The Experts below are selected from a list of 57 Experts worldwide ranked by ideXlab platform
Ron Goldman - One of the best experts on this subject based on the ideXlab platform.
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A Polynomial Blossom for the Askey–Wilson Operator
Constructive Approximation, 2019Co-Authors: Plamen Simeonov, Ron GoldmanAbstract:We introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a corresponding Askey–Wilson Bernstein basis for which this new blossom provides the Dual Functionals. We derive a partition of unity property and a Marsden identity for this Askey–Wilson Bernstein basis, which turn out to be the terminating versions of Rogers’ $${}_6\phi _5$$ 6 ϕ 5 summation formula and a very-well-poised $${}_8\phi _7$$ 8 ϕ 7 summation formula. Recurrence and symmetry relations and differentiation and degree elevation formulas for the Askey–Wilson Bernstein bases, as well as degree elevation formulas for Askey–Wilson Bézier curves, are also given.
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a polynomial blossom for the askey wilson operator
Constructive Approximation, 2019Co-Authors: Plamen Simeonov, Ron GoldmanAbstract:We introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a corresponding Askey–Wilson Bernstein basis for which this new blossom provides the Dual Functionals. We derive a partition of unity property and a Marsden identity for this Askey–Wilson Bernstein basis, which turn out to be the terminating versions of Rogers’ $${}_6\phi _5$$ summation formula and a very-well-poised $${}_8\phi _7$$ summation formula. Recurrence and symmetry relations and differentiation and degree elevation formulas for the Askey–Wilson Bernstein bases, as well as degree elevation formulas for Askey–Wilson Bezier curves, are also given.
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Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming
Numerical Algorithms, 2016Co-Authors: Ron Goldman, Plamen SimeonovAbstract:We introduce the G -blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G = G ( t ). By invoking the G -blossom, we construct G -Bernstein bases and G -Bézier curves and study their algebraic and geometric properties. We show that the G -blossom provides the Dual Functionals for the G -Bernstein basis functions and we use this Dual functional property to prove that G -Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G -Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G -Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G -Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G -Bézier curve, and we derive the geometric rate of convergence of this algorithm.
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Quantum Bernstein bases and quantum Bézier curves
Journal of Computational and Applied Mathematics, 2015Co-Authors: Ron Goldman, Plamen SimeonovAbstract:The purpose of this paper is to investigate the most general quantum Bernstein bases and quantum Bezier curves. Classical Bernstein bases satisfy a two-term formula for their classical derivatives; quantum Bernstein bases satisfy a two-term formula for their quantum derivatives. To study the properties of these general quantum polynomial schemes, a new variant of the blossom, the quantum blossom, is introduced by altering the diagonal property of the classical blossom. The significance of the quantum blossom is that the quantum blossom provides the Dual Functionals for quantum Bezier curves over arbitrary intervals. Using the quantum blossom, several fundamental identities involving the quantum Bernstein bases are developed, including a quantum variant of the Marsden identity and the partition of unity property. Based on these properties of quantum Bernstein bases, quantum Bezier curves are shown to be affine invariant, and under certain conditions lie in the convex hull of their control points. In addition, for each quantum Bezier curve of degree n , a collection of n ! , affine invariant, recursive evaluation algorithms are derived. Using two of these recursive evaluation algorithms, a recursive subdivision procedure for quantum Bezier curves is constructed. This subdivision procedure generates a sequence of control polygons that converges rapidly to the original quantum Bezier curve.
Plamen Simeonov - One of the best experts on this subject based on the ideXlab platform.
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A Polynomial Blossom for the Askey–Wilson Operator
Constructive Approximation, 2019Co-Authors: Plamen Simeonov, Ron GoldmanAbstract:We introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a corresponding Askey–Wilson Bernstein basis for which this new blossom provides the Dual Functionals. We derive a partition of unity property and a Marsden identity for this Askey–Wilson Bernstein basis, which turn out to be the terminating versions of Rogers’ $${}_6\phi _5$$ 6 ϕ 5 summation formula and a very-well-poised $${}_8\phi _7$$ 8 ϕ 7 summation formula. Recurrence and symmetry relations and differentiation and degree elevation formulas for the Askey–Wilson Bernstein bases, as well as degree elevation formulas for Askey–Wilson Bézier curves, are also given.
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a polynomial blossom for the askey wilson operator
Constructive Approximation, 2019Co-Authors: Plamen Simeonov, Ron GoldmanAbstract:We introduce a blossoming procedure for polynomials related to the Askey–Wilson operator. This new blossom is symmetric, multiaffine, and reduces to the complex representation of the polynomial on a certain diagonal. This Askey–Wilson blossom can be used to find the Askey–Wilson derivative of a polynomial of any order. We also introduce a corresponding Askey–Wilson Bernstein basis for which this new blossom provides the Dual Functionals. We derive a partition of unity property and a Marsden identity for this Askey–Wilson Bernstein basis, which turn out to be the terminating versions of Rogers’ $${}_6\phi _5$$ summation formula and a very-well-poised $${}_8\phi _7$$ summation formula. Recurrence and symmetry relations and differentiation and degree elevation formulas for the Askey–Wilson Bernstein bases, as well as degree elevation formulas for Askey–Wilson Bezier curves, are also given.
