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

Approximating a function from its values f(xi)f(xi) at a set of evenly spaced points xixi through (N+1)(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 NN. Normalizing the span of the points to [−1,1][−1,1], the new strategy applies a filtered trigonometric interpolant on the subinterval x∈[−1+D,1−D]x∈[−1+D,1−D] and ordinary polynomial interpolation in the two remaining subintervals. Convergence is guaranteed because the width DD of the polynomial interpolation subintervals decreases   as N→∞N→∞, being proportional to 1/N. Applications to the Gibbs Phenomenon and hydrodynamic shocks are discussed.