Adaptive interpolation with maximum order close to discontinuities

• We present a new general method to approximate functions with maximum order at the points close to the isolated discontinuities. • We present a family of non-linear weights and a general formula to calculate them. • The approximation using this new method presents a more approximate behaviour than when WENO or linear method are used. • We perform several experiments to check the theoretical results. Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to this method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique but the design of the weights in this case is more simple. Also, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are performed to demonstrate our results and to compare them with standard methods.