Tchebycheffian B-Splines Revisited: An Introductory Exposition

Abstract

Tchebycheffian splines are smooth piecewise functions where the different pieces are drawn from extended Tchebycheff spaces. They are a natural generalization of polynomial splines and can be represented in terms of an interesting set of basis functions, the so-called Tchebycheffian B-splines, which generalize the standard polynomial B-splines. We provide an accessible and self-contained exposition of Tchebycheffian B-splines and their main properties. Our construction is based on an integral recurrence relation and allows for the use of different extended Tchebycheff spaces on different intervals. The special class of generalized B-splines is also discussed in detail.