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''reciprocal kernels''
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Two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m0655901.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m0655902.png" /> of real variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m0655903.png" /> (or, in general, of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m0655904.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m0655905.png" /> of a Euclidean space), defined on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m0655906.png" /> and satisfying the condition
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{{TEX|auto}}
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{{TEX|done}}
  
<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/m/m065/m065590/m0655907.png" /></td> </tr></table>
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''reciprocal kernels''
  
<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/m/m065/m065590/m0655908.png" /></td> </tr></table>
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Two functions  $  K ( x, s) $
 +
and  $  K _ {1} ( x, s) $
 +
of real variables  $  x, s $(
 +
or, in general, of points  $  P $
 +
and  $  Q $
 +
of a Euclidean space), defined on the square  $  [ a, b] \times [ a, b] $
 +
and satisfying the condition
  
If a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m0655909.png" /> reciprocal with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559010.png" /> exists, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559011.png" /> is the resolvent kernel of the integral [[Fredholm equation|Fredholm equation]]
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$$
 +
K _ {1} ( x, s) - K ( x, s) =
 +
$$
  
<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/m/m065/m065590/m06559012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$
 +
= \
 +
\int\limits _ { a } ^ { b }  K ( x, t) K _ {1} ( t, s)  dt
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= \int\limits _ { a } ^ { b }  K _ {1} ( x, t) K ( t, s) dt.
 +
$$
  
 +
If a kernel  $  K _ {1} ( x, s) $
 +
reciprocal with  $  K ( x, s) $
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exists, then  $  K _ {1} ( x, s) $
 +
is the resolvent kernel of the integral [[Fredholm equation|Fredholm equation]]
  
 +
$$ \tag{* }
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\phi ( x) - \int\limits _ { a } ^ { b }  K ( x, s) \phi ( s)  ds  =  f ( x).
 +
$$
  
 
====Comments====
 
====Comments====
Indeed, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559014.png" /> are reciprocal kernels, the solution of equation (*) above is given by
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Indeed, when $  K( x, s) $
 +
and $  K _ {1} ( x, s) $
 +
are reciprocal kernels, the solution of equation (*) above is given 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/m/m065/m065590/m06559015.png" /></td> </tr></table>
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$$
 +
\phi ( x)  = f( x) + \int\limits _ { a } ^ { b }  K _ {1} ( x, t) f( t)  dt.
 +
$$
  
 
Consider the Fredholm equation
 
Consider the Fredholm 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/m/m065/m065590/m06559016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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$$ \tag{a1 }
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\phi ( x)  = f( x) + \lambda \int\limits _ { a } ^ { b }  K ( x, t) \phi ( t) dt
 +
$$
  
and the iterated kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559017.png" />,
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and the iterated kernels $  K  ^ {(} 1) ( x, t) = K ( x, t) $,
  
<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/m/m065/m065590/m06559018.png" /></td> </tr></table>
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$$
 +
K  ^ {(} n) ( x, t)  = \int\limits _ { a } ^ { b }  K  ^ {(} n- 1) ( x, s) K( s, t)  ds,
 +
\  n = 2, 3 ,\dots.
 +
$$
  
 
Form the [[Neumann series|Neumann series]]
 
Form the [[Neumann series|Neumann series]]
  
<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/m/m065/m065590/m06559019.png" /></td> </tr></table>
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$$
 +
R( x, t; \lambda )  = K  ^ {(} 1) ( x, t) + \lambda K  ^ {(} 2) ( x, t) + \dots =
 +
$$
  
<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/m/m065/m065590/m06559020.png" /></td> </tr></table>
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$$
 +
= \
 +
\sum _ {n= 1 } ^  \infty  \lambda  ^ {n-} 1 K  ^ {(} n) ( x, t).
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559021.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559022.png" />, this series is uniformly convergent for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559023.png" /> small. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065590/m06559024.png" /> satisfies
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If $  K( x, t) $
 +
is continuous on $  [ a, b] \times [ a, b] $,  
 +
this series is uniformly convergent for $  \lambda $
 +
small. Then $  R( x, t;  \lambda ) $
 +
satisfies
  
<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/m/m065/m065590/m06559025.png" /></td> </tr></table>
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$$
 +
\lambda \int\limits _ { a } ^ { b }  R( x, t; \lambda ) K( t, s)  dt  = R ( x, s; \lambda ) - K( x, s),
 +
$$
  
 
and
 
and
  
<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/m/m065/m065590/m06559026.png" /></td> </tr></table>
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$$
 +
\phi ( x)  = f( x) + \lambda \int\limits _ { a } ^ { b }  R ( x, t; \lambda ) f ( t)  dt
 +
$$
  
 
solves (a1).
 
solves (a1).

Revision as of 08:02, 6 June 2020


reciprocal kernels

Two functions $ K ( x, s) $ and $ K _ {1} ( x, s) $ of real variables $ x, s $( or, in general, of points $ P $ and $ Q $ of a Euclidean space), defined on the square $ [ a, b] \times [ a, b] $ and satisfying the condition

$$ K _ {1} ( x, s) - K ( x, s) = $$

$$ = \ \int\limits _ { a } ^ { b } K ( x, t) K _ {1} ( t, s) dt = \int\limits _ { a } ^ { b } K _ {1} ( x, t) K ( t, s) dt. $$

If a kernel $ K _ {1} ( x, s) $ reciprocal with $ K ( x, s) $ exists, then $ K _ {1} ( x, s) $ is the resolvent kernel of the integral Fredholm equation

$$ \tag{* } \phi ( x) - \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = f ( x). $$

Comments

Indeed, when $ K( x, s) $ and $ K _ {1} ( x, s) $ are reciprocal kernels, the solution of equation (*) above is given by

$$ \phi ( x) = f( x) + \int\limits _ { a } ^ { b } K _ {1} ( x, t) f( t) dt. $$

Consider the Fredholm equation

$$ \tag{a1 } \phi ( x) = f( x) + \lambda \int\limits _ { a } ^ { b } K ( x, t) \phi ( t) dt $$

and the iterated kernels $ K ^ {(} 1) ( x, t) = K ( x, t) $,

$$ K ^ {(} n) ( x, t) = \int\limits _ { a } ^ { b } K ^ {(} n- 1) ( x, s) K( s, t) ds, \ n = 2, 3 ,\dots. $$

Form the Neumann series

$$ R( x, t; \lambda ) = K ^ {(} 1) ( x, t) + \lambda K ^ {(} 2) ( x, t) + \dots = $$

$$ = \ \sum _ {n= 1 } ^ \infty \lambda ^ {n-} 1 K ^ {(} n) ( x, t). $$

If $ K( x, t) $ is continuous on $ [ a, b] \times [ a, b] $, this series is uniformly convergent for $ \lambda $ small. Then $ R( x, t; \lambda ) $ satisfies

$$ \lambda \int\limits _ { a } ^ { b } R( x, t; \lambda ) K( t, s) dt = R ( x, s; \lambda ) - K( x, s), $$

and

$$ \phi ( x) = f( x) + \lambda \int\limits _ { a } ^ { b } R ( x, t; \lambda ) f ( t) dt $$

solves (a1).

The terminology "mutual kernels" and "reciprocal kernels" is rarely used.

References

[a1] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian)
[a2] P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)
[a3] B.L. Moiseiwitsch, "Integral equations" , Longman (1977)
How to Cite This Entry:
Mutual kernels. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mutual_kernels&oldid=12529
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article