View source for Iterated kernel

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]])

<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>

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>

<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

<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>

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>