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2021

Polynomial Approximation of Symmetric Functions

Bachmayr, Markus, Dusson, Geneviève, Ortner, Christoph et al.

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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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