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Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming
Numerical Algorithms, 2016Co-Authors: Ron Goldman, Plamen SimeonovAbstract:We introduce the G -blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G = G ( t ). By invoking the G -blossom, we construct G -Bernstein bases and G -Bézier curves and study their algebraic and geometric properties. We show that the G -blossom provides the Dual Functionals for the G -Bernstein basis functions and we use this Dual functional property to prove that G -Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G -Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G -Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G -Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G -Bézier curve, and we derive the geometric rate of convergence of this algorithm.
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Quantum Bernstein bases and quantum Bézier curves
Journal of Computational and Applied Mathematics, 2015Co-Authors: Ron Goldman, Plamen SimeonovAbstract:The purpose of this paper is to investigate the most general quantum Bernstein bases and quantum Bezier curves. Classical Bernstein bases satisfy a two-term formula for their classical derivatives; quantum Bernstein bases satisfy a two-term formula for their quantum derivatives. To study the properties of these general quantum polynomial schemes, a new variant of the blossom, the quantum blossom, is introduced by altering the diagonal property of the classical blossom. The significance of the quantum blossom is that the quantum blossom provides the Dual Functionals for quantum Bezier curves over arbitrary intervals. Using the quantum blossom, several fundamental identities involving the quantum Bernstein bases are developed, including a quantum variant of the Marsden identity and the partition of unity property. Based on these properties of quantum Bernstein bases, quantum Bezier curves are shown to be affine invariant, and under certain conditions lie in the convex hull of their control points. In addition, for each quantum Bezier curve of degree n , a collection of n ! , affine invariant, recursive evaluation algorithms are derived. Using two of these recursive evaluation algorithms, a recursive subdivision procedure for quantum Bezier curves is constructed. This subdivision procedure generates a sequence of control polygons that converges rapidly to the original quantum Bezier curve.
Phillip J Barry - One of the best experts on this subject based on the ideXlab platform.
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de boor fix Dual Functionals and algorithms for tchebycheffian b spline curves
Constructive Approximation, 1996Co-Authors: Phillip J BarryAbstract:The de Boor-Fix Dual Functionals are a potent tool for deriving results about piecewise polynomial B-spline curves. In this paper we extend these Functionals to Tchebycheffian B-spline curves and then use them to derive fundamental algorithms that are natural generalizations of algorithms for piecewise polynomial B-spline algorithms. Then, as a further example of the utility of this approach, we introduce “geometrically continuous Tchebycheffian spline curves,” and show that a further generalization works for them as well.
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Knot insertion algorithms for piecewise polynomial spaces determined by connection matrices
Advances in Computational Mathematics, 1993Co-Authors: Phillip J Barry, Ronald N. Goldman, Charles A. MicchelliAbstract:We show that many fundamental algorithms and techniques for B-spline curves extend to geometrically continuous splines. The algorithms, which are all related to knot insertion, include recursive evaluation, differentiation, and change of basis. While the algorithms for geometrically continuous splines are not as computationally simple as those for B-spline curves, they share the same general structure. The techniques we investigate include knot insertion, Dual Functionals, and polar forms; these prove to be useful theoretical tools for studying geometrically continuous splines.
Lim Yohanes Stefanus - One of the best experts on this subject based on the ideXlab platform.
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De Boor-Fix Dual Functionals for transformation from polynomial basis to convolution basis
ACM Communications in Computer Algebra, 2009Co-Authors: Lim Yohanes StefanusAbstract:Basis transformation plays a central role in data representations and in many problem solving techniques such as Fourier transform and wavelet transform. A pertinent basis gives pertinent information. Dual Functionals are a mathematical tool for basis transformation. This work presents, from the perspective of algebraic computing, a generalization of the de Boor-Fix Dual Functionals for the Bernstein basis functions to the case of the convolution basis functions. The convolution basis functions can be characterized as polar forms of the corresponding Bernstein basis functions. These new Dual Functionals can be used for computing algebraically the expansion coefficients of any polynomial over the convolution basis which has applications in computer-aided geometric design and secret information encoding.
Su Zhixun - One of the best experts on this subject based on the ideXlab platform.
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A Dual functional to the univariate B-spline
Journal of Computational and Applied Mathematics, 2005Co-Authors: Zhao Guohui, Liu Xiu-ping, Su ZhixunAbstract:In this paper, Dual Functionals with local supports to the univariate B-splines are constructed by symmetrization, which are a linear combination of function values.
