On a build-up polynomial frame for the detection of singularities

Let f : [−1, 1] → IR, x0 ∈ (−1, 1), r ≥ 0 be an integer. The point x0 is called a singularity of f of order r if the derivative f (r) has a jump discontinuity at x0, but is continuous at every other point of some neighborhood of x0. In this paper, we propose a sequence of polynomial operators {τj} with the following properties. Each τj is computed using the values f(cos(kπ/2 j)), k = 1, · · · , 2j − 1, and the quantity τj(f, x) is “large” near a singularity, and “small” away from it. Precise quantitative estimates are given.