Legendre-like Basis for the Uniform Four-order Algebraic Trigonometric Spline Space

In order to develop the theory of orthogonal basis for algebraic-trigonometric spline space,a novel approach is presented to construct an orthogonal basis for the uniform four-order algebraic trigonometric spline space.Based on the C-B spline functions of order six,a set of auxiliary functions is constructed.And the proposed orthogonal splines are given as the second-order derivatives of these auxiliary functions.This orthogonal basis is also called Legendre-like basis.The result of the practical examples shows that using this basis can simplify inner product computation and facilitate solving least-squares approximation.