Dual polynomial spline bases

TL;DR

This paper introduces methods for constructing dual bases for B-spline and truncated power bases, providing explicit formulas and algorithms, with applications demonstrated through examples.

Contribution

It offers explicit formulas and iterative algorithms for dual bases of B-spline and truncated power polynomial spaces, advancing computational spline theory.

Findings

Explicit dual B-spline basis formulas derived

Iterative algorithm for dual truncated power basis developed

Applications demonstrated with illustrative examples

Abstract

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 Wo\'zny, 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 (Wo\'zny, 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.