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Let $K$ be the integral operator defined by

$ (Kf)(x)=\int_0^1 u(x)v(y)f(y) dy $

for some continuous functions $u,v$ in the Hilbert space with inner product $\langle f,g \rangle = \int_0^1 f(x)^* g(x) dx$ on $(0,1)$. I want to find the eigenfunctions and eigenvalues corresponding to $K$. (this is problem 3.4 in http://www.mat.univie.ac.at/~gerald/ftp/book-fa/ )

The exercise is from a chapter about compact symmetric operators (which this operator is), but it only contains existence theorems.

If I could get some helpful hints on how to get started, I'd be thankful. (I have a suspicion this is easier than it looks)

Thanks in advance.

1 Answers 1

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You asked for a hint: Notice you can take $u(x)$ out of the integral!

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    :) Of course! Thanks!2011-07-11