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 <= alpha h, where alpha is an element of[0, 1/2) is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be trouble for alpha >= 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 are enough. Connections with the Fejer-Kalmar theorem are discussed.