Complex Systems Representation by C ^k Spline Functions

Abstract

This work presents the principal algebraic, arithmetic and geometrical properties of the C ^k spline functions in the temporal space as well as in the frequencial space. Thanks to their good properties of regularity, of smoothness and compactness in both spaces, precise and powerful computations implying C ^k spline functions can be considered. The main algebraic property of spline functions is to have for coefficients of their functional expansion of a considered function, the whole set of partial or total derivatives up to the order k of the considered function. In this way C ^k spline function can be defined as the interpolating functions of the set of the all Taylor Mac Laurin expansion up to the degree k defined at each point of discretization of the considered studying function. This fundamental property allows a much easier representation of complex systems in the linear case as well as in the nonlinear case. Then traditional differential and integral calculus lead in the C ^k spline functional spaces to new functional and invariant calculus.