Optimal interpolation

Abstract

The classical interpolation problem is considered of estimating a function of one variable, f(.), given a number of function values f(x_i), i=1,2,...,m. If a bound on ‖f^(k)(.)‖_∞ is given also, k≤m, then bounds on f(ζ) can be found for any ζ. A method of calculating the closest bounds is described, which is shown to be relevant to the problem of finding the interpolation formula whose error is bounded by the smallest possible multiple of ‖f^(k)(.)‖_∞, when ‖f^(k)(.)‖_∞ is unknown. This formula is identified and is called the optimal interpolation formula. The corresponding interpolating function is a spline of degree (k-1) with (m-k) knots, so it is very suitable for practical computation.