Local Spline Approximation of Discontinuous Functions and Location of Discontinuities, Given Low Order Fourier Coefficient Information

In recent years several workers have published methods for accurately approximating a function with discontinuities, given Fourier coefficient information. Our own contribution concentrates on robustness and use of low order coefficients. In this paper we present a version of our method which is local in that it approximates the orthogonal projection of the function onto a space of splines which in a period interval is supported on a proper subinterval. The approximation is only slightly influenced by behavior of the function outside the subinterval. In particular, discontinuity locations need not be known outside the subinterval. We establish modest bounds for the approximation operator, which proves robustness in the presence of noise. One advantage of this local method is that, compared with the earlier non-local version, it can be less expensive: the cost is related to the dimension of the space of splines and this dimension can be small if the subinterval is small. Another advantage is that this version can be used to locate discontinuities. We choose the support of the spline space so that only one discontinuity in a period interval can lie in the support. We then shift the support systematically, and monitor quantities related to the approximation as functions of the shift. The spline space and monitored quantities are chosen so that easily identified values indicate the location of the discontinuity.