Approximation of Polynomial Singularity Curves of Piecewise Polynomial Functions of Two Variables by Their Fourier-Jacobi Coefficients

Abstract

In the present paper, we generalize the result obtained in an earlier paper by the author with respect to recovering the singularity curves of a piecewise constant function of two variables by its Fourier-Jacobi coefficients. The method is based on a modified Prony’s method suggested in an earlier paper by us for recovering (approximating) the locations of discontinuities of a piecewise constant (smooth) function of one variable. It is shown that if the singularity curves of a piecewise polynomial function of two variables are polynomials and some indices of the Jacobi system are integers, then the curves can be approximated within $O(n^{−2})$ precision and the associated jump magnitudes within $O(n^{−1})$, given $O(n)$ Fourier-Jacobi coefficients of the function. For some nonconstant piecewise polynomial functions the singularity curves and the associated jump magnitudes may be recovered exactly. In addition, numerical examples are presented to illustrate the accuracy of the method.