Trigonometric Interpolation and Quadrature in Perturbed Points

The trigonometric interpolants to a periodic function $f$ in equispaced
points converge if $f$ is Dini-continuous, and the associated quadrature
formula, the trapezoidal rule, converges if $f$ is continuous. What if the
points are perturbed? With equispaced grid spacing $h$, let each point be
perturbed by an arbitrary amount $\le \alpha h$, where $\alpha\in [\kern .5pt
0,1/2)$ is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests
there may be be trouble for $\alpha\ge 1/4$. We show that convergence of both
the interpolants and the quadrature estimates is guaranteed for all
$\alpha<1/2$ if $f$ is twice continuously differentiable, with the convergence
rate depending on the smoothness of $f$. More precisely it is enough for $f$ to
have $4\alpha$ derivatives in a certain sense, and we conjecture that $2\alpha$
derivatives is enough. Connections with the Fej\'er--Kalm\'ar theorem are
discussed.