Tchebycheffian splines are smooth piecewise functions whose pieces drawn from (possibly different) Tchebycheff spaces, a natural generalization of algebraic polynomial spaces. They enjoy most the properties known in spline case. In particular, under suitable assumptions, admit representation terms basis functions, called B-splines (TB-splines), completely analogous to B-splines. A particularly ...

This paper studies the merits of using knot interval notation for B-spline curves, and presents formulae in terms of knot intervals for common B-spline operations such as knot insertion, differentiation, and degree elevation. Using knot interval notation, the paper introduces MD-splines, which are B-spline-like curves that are comprised of polynomial segments of various degrees (MD stands for “...

A dyadic algorithm for diagonalizing an arbitrary positive definite band matrix, referred to as a Gramian, is obtained efficiently orthogonalize the B-splines. The can be also used fast inversion method Gramian characterized by remarkable sparsity of matrix. There are two versions algorithm: first one more efficient and applicable Toeplitz while second general, works with any but computationall...