Sharp stability bounds for perturbed Fourier interpolation

The manuscript proves two bounds conjectured by Austin and Trefethen for
trigonometric interpolation of degree $N$
in $m = 2N + 1$ points, each displaced from the equispaced grid by at most
$\alpha$ grid spacings: the Lebesgue constant is at most
$C(N^{2\alpha} - 1)/(\alpha(1 - 2\alpha))$ with $C$ absolute, for all $N \ge 2$
and $0 < \alpha < 1/2$, and the normalised Fourier matrix $F$ of the nodes
satisfies $\|F^{-1}\|_2 \le C_\alpha N^{4\alpha - 1}$ for each fixed
$1/4 < \alpha < 1/2$, with $C_\alpha$ independent of $N$ and the nodes. The first bound rests on the node polynomial divided by
$z^m + 1$, which, suitably rotated, has positive real part in the disk, as does
its reciprocal; the Poisson representation of the reciprocal and a radial
comparison bound each cardinal function, and these bounds are summed over
dyadic annuli. For the second, the node polynomial is compared with the
exponential of a Herglotz integral of the shifts, the weak-type $(1,1)$ bound
for the periodic Hilbert transform gives local second-moment estimates, and a
dyadic block decomposition bounds the matrix of interpolation from the
perturbed to the equispaced nodes, whose norm is $\|F^{-1}\|_2$.