Approximation of the discontinuities of a function by its classical orthogonal polynomial Fourier coefficients

In the present paper, we generalize the method suggested in an earlier paper by the author and overcome its main deficiency. First, we modify the well-known Prony method, which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its Fourier coefficients with respect to any system of the classical orthogonal polynomials. Next, we will show that the method is applicable to a wider class of functions, namely, to the class of piecewise smooth functions—for functions which piecewise belong to C2[−1, 1], the locations of discontinuities are approximated to within O(1/n) by means of their Fourier-Jacobi coefficients. Unlike the previous one, the generalized method is robust, since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial. In addition, the error estimate is uniform for any [c, d] ⊂ (−1, 1). To the end, we discuss the accuracy, stability, and complexity of the method and present numerical examples.