View source for Iterated kernel

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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]])

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

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====
<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">  S.G. Mikhlin,   "Linear integral equations" , Hindushtan Publ. Comp. , Delhi  (1960)  (Translated from Russian)</TD></TR></table>



====Comments====


====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">  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)</TD></TR></table>

{{SourceattributionEOM|Iterated kernel}}
[[Category:Mathematics]]