# Truncation Dimension for Linear Problems on Multivariate Function Spaces

## TL;DR

This paper investigates the truncation dimension in multivariate function spaces, determining when solutions can be approximated by functions with fewer variables, and finds that the necessary number of variables can be surprisingly small.

## Contribution

It introduces the concept of truncation dimension for linear problems on weighted multivariate spaces and analyzes conditions under which high-dimensional problems can be effectively approximated.

## Key findings

- Truncation dimension can be very small even with modest weight decay.
- Approximation error can be controlled by truncating to a small number of variables.
- The results provide insights into high-dimensional problem simplification.

## Abstract

The paper considers linear problems on weighted spaces of multivariate functions of many variables. The main questions addressed are: When is it possible to approximate the solution for the original function of very many variables by the solution for the same function; however with all but the first $k$ variables set to zero, so that the corresponding error is small? What is the truncation dimension, i.e., the smallest number $k=k(\varepsilon)$ such that the corresponding error is bounded by a given error demand $\varepsilon$? Surprisingly, $k(\varepsilon)$ could be very small even for weights with a modest speed of convergence to zero.