View source for Mercer theorem

The bilinear series

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of a Hermitian positive-definite continuous kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634402.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634403.png" /> (cf. [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]]; [[Kernel of an integral operator|Kernel of an integral operator]]), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634404.png" /> is the closure of a bounded domain in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634405.png" />, converges absolutely and uniformly in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634406.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634407.png" />. Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634408.png" /> are the characteristic numbers of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m0634409.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344010.png" /> are the corresponding orthonormalized eigen functions. If a kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344011.png" /> satisfies the conditions of Mercer's theorem, then the integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344012.png" />,

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is nuclear (cf. [[Nuclear operator|Nuclear operator]]) and its [[Trace|trace]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m063/m063440/m06344014.png" /> can be calculated by the formula

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Mercer's theorem can be generalized to the case of a bounded discontinuous kernel.

The theorem was proved by J. Mercer [[#References|[1]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Mercer,   ''Philos. Trans. Roy. Soc. London Ser. A'' , '''209'''  (1909)  pp. 415–446</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Mercer,   "Functions of positive and negative type, and their connection with the theory of integral equations"  ''Proc. Roy. Soc. London Ser. A'' , '''83'''  (1908)  pp. 69–70</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.G. Petrovskii,   "Lectures on the theory of integral equations" , Graylock  (1957)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.G. Tricomi,   "Integral equations" , Interscience  (1957)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.A. Krasnosel'skii,   et al.,   "Integral operators in spaces of summable functions" , Noordhoff  (1976)  (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  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A.C. Zaanen,   "Linear analysis" , North-Holland  (1956)</TD></TR></table>