Convergence Rates for Fourier Partial Sums of Polygons and Periodic Splines

Abstract

Polygons can be seen as closed parameterized curves. Their parameterizations can be chosen as continuous, piecewise linear, periodic functions. Such functions possess a convergent Fourier series. Often polygons are classified with Fourier descriptors defined via Fourier coefficients of the parameterization. This fact motivates the discussion of the approximation error of Fourier partial sums of piecewise linear functions. More generally, the paper investigates convergence rates for periodic splines using elementary techniques of calculus. For example, such splines are used as curve parameterizations for active contours. Error bounds are shown to be best possible. An interesting effect is that the convergence rate at knots is different for odd and even degrees of piecewise polynomials. The slower rate for polynomials of odd degree can be used to detect dominant corners of contours.