Shape preserving alternatives to the rational Bézier model

Mainar, E.; Peña, J. M.; Sánchez-Reyes, Javier

doi:10.1016/s0167-8396(01)00011-5

Cited by 145 publications

(91 citation statements)

References 17 publications

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“…Following the notation of [9], [10], [23], we can deduce from the previous result that a non-negative Bernstein basis of an extended Chebyshev system over [a, b] is totally positive on [a, b], i.e., a B-basis.…”

Section: Generalized Convexitymentioning

confidence: 99%

“…If one generalizes the space of polynomials of degree at most n by retaining the bound on the number of zeros, one is led to the notion of an extended Chebyshev space (or system) U n of dimension n + 1 over the interval [a, b]: U n is an n + 1 dimensional subspace of C n ([a, b]) such that each f ∈ U n has at most n zeros in [a, b], counting multiplicities, unless f vanishes identically. Recently, a rich mathematical literature has emerged concerning generalized Bernstein bases in the framework of extended Chebyshev spaces, see [9], [10], [11], [12], [23], [24], [25], [26], [27], [28], [30], [31], [35].…”

Section: Introductionmentioning

confidence: 99%

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Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

2009

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Abstract. We study the existence and shape preserving properties of a generalized Bernstein operator B n fixing a strictly positive function f 0 , and a second function f 1 such that f 1 /f 0 is strictly increasing, within the framework of extended Chebyshev spaces U n . The first main result gives an inductive criterion for existence: suppose there exists a Bernstein operator B n : C[a, b] → U n with strictly increasing nodes, fixing f 0 , f 1 ∈ U n . If U n ⊂ U n+1 and U n+1 has a non-negative Bernstein basis, then there exists a Bernstein operator B n+1 : C[a, b] → U n+1 with strictly increasing nodes, fixing f 0 and f 1 . In particular, if f 0 , f 1 , ..., f n is a basis of U n such that the linear span of f 0 , .., f k is an extended Chebyshev space over [a, b] for each k = 0, ..., n, then there exists a Bernstein operator B n with increasing nodes fixing f 0 and f 1 . The second main result says that under the above assumptions the following inequalities hold

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Section: Generalized Convexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Shape preserving properties of generalized Bernstein operators on Extended Chebyshev spaces

2009

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“…On the one hand, special systems of exponential polynomials are considered, such as 1, x, ..., x n−1 , cos x, sin x, (which corresponds to the case λ 0 = ... = λ n−2 = 0, λ n−1 = i, and λ n = −i), cf. [5], [22], [33] and [6], [4] for further generalizations. On the other hand, a remarkable result is the existence of a so-called normalized Bernstein basis in certain classes of extended Chebyshev systems, see [3], [24].…”

Section: Introductionmentioning

confidence: 99%

Bernstein Operators for Exponential Polynomials

2008

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Abstract. Let L be a linear differential operator with constant coefficients of order n and complex eigenvalues λ 0 , ..., λ n . Assume that the set U n of all solutions of the equation Lf = 0 is closed under complex conjugation. If the length of the interval [a, b] is smaller than π/M n , where M n := max {|Imλ j | : j = 0, ..., n}, then there exists a basis p n,k , k = 0, ...n, of the space U n with the property that each p n,k has a zero of order k at a and a zero of order n − k at b, and each p n,k is positive on the open interval (a, b) . Under the additional assumption that λ 0 and λ 1 are real and distinct, our first main result states that there exist points a = t 0 < t 1 < ... < t n = b and positive numbers α 0 , .., α n , such that the operator

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“…They belong piecewise to the space span{1, t, sin t, cos t}. These splines generalize the well-known cubic splines and have many applications in numerical analysis for the shape preserving approximation, the description of curves and their parametrization, and other problems (see, for instance, [14], [15], [16], and references therein). In particular (see [15]) these splines are attractive from a geometrical point of view, because they are able to provide parameterizations of conic sections with respect to the arc length so that equally spaced points in the parameter domain correspond to equally spaced points on the described curve.…”

Section: Proofs Of Theoremsmentioning

confidence: 99%