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Difference between revisions of "Iterated kernel"

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A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i0530001.png" /> that is formed from the given kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i0530002.png" /> of an [[Integral operator|integral operator]] (cf. [[Kernel of an integral operator|Kernel of an integral operator]])
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A function $(x,s)\mapsto K_n(x,s)$ that is formed from the given kernel $K$ of an [[Integral operator|integral operator]] (cf. [[Kernel of an integral operator|Kernel of an integral operator]])
  
<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/i/i053/i053000/i0530003.png" /></td> </tr></table>
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$$A\phi(x)=\int\limits_a^bK(x,t)\phi(t)dt,$$
  
 
by the recurrence relations
 
by the recurrence relations
  
<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/i/i053/i053000/i0530004.png" /></td> </tr></table>
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$$K_1(x,s)=K(x,s),\quad K_n(x,s)=\int\limits_a^bK_{n-1}(x,t)K(t,s)dt.$$
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i0530005.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i0530006.png" />-th iterate, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i0530008.png" />-th iterated kernel, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i0530009.png" />. An iterated kernel is sometimes called a repeated kernel. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i05300010.png" /> is a continuous or square-integrable kernel, then all its iterates are continuous, respectively, square integrable. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i05300011.png" /> is a symmetric kernel, so are all its iterates. The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i05300012.png" /> is the kernel of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i053/i053000/i05300013.png" />. The equality
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$K_n$ is called the $n$-th iterate, or $n$-th iterated kernel, of $K$. An iterated kernel is sometimes called a repeated kernel. If $K$ is a continuous or square-integrable kernel, then all its iterates are continuous, respectively, square integrable. If $K$ is a symmetric kernel, so are all its iterates. The kernel $K_n$ is the kernel of the operator $A^n$. 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/i/i053/i053000/i05300014.png" /></td> </tr></table>
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$$K_n(x,s)=\int\limits_a^bK_{n-m}(x,t)K_m(t,s)dt,\quad1\leq m\leq n-1,$$
  
 
holds.
 
holds.

Latest revision as of 19:24, 9 October 2014

A function $(x,s)\mapsto K_n(x,s)$ that is formed from the given kernel $K$ of an integral operator (cf. Kernel of an integral operator)

$$A\phi(x)=\int\limits_a^bK(x,t)\phi(t)dt,$$

by the recurrence relations

$$K_1(x,s)=K(x,s),\quad K_n(x,s)=\int\limits_a^bK_{n-1}(x,t)K(t,s)dt.$$

$K_n$ is called the $n$-th iterate, or $n$-th iterated kernel, of $K$. An iterated kernel is sometimes called a repeated kernel. If $K$ is a continuous or square-integrable kernel, then all its iterates are continuous, respectively, square integrable. If $K$ is a symmetric kernel, so are all its iterates. The kernel $K_n$ is the kernel of the operator $A^n$. The equality

$$K_n(x,s)=\int\limits_a^bK_{n-m}(x,t)K_m(t,s)dt,\quad1\leq m\leq n-1,$$

holds.

References

[1] V.I. Smirnov, "A course of higher mathematics" , 4 , Addison-Wesley (1964) pp. Chapt. 1 (Translated from Russian)
[2] S.G. Mikhlin, "Linear integral equations" , Hindushtan Publ. Comp. , Delhi (1960) (Translated from Russian)


Comments

References

[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[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)
How to Cite This Entry:
Iterated kernel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Iterated_kernel&oldid=33522
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article