Exponentially accurate Runge-free approximation of non-periodic functions from samples on an evenly spaced grid

AbstractApproximating a function from its values f(xi) at a set of evenly spaced points xi through (N+1)-point polynomial interpolation often fails because of divergence near the endpoints, the “Runge Phenomenon”. This report shows how to achieve an error that decreases exponentially fast with N. Normalizing the span of the points to [−1,1], the new strategy applies a filtered trigonometric interpolant on the subinterval x∈[−1+D,1−D] and ordinary polynomial interpolation in the two remaining subintervals. Convergence is guaranteed because the width D of the polynomial interpolation subintervals decreases as N→∞, being proportional to 1/N. Applications to the Gibbs Phenomenon and hydrodynamic shocks are discussed