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arXiv · 2109.14771

Polynomial Approximation of Symmetric Functions

Markus Bachmayr·Geneviève Dusson·Christoph Ortner·Jack Thomas
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Abstract

We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$ arguments. We demonstrate how these symmetries can be exploited to improve the cost versus error ratio in a polynomial approximation of the function $f$, and in particular study the dependence of that ratio on $d, N$ and the polynomial degree. These results are then used to construct approximations and prove approximation rates for functions defined on multi-sets where $N$ becomes a parameter of the input.

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Markus Bachmayr, Geneviève Dusson, Christoph Ortner, Jack Thomas. 2021-09-30. Polynomial Approximation of Symmetric Functions. https://arxiv.org/abs/2109.14771

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