Difference between revisions of "Kernel of an integral operator"
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A function $K(x,y)$ in two variables that defines an [[Integral operator|integral operator]] $A$ by the equality | A function $K(x,y)$ in two variables that defines an [[Integral operator|integral operator]] $A$ by the equality | ||
| − | $$\psi(y)=A[\phi(x)]=\int K(x,y)\phi(x)d\mu(x),$$ | + | $$\psi(y)=A[\phi(x)]=\int K(x,y)\phi(x)\,d\mu(x),$$ |
where $x$ ranges over a [[Measure space|measure space]] $(X,d\mu)$ and $\phi$ belongs to a certain space of functions defined on $X$. | where $x$ ranges over a [[Measure space|measure space]] $(X,d\mu)$ and $\phi$ belongs to a certain space of functions defined on $X$. | ||
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=43607
Kernel of an integral operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_of_an_integral_operator&oldid=43607
This article was adapted from an original article by G.L. Litvinov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article