian kernel is self-adjoint. A Hermitian kernel is called complete (or closed) if the system of its eigen functions is complete in $ L _ {2} [ a , b ] $; otherwise it is called incomplete. A Hermitian kernel is called positive (negative) if all its eigen values are positive (negative). A complete positive (negative) kernel is called positive (negative) definite.

The interval $ [ a , b ] $ can be replaced by a domain $ \Omega $ in an $ n $- dimensional Euclidean space.

References

[1] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) pp. Chapt. 1 (Translated from Russian)
[2] V.S. Vladimirov, "Gleichungen der mathematischen Physik" , MIR (1984) (Translated from Russian)

Comments

Another result connected with the series (2) is Mercer's theorem: Let the kernel be continuous on $ [ a , b ] \times [ a , b ] $ and suppose that for all $ f $ in $ L _ {2} [ a , b ] $,

$$ \int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } K ( x , y ) f ( y) \overline{ {f ( x) }}\; d y d x \geq 0 . $$

If $ \phi _ {1} , \phi _ {2} ,\dots $ is an orthonormal sequence of (continuous) eigen functions corresponding to the eigen values $ \lambda _ {1} , \lambda _ {2} \dots $ then the series (2) converges absolutely and uniformly on $ [ a , b ] \times [ a , b ] $ and the sum of (2) is $ K ( x , y ) $.

References

[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a2] F.G. Tricomi, "Integral equations" , Interscience (1957)
How to Cite This Entry:
Hermitian kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hermitian_kernel&oldid=47219
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article