Truncation in Average and Worst Case Settings for Special Classes of $\infty$-Variate Functions

TL;DR

This paper derives sharp bounds on truncation errors for a class of infinite-variate functions in both average and worst case scenarios, with applications in high-dimensional approximation.

Contribution

It provides new bounds on truncation errors for functions of the form g(∑ x_j ξ_j), covering both Hilbert and Hölder spaces, advancing high-dimensional approximation theory.

Findings

Established sharp bounds for average case truncation errors.

Derived worst case error bounds for functions in reproducing kernel Hilbert spaces.

Applicable to functions with rapidly converging coefficients ξ_j.

Abstract

The paper considers truncation errors for functions of the form $f(x_1,x_2,\dots)=g(\sum_{j=1}^\infty x_j\,\xi_j)$, i.e., errors of approximating $f$ by $f_k(x_1,\dots,x_k)=g(\sum_{j=1}^k x_j\,\xi_j)$, where the numbers $\xi_j$ converge to zero sufficiently fast and $x_j$'s are i.i.d. random variables. As explained in the introduction, functions $f$ of the form above appear in a number of important applications. To have positive results for possibly large classes of such functions, the paper provides sharp bounds on truncation errors in both the average and worst case settings. In the former case, the functions $g$ are from a Hilbert space $G$ endowed with a zero mean probability measure with a given covariance kernel. In the latter case, the functions $g$ are from a reproducing kernel Hilbert space, or a space of functions satisfying a H\"older condition.