Dual polynomial spline bases
For researchers in computer-aided design and approximation theory, this offers explicit dual bases that simplify computations, though the methods are incremental extensions of prior work.
The paper provides explicit formulas and construction methods for dual bases of the B-spline and truncated power bases, enabling efficient computation in spline spaces.
In the paper, we give methods of construction of dual bases for the B-spline basis and truncated power basis. Explicit formulas for the dual B-spline basis are obtained using the Legendre-like orthogonal basis of the polynomial spline space presented in (Wei et al., Comput.-Aided Des. 45 (2013), 85-92) and a connection between orthogonal and dual bases of any space given in (Lewanowicz and Woźny, J. Approx. Theory 138 (2006), 129-150). Construction of the dual truncated power basis is performed in two phases. We start with explicit formulas for the dual power basis of the space of polynomials. Then, we expand this basis using an iterative algorithm proposed in (Woźny, J. Comput. Appl. Math. 260 (2014), 301-311). As a result, we obtain the dual truncated power basis. We also present some applications of the proposed dual polynomial spline bases and illustrative examples.