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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. [ABSTRACT FROM AUTHOR]

In this article, we propose a new numerical method and its analysis to solve eigenvalue problems for self-adjoint Schrödinger operators, by combining the Feshbach–Schur perturbation theory with the spectral Fourier discretization. In order to analyze the method, we establish an abstract framework of Feshbach–Schur perturbation theory with minimal regularity assumptions on the potential that is then applied to the setting of the new spectral Fourier discretization method. Finally, we present some numerical results that underline the theoretical findings. [ABSTRACT FROM AUTHOR]