The weighted dual functionals for the univariate Bernstein basis

Rababah, Abedallah; Al-Natour, Mohammad Q.

doi:10.1016/j.amc.2006.08.065

Cited by 20 publications

(7 citation statements)

References 13 publications

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“…Details on conversions between the Bernstein basis and other orthogonal polynomial bases may be found in [29,40,160,161,162].…”

Section: The Legendre Basismentioning

confidence: 99%

Bernstein Polynomials

2021

MJS

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Bernstein polynomials (aka, B-polys) have excellent properties allowing them to be used as basis functions in many applications of physics. In this paper, a brief tutorial description of their properties is given and then their use in obtaining B-polys, B-splines or Basis spline functions, Bezier curves and ODE solution curves, is computationally demonstrated. An example is also described showing their application to solving a fourth-order BVP relating to the bending at the free end of a cantilever.

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“…Details on conversions between the Bernstein basis and other orthogonal polynomial bases may be found in [29,40,160,161,162].…”

Section: The Legendre Basismentioning

confidence: 99%

Bernstein Polynomials

2021

MJS

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“…Dual Bernstein polynomials associated with the Legendre inner product were introduced by Ciesielski in 1987 [4]. Their properties and generalizations were studied, e.g., by Jüttler [12], Rababah and Al-Natour [19,20], as well as by Lewanowicz and Woźny [14,15,24]. It is worth noticing that dual Bernstein polynomials introduced in [14], which are associated with the shifted Jacobi inner product, have recently found many applications in numerical analysis and computer graphics (curve intersection using Bézier clipping, degree reduction and merging of Bézier curves, polynomial approximation of rational Bézier curves, etc.).…”

Section: Introductionmentioning

confidence: 99%

Differential-recurrence properties of dual Bernstein polynomials

Chudy

Woźny

2018

Applied Mathematics and Computation

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New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials. Using these results, a fourth-order differential equation satisfied by dual Bernstein polynomials has been constructed. Also, a fourth-order recurrence relation for these polynomials has been obtained; this result may be useful in the efficient solution of some computational problems.

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“…The unconstrained dual basis D n i (x; α, β) corresponds to the case k = l = 0, hence D n i (x; α, β) = D (n,0,0) i (x; α, β). Dual Bernstein polynomials D (n,k,l) i (x; α, β) were studied by Ciesielski [4] (case α = β = 0 and k = l = 0), also by Jüttler [6] (case α = β = 0 and general k = l), Rababah and Al-Natour [11] (k = l = 0 and general α, β) and [12] (general α, β and k = l).…”

Section: Introductionmentioning

confidence: 99%