Let $K(x,y)=K(y,x)$ be a continuous symmetric function. The integral equation $$\varphi(x)=\lambda\int_a^b K(x,y)\varphi(y)dy$$ has eigenvalues $\lambda_n$ and eigenfunctions $\varphi_n(x)$. It is known that if $$\sum\limits_{n=1}^{\infty}\frac{\varphi_n(x)\varphi_n(y)}{\lambda_n}$$ converges uniformly then it converges to $K(x,y)$.

What is an example of $K(x,y)$ where the series does not converge to $K(x,y)$ ?

The interval $[a,b]$ is finite.

Perhaps an example with $$\sum\frac{\sin(nx)\sin(ny)}{n} ?$$


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