# Analysis of tensor approximation schemes for continuous functions

## TL;DR

This paper investigates tensor approximation methods for continuous functions within Sobolev spaces, demonstrating that certain tensor formats can achieve dimension-robust approximation costs under specific conditions.

## Contribution

It provides a detailed analysis of tensor approximation schemes for continuous functions, highlighting conditions for dimension-robustness in Tucker and tensor train formats.

## Key findings

- Cost of tensor approximations is dimension-robust with proper weights.
- Analysis applies to functions in isotropic Sobolev spaces.
- Both Tucker and tensor train formats are effective under these conditions.

## Abstract

In this article, we analyze tensor approximation schemes for continuous functions. We assume that the function to be approximated lies in an isotropic Sobolev space and discuss the cost when approximating this function in the continuous analogue of the Tucker tensor format or of the tensor train format. We especially show that the cost of both approximations are dimension-robust when the Sobolev space under consideration provides appropriate weights.