Trigonometric interpolation and quadrature in perturbed points

Journal article

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 ≤ α h, where α ∈ [0,1/2) is a fixed constant. The Kadec 1/4 theorem of sampling theory suggests there may be be trouble for α ≥ 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.

Authors: Austin, A; Trefethen, L

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Society for Industrial and Applied Mathematics
• Journal: SIAM Journal on Numerical Analysis
• Volume: 55
• Issue: 5
• Pages: 2113–2122
• Publication date: 2017-09-01
• Acceptance date: 2017-06-27
• DOI: 10.1137/16M1107760
• Keywords: Lebesgue constant; Kadec 1/4 theorem; quadrature; sampling theory; trigonometric interpolation; Fejér-Kalmár theorem