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| Main Authors: | , , , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.19219 |
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| _version_ | 1866909773007421440 |
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| author | Dressler, Mareike Foucart, Simon Joldes, Mioara de Klerk, Etienne Lasserre, Jean-Bernard Xu, Yuan |
| author_facet | Dressler, Mareike Foucart, Simon Joldes, Mioara de Klerk, Etienne Lasserre, Jean-Bernard Xu, Yuan |
| contents | We consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval $[-1,1]$. Let $Π^*_n$ be the subset of polynomials of degree at most $n$ in $d$ variables, whose homogeneous part of degree $n$ has coefficients summing up to $1$. The problem is determining a polynomial in $Π^*_n$ with the smallest uniform norm on a domain $Ω$, which we call a least Chebyshev polynomial (associated with $Ω$). Our main result solves the problem for $Ω$ belonging to a non-trivial class of sets that we call diagonally-determined, and establishes the remarkable result that a least Chebyshev polynomial can be given via the classical, univariate, Chebyshev polynomial. In particular, the solution can be independent of the dimension. Diagonally-determined domains include centered balls in $\mathbb{R}^d$ in any norm, but can be non-convex and even non-simply connected. We also introduce a computational procedure, based on semidefinite programming hierarchies, to detect if a given semi-algebraic set is diagonally-determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_19219 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Least multivariate Chebyshev polynomials on diagonally determined sets Dressler, Mareike Foucart, Simon Joldes, Mioara de Klerk, Etienne Lasserre, Jean-Bernard Xu, Yuan Optimization and Control Primary: 41A10, 41A63, 90C22 We consider a new multivariate generalization of the classical monic (univariate) Chebyshev polynomial that minimizes the uniform norm on the interval $[-1,1]$. Let $Π^*_n$ be the subset of polynomials of degree at most $n$ in $d$ variables, whose homogeneous part of degree $n$ has coefficients summing up to $1$. The problem is determining a polynomial in $Π^*_n$ with the smallest uniform norm on a domain $Ω$, which we call a least Chebyshev polynomial (associated with $Ω$). Our main result solves the problem for $Ω$ belonging to a non-trivial class of sets that we call diagonally-determined, and establishes the remarkable result that a least Chebyshev polynomial can be given via the classical, univariate, Chebyshev polynomial. In particular, the solution can be independent of the dimension. Diagonally-determined domains include centered balls in $\mathbb{R}^d$ in any norm, but can be non-convex and even non-simply connected. We also introduce a computational procedure, based on semidefinite programming hierarchies, to detect if a given semi-algebraic set is diagonally-determined. |
| title | Least multivariate Chebyshev polynomials on diagonally determined sets |
| topic | Optimization and Control Primary: 41A10, 41A63, 90C22 |
| url | https://arxiv.org/abs/2405.19219 |