A note on singularity approximation

A simple method is considered for approximating the locations of singularities and the associated jumps of a piecewise constant function. The locations of jump discontinuities of a function are recovered approximately, one by one, by means of ratios of so called higher order Fourier–Jacobi coefficients of the function.It is shown that the location of singularity of a piecewise constant function with one discontinuity is recovered exactly and the locations of singularities of a piecewise constant function with multiple discontinuities are recovered with exponential accuracy. The method is applicable to piecewise smooth functions as well, however the accuracy of the approximation sharply declines.In addition, the stability and complexity of the method is discussed and some numerical examples are presented.