Suppose that there is a Fourier series: \begin{equation} K(x) = \frac{1}{L} \sum_{k \in \mathbb{Z}} m \left( \frac{k\pi}{L} \right) \cos\left( \frac{k\pi x}{L} \right), \quad x \in (-L,L), \end{equation} where $m(\xi) = (1 + \xi^2)^{-s/2}$, $\xi \in \mathbb{R}$, is a Bessel potential of order $s > 1$, and $L \in (0,\infty)$. I know that the sum is uniformly convergent.
How can I show that this series is strictly decreasing on $(0,L)$? Is there an explicit formula for $K'(x)$?
Can we say something about the regularity of $K$? I think $K$ is in the Sobolev space $W^{s-\epsilon,1}(-L,L)$ for any $\epsilon \in (0,1)$. That is, $K$ is Hölder continuous when $s \le 2$ and continuously differentiable when $s > 2$.
Thank you for any reference/help.