Weighted dual functions for Bernstein basis satisfying boundary constraints Abedallah Rababah * , Mohammad Al-Natour Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan Abstract In this paper, we consider the issue of dual functions for the Bernstein basis which satisfy boundary conditions. The Jacobi weight function with the usual inner product in the Hilbert space are used. Some examples of the transformation matrices are given. Some figures for the weighted dual functions of the Bernstein basis with respect to the Jacobi weight function satisfying boundary conditions are plotted. We discuss special cases of the Jacobi weight function as the Legendre weight function and the Chebyshev weight functions of the first, second, and third kinds. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Bernstein basis; Dual basis; Boundary constraints 1. Introduction and definitions The Bernstein polynomials have many important properties which make them the most commonly used basis in approximation theory and computer aided geometric design (CAGD), see [4,7,11]. The Bernstein polynomials are not orthogonal, see [1,3,17], so it is not appropriate to use them in the least squares approx- imation. To enable the use of the Bernstein polynomials in the least squares approximation, two approaches are suggested. The first approach is basis transformation, see [5,6,13–15], and the second approach is finding the dual basis functions for the Bernstein polynomials, see [2,9,16]. Associated with the Bernstein polynomials, there are the corresponding dual basis functions. A recurrence relation for the dual basis functions of B-Splines of degree n by using the dual basis functions of degree n  1 and the Legendre polynomials of degree n is studied in [2]. Explicit formula for the dual basis of the Bernstein polynomials with respect to the usual inner product of the Hilbert space L 2 ½0; 1 is given in [9]. These results have been generalized to the dual basis of the Bernstein polynomials with respect to the Jacobi weight function in [16]. More results can be found in [12,15]. Let P n denote the ðn þ 1Þ-dimensional real linear space of all polynomials of degree at most n, in the var- iable t, i.e., 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.10.006 * Corresponding author. E-mail address: rababah@just.edu.jo (A. Rababah). Available online at www.sciencedirect.com Applied Mathematics and Computation 199 (2008) 456–463 www.elsevier.com/locate/amc