Trigonometric Interpolation and Quadrature in Perturbed Points
This work extends classical results on trigonometric interpolation and quadrature to perturbed grids, providing theoretical guarantees for a practical scenario where points are not exactly equispaced.
The paper studies trigonometric interpolation and quadrature for periodic functions when evaluation points are perturbed by up to αh, with α < 1/2. It proves convergence for twice continuously differentiable functions and shows that the required smoothness depends on α, conjecturing that 2α derivatives suffice.
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 $\le αh$, where $α\in [\kern .5pt 0,1/2)$ is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be be trouble for $α\ge 1/4$. We show that convergence of both the interpolants and the quadrature estimates is guaranteed for all $α<1/2$ 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 $4α$ derivatives in a certain sense, and we conjecture that $2α$ derivatives is enough. Connections with the Fejér--Kalmár theorem are discussed.