Methods to evaluate the largest eigenvalue of an integral operator Consider the following integral operator
$$
(\mathcal Ku)(x)= \int_{-\pi}^\pi K(y-x) u(y) dy
$$
For example, $K(s) = |s|$ is of my particular interest. I'm searching for a method to obtain the largest eigenvalue $\lambda_0$ of this operator.
What have I tried: Fourier transform/series - no luck unless $K(s)$ is $2\pi$ periodic, but this is not the case, since $K(s)$ is used on $[-2\pi, 2\pi]$ and cannot be $2\pi$ periodic extended.
PS. Numerical experiments show that $\frac{1}{\lambda_0} \approx 0.0729122$. That was obtained using Galerkin method with scaled Legendre polynomials as basis.
 A: For $K(s) = |s|$ the solution was not so hard: 
$$
(\mathcal Ku)(x) = \int_{-\pi}^x (x-y) u(y) dy + \int_x^\pi (y-x) u(y) dy
$$
Differentiating twice yield
$$
(\mathcal Ku)'(x) = \int_{-\pi}^x u(y) dy - \int_x^\pi u(y) dy,\\
(\mathcal Ku)''(x) = 2u(x).
$$
This is because $K(s)$ is (up to a constant factor) fundamental solution for the 1D Laplace operator, i.e.
$$
K''(z) = 2\delta(z).
$$
So the equation $\mathcal Ku = \lambda u$ leads to
$$
2u(x) = \lambda u''(x).
$$
Moreover, 
$$
\lambda u(\pm \pi) = (\mathcal{Ku})(\pm \pi) = \int_{-\pi}^\pi (\pi\mp y)u(y) dy = \pi\int_{-\pi}^\pi u(y)dy \mp \int_{-\pi}^\pi yu(y)dy,\\
\lambda u'(\pm \pi) = (\mathcal{Ku})'(\pm \pi) = \pm\int_{-\pi}^\pi u(y) dy \implies u'(-\pi) = -u'(\pi),\\
\lambda( \pi u'(-\pi) + u(-\pi) ) = -\int_{-\pi}^\pi yu(y) dy = 
\lambda( \pi u'(\pi) - u(\pi) ).
$$
So now we have a Sturm-Liouville problem
$$
u''(x) = \frac{2}{\lambda} u(x)\\
u'(-\pi) + u'(\pi) = 0\\
\pi u'(-\pi) - \pi u'(\pi) + u(-\pi) + u(\pi) = 0
$$
The solution to the equation is 
$$
u(x) = A \cosh k x + B \sinh k x, \qquad k = \sqrt\frac{2}{\lambda}
$$
Consider only $k > 0$ (I'm interested in $\lambda > 0$). From the first boundary condition $B = 0$. From the second we obtain a transcendent equation for $k$:
$$
\cosh k \pi - k \pi \sinh k \pi = 0 \implies \coth k \pi = k \pi, \qquad 
k \approx 0.381869571437564,\\
\lambda^{-1} = \frac{k^2}{2} \approx 0.0729121847949544.
$$
