On linear independence of linear and bilinear point-based splines

Abstract

The basis of T-splines are the point-based splines (PB splines) that are unstructured meshless splines. In this paper, we study associated PB splines with local knot vectors that are arbitrarily distributed in $$([0,1]\\cap {\\mathbb {Q}})^d, d=1,2$$ ( [ 0 , 1 ] ∩ Q ) d , d = 1 , 2 , where $${\\mathbb {Q}}$$ Q is the set of rational numbers. We prove the linear independence of linear PB splines under a mild assumption that their central knots are all distinct. The linearly independent property is one of important prerequisites for isogeometric analysis. Moreover, we illustrate that the same assumption can not be extended to two-dimensional case, by giving a set of linearly dependent bilinear PB splines.