Trigonometric Interpolation and Quadrature in Perturbed Points

The trigonometric interpolants to a periodic function in equispaced grids converge if the function is Dini-continuous, and the associated quadrature rule, the trapezoid rule, converges if the function is continuous. We investigate the robustness of these results in the presence of perturbations to the grid points. We present theorems that quantify the effects of perturbing the points on the rates of convergence of both the approximation and quadrature schemes and explore connections with sampling theory, the Kadec 1/4 theorem, and the Fejér-Kalmár-Walsh theorem.