Symmetric Iterative Interpolation Processes

Abstract

Using a base b and an even number of knots, we define a symmetric iterative interpolation process. The main properties of this process come from an associated function F. The basic functional equation for F is that F(t/b) = $[\sum\nolimits_n {F(n/b)F(t - n)} ]$. We prove that F is a continuous positive definite function. We find almost precisely in which Lipschitz classes derivatives of F belong. If a function y is defined only on integers, this process extends y continuously to the real axis as $[y(t) = \sum\nolimits_n {y(n)F(t - n)} ]$. Error bounds for this iterative interpolation are given.