On rational B-splines with prescribed poles
Let \([\alpha,\beta]\) a real interval and \(P_2,P_3,\dots,P_m \notin [\alpha,\beta]\) (\(P_1\) is considered \(\infty\)). The authors of this paper consider Cauchy-Vandermonde systems (CV systems) \[ (1,x,\dots,x^{n_1-1},{{1}\over {x-P_2}},\dots,{{1}\over {(x-P_2)^{n_2}}},\dots,{{1}\over {x-P_m}},\dots,{{1}\over {(x-P_m)^{n_m}}}) \] giving rise to rational functions with prescribed poles. Here \(n_1\geq 1, \,n_i\geq 0 \quad \forall i>1\). The authors show how to construct an extended complete Chebyshev system (ECT system) in canonical form from the CV system above. Then they construct a space of rational splines with prescribed poles for a sequence of simple knots, and a basis of B-splines formed by splines with minimal compact support normalized to have integral 1 along the real line. All the constructions depend on a parameter \(\epsilon>0\) because the CV system used in the interval beteeen consecutive knots \([xi,x_{i+1}]\) is \[ (1,x-x_i,\dots,(x-x_i)^{d-3},{{1}\over {x-x_i+\epsilon}},{{1}\over {x-x_{i+1}-\epsilon}}). \] In the main result the authors prove that in the limit \(\epsilon \to 0\) the rational B-splines tend to certain polynomial B-splines.
scientific article; zbMATH DE number 2085542
scientific article