A New Approach to Tchebycheffian B-Splines

. Originally, Tchebycheffian B-splines have been defined by generalized divided differences. In this paper, we define Tchebycheffian B-splines by integration. Based upon this definition, all basic algorithms for Tchebycheffian splines can be derived in a straightforward manner. As an example, a knot insertion algorithm for Tchebycheffian splines is constructed. x1. Introduction The class of Tchebycheffian splines contains many different kinds of splines: for example B-splines, exponential splines, and hyberbolic splines, see [11]. Algorithms for Tchebycheffian splines have been constructed by generalized divided differences, see e.g. [5], by generalized polar forms [8,6], and by generalized de-Boor-Fix dual functionals [1]. A fourth possibility based upon a new construction method for Tchebycheffian B-splines is presented in this paper. This construction method, which can be considered as a generalized convolution having its origin in the derivative formula for B-splines, makes it pos..