Learning high-dimensional functions on the torus
In this paper we study the multivariate ANOVA decomposition for $1$-periodic functions on the torus. In particular we use the integral projection operator that leads to the classical ANOVA decomposition. Relationships between the Fourier coefficients of the function and its ANOVA terms lead to special frequency index sets and give an understanding of the decomposition working in the frequency domain. Moreover, we consider the truncated ANOVA decomposition and provide error bounds for approximation in $\mathrm{L}_\infty$ and $\mathrm{L}_2$. We present an approximation method based on the truncated decomposition with regard to a superposition dimension $d_s$.
In this article we exhibit some balls lying in the quasi-Fuchsian space of once punctured tori, which are maximal in the class of balls with the same centers. The centers of our maximal balls lie on the slice determined by the trace equation y = ¯ x.