This paper addresses theoretical considerations behind the algorithmic computation of polynomial multi-degree spline basis functions as presented in Toshniwal et al. (2017). The approach in Toshniwal et al. (2017) breaks from the reliance on computation of integrals recursively for building B-spline-like basis functions that span a given multi-degree spline space. The gains in efficiency are indisputable; however, the theoretical robustness needs to be examined. In this paper, we show that the construction of Toshniwal et al. (2017) yields linearly independent functions with the minimal support property that span the entire multi-degree spline space and form a convex partition of unity.

In this paper, we present an efficient and robust approach to compute a normalized B-spline-like basis for spline spaces with pieces drawn from extended Tchebycheff spaces. The extended Tchebycheff spaces and their dimensions are allowed to change from interval to interval. The approach works by constructing a matrix that maps a generalized Bernstein-like basis to the B-spline-like basis of interest. The B-spline-like basis shares many characterizing properties with classical univariate B-splines and may easily be incorporated in existing spline codes. This may contribute to the full exploitation of Tchebycheffian splines in applications, freeing them from the restricted role of an elegant theoretical extension of polynomial splines. Numerical examples are provided that illustrate the procedure described.