A robust method for accurately representing nonperiodic functions given Fourier coefficient information

When a function is smooth but not smoothly periodic with a particular period, and nonetheless is represented by partial sums of a Fourier series calculated using that period, the well-known Gibbs phenomenon defeats uniform convergence of the sums, and convergence is slow. In recent years, several workers have developed methods for recovering accurate and fast converging representations for functions in this situation. These efforts have not concentrated on bounds for the operators corresponding to the methods, and thus have not explicitly proven robustness in the presence of noise. In this paper we present a method for which explicit bounds are established for the operator. The method is, in effect, least-squares fitting of the given Fourier coefficients by the coefficients of polynomial splines with appropriate discontinuities. We obtain bounds by exact calculations of projections in spline spaces, using a computer algebra system. We give examples of the method and two other published methods working with noisy data.