Polynomial frames for the detection of singularities

Abstract We propose a class of algebraic polynomial frames, which are computationally easier to implement than polynomial bases. We also discuss the weighted Lstability of our frames for 1 ≤ p ≤ ∞. Our analysis is based on orthogonal polynomials with respect to the weight in question, but the frame bounds are independent of the system of orthogonal polynomials used. In spite of the fact that algebraic polynomials are inherently nonlocal, our frames provide good localization properties. In particular, they can be used to detect discontinuities in derivatives of all orders of a function. We describe asymptotic expressions for the frame coefficients in the vicinity of a discontinuity.