Iterated kernel

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)

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)