Local monotone interpolation and one-parameter groups

This paper discusses the use of one-parameter groups consisting of the diffeomorphisms of the unit interval [ 0, 1] for constructing the spline interpolants preserving the monotonicity of the given data sets. The problems where the violation of the monotonicity of the original data is inappropriate, arise in many application areas of the spline interpolation.We show that the proposed choice of local Hermite interpolants in the form of superpositions of the diffeomorphisms be- longing to definite one-parameter groups guarantees the shape preserving for strictly monotone data. To select suitable groups and study their properties, it is convenient to define their elements implicitly, using as the starting point the corre- sponding Schröder’s equation or the infinitesimal operator of the group. The efficiency of the proposed approach is verified in a series of computational experiments.