View source for Hermitian kernel

A complex-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470601.png" /> that is square-integrable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470602.png" /> and satisfies the condition (of Hermitian symmetry)

<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/h/h047/h047060/h0470603.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470604.png" />. The bar in (1) denotes transition to the complex-conjugate value. If a Hermitian kernel vanishes almost nowhere, then it has at least one characteristic (eigen) value, that is, there exists a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470605.png" /> such that the equation

<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/h/h047/h047060/h0470606.png" /></td> </tr></table>

has a non-zero solution (an eigen function corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470607.png" />). All eigen values are real and on any interval there are only finitely many. The eigen functions corresponding to distinct eigen values are orthogonal to each other.

There is an orthonormal (finite or infinite) sequence of eigen functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470608.png" /> corresponding to the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h0470609.png" />. The series

<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/h/h047/h047060/h04706010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

converges in the mean on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706011.png" /> to the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706012.png" />. If the kernel is continuous and the series (2) converges uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706013.png" />, then its sum is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706014.png" />. The system of eigen values and eigen functions of a Hermitian kernel is finite if and only if the kernel is degenerate (cf. [[Degenerate kernel|Degenerate kernel]]).

All iterated kernels (cf. [[Iterated kernel|Iterated kernel]]) of a Hermitian kernel are also Hermitian.

The linear integral operator generated by a Hermitian kernel is self-adjoint. A Hermitian kernel is called complete (or closed) if the system of its eigen functions is complete in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706015.png" />; otherwise it is called incomplete. A Hermitian kernel is called positive (negative) if all its eigen values are positive (negative). A complete positive (negative) kernel is called positive (negative) definite.

The interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706016.png" /> can be replaced by a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706017.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706018.png" />-dimensional Euclidean space.

====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">  V.S. Vladimirov,   "Gleichungen der mathematischen Physik" , MIR  (1984)  (Translated from Russian)</TD></TR></table>



====Comments====
Another result connected with the series (2) is Mercer's theorem: Let the kernel be continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706019.png" /> and suppose that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706021.png" />,

<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/h/h047/h047060/h04706022.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706023.png" /> is an orthonormal sequence of (continuous) eigen functions corresponding to the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706024.png" /> then the series (2) converges absolutely and uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706025.png" /> and the sum of (2) is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047060/h04706026.png" />.

====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">  F.G. Tricomi,   "Integral equations" , Interscience  (1957)</TD></TR></table>