Accurate discontinuity detection using limited resolution information

Let low resolution spline wavelet or Fourier coefficient information be available for a function f=g+ϵf=g+ϵ where gg is a piecewise polynomial with jump discontinuities of itself and its derivatives and ϵϵ is the noise. We construct a function rr such that the convolution r∗gr∗g is a polynomial in the neighborhood of the jump and has the jump location as root, and such that the convolution can be calculated using only the available information and a rectangle rule quadrature. Applying this calculation to f=g+ϵf=g+ϵ yields a polynomial which is perturbed from r∗gr∗g by an amount proportional to the L2L2-norm of ϵϵ. Some methods lose accuracy when large derivative jumps coincide with function jumps and resolution is limited, especially in the presence of noise. The present method maintains reasonable accuracy even with large derivative jumps and noise ‖ϵ‖2≈.02‖f‖2‖ϵ‖2≈.02‖f‖2. The present method is a local method, and requires some other strategy to locate the proper polynomial regions. We present a simple method which produces approximate jump locations close enough to actual ones to locate the desired polynomial regions.