On Smooth Interpolation

For a given set of n distinct points (x1, y1), . . . , (xn, yn) and a given system of continuously differentiable functions U = {ui} ∞ i=0, u0 ≡ 1, such that the system of the derivatives V = {vi} ∞ i=1, vi := u ′ i, constitutes a Complete Tchebycheff system on [x1, xn], we prove the existence and uniqueness of a generalized polynomial u∗n+k of minimal arclength amongst all generalized polynomials of the form u = c0u0 + · · · + cn+kun+k , ci ∈ R , interpolating to the points. Furthermore, it is shown that the arc-length of u∗n+k on [x1, xn] approaches the arc-length of the polygonal line p passing through the points, as k → ∞. This supplies a generalization of a result of T.J. Rivlin about interpolation by algebraic polynomials. A new characterization of the extremal generalized polynomials u∗n+k is presented as well.