Nested sequences of rational spaces: Bernstein approximation, dimension elevation, and Pólya-type theorems on positive polynomials

Ait‐Haddou, Rachid; Mazure, Marie-Laurence; Render, Hermann

doi:10.1016/j.jat.2018.04.010

Cited by 2 publications

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“…This result would be true in particular for any polynomial P positive on ]0, +∞[. In other words, if such an extension were true, the infinite sequence q −i , j ≥ 0, would be Pólya positive in the sense of [3] or equivalently, the infinite sequence of degree one polynomials (x + q −i ), j ≥ 0, would be strongly positive in the sense of [4]. This property is known to be true if and only if…”

Section: A Quantum Pólya-type Theoremmentioning

confidence: 91%

“…It has been shown, for instance in [3], that Bernstein's theorem is equivalent to the celebrated univariate Pólya theorem on positive polynomials [26]. This theorem states that for any polynomial P positive on the interval [0, ∞[, there exists an integer N such that the coefficients in the monomial basis of the polynomial (1 + x) N P(x) are nonnegative.…”

Section: Introductionmentioning

confidence: 99%

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Quantum Lorentz degrees of polynomials and a Pólya theorem for polynomials positive on q-lattices

Ait‐Haddou

Goldman

Mazure

2021

Applied Numerical Mathematics

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We establish the uniform convergence of the control polygons generated by repeated degree elevation of q-Bézier curves ( i.e., polynomial curves represented in the q-Bernstein bases of increasing degrees) on [0, 1], q > 1, to a piecewise linear curve with vertices on the original curve. A similar result is proved for q < 1, but surprisingly the limit vertices are not on the original curve, but on the q −1 -Bézier curve with control polygon taken in the reverse order. We introduce a q-deformation (quantum Lorentz degree) of the classical notion of Lorentz degree for polynomials and we study its properties. As an application of our convergence results, we introduce a notion of q-positivity which guarantees that the q-Lorentz degree is finite. We also obtain upper bounds for the quantum Lorentz degrees. Finally, as a by-product we provide a generalization to polynomials positive on q-lattices of the univariate Pólya theorem concerning polynomials positive on the non-negative axis.

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Section: A Quantum Pólya-type Theoremmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%