2021/12/07 by Anna Kamont, Kamont, Anna, Markus Passenbrunner +1
Engineering · #Advanced Numerical Analysis Techniques
paper · pdf · doi:10.48550/arxiv.2112.03664
B-splines of order k can be viewed as a mapping N taking a (k+1)-tuple of increasing real numbers a0 < ⋯ < ak and giving as a result a certain piecewise polynomial function. Looking at this mapping N as a whole, basic roperties of B-spline functions imply that it has the following algebraic properties: (1) N(a0,…,ak) has local support; (2) N(a0,…,ak) allows refinement, i.e. for every a∈ ∪j=0k-1 (aj,aj+1) we have that if (α0,…, αk+1) is the increasing rearrangement of the points \a0,…,ak,a\, the 'old' function N(a0,…,ak) is a linear combination of the 'new' functions N(α0,…,αk) and N(α1,…,αk+1); (3) N is translation and dilation invariant. It is easy to see that derivatives of N(a0,…,ak) satisfy properties (1)-(3) as well. In this paper we investigate if properties (1)-(3) are already sufficient to characterize B-splines and their derivatives.