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Difference between revisions of "Kernel of an integral operator"

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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055320/k0553201.png" /> in two variables that defines an [[Integral operator|integral operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055320/k0553202.png" /> by the equality
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A function $K(x,y)$ in two variables that defines an [[Integral operator|integral operator]] $A$ by the equality
  
<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/k/k055/k055320/k0553203.png" /></td> </tr></table>
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$$\psi(y)=A[\phi(x)]=\int K(x,y)\phi(x)\,d\mu(x),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055320/k0553204.png" /> ranges over a [[Measure space|measure space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055320/k0553205.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055320/k0553206.png" /> belongs to a certain space of functions defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055320/k0553207.png" />.
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where $x$ ranges over a [[Measure space|measure space]] $(X,d\mu)$ and $\phi$ belongs to a certain space of functions defined on $X$.
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)  {{MR|0632943}} {{ZBL|0458.47001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  V.S. Sunder,  "Bounded integral operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055320/k0553208.png" /> spaces" , Springer  (1978)  {{MR|517709}} {{ZBL|0389.47001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)  {{MR|0461049}} {{ZBL|0207.44602}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)  {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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)  {{MR|}} {{ZBL|}} </TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)  {{MR|0632943}} {{ZBL|0458.47001}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  V.S. Sunder,  "Bounded integral operators on $L^2$ spaces" , Springer  (1978)  {{MR|517709}} {{ZBL|0389.47001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)  {{MR|0461049}} {{ZBL|0207.44602}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''4''' , Addison-Wesley  (1964)  (Translated from Russian)  {{MR|0182690}} {{MR|0182688}} {{MR|0182687}} {{MR|0177069}} {{MR|0168707}} {{ZBL|0122.29703}} {{ZBL|0121.25904}} {{ZBL|0118.28402}} {{ZBL|0117.03404}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  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)  {{MR|}} {{ZBL|}} </TD></TR></table>

Latest revision as of 17:02, 30 December 2018

A function $K(x,y)$ in two variables that defines an integral operator $A$ by the equality

$$\psi(y)=A[\phi(x)]=\int K(x,y)\phi(x)\,d\mu(x),$$

where $x$ ranges over a measure space $(X,d\mu)$ and $\phi$ belongs to a certain space of functions defined on $X$.


Comments

References

[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981) MR0632943 Zbl 0458.47001
[a2] P.R. Halmos, V.S. Sunder, "Bounded integral operators on $L^2$ spaces" , Springer (1978) MR517709 Zbl 0389.47001
[a3] K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970) MR0461049 Zbl 0207.44602
[a4] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) (Translated from Russian) MR0182690 MR0182688 MR0182687 MR0177069 MR0168707 Zbl 0122.29703 Zbl 0121.25904 Zbl 0118.28402 Zbl 0117.03404
[a5] 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)
How to Cite This Entry:
Kernel of an integral operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_an_integral_operator&oldid=28227
This article was adapted from an original article by G.L. Litvinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article