# Approximation with Neural Networks in Variable Lebesgue Spaces

arXiv: https://arxiv.org/abs/2007.04166 · HTML: https://mlsanity.com/paper/2007.04166/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Jul 2020.

## Built on by (most cited first, 2 of 2)

- [Non-convergence to global minimizers for Adam and stochastic gradient descent optimization and constructions of local minimizers in the training of artificial neural networks](https://mlsanity.com/paper/2402.05155/) (2024)
- [Non-convergence to the optimal risk for Adam and stochastic gradient descent optimization in the training of deep neural networks](https://mlsanity.com/paper/2503.01660/) (2025)

## Abstract (arXiv)

This paper concerns the universal approximation property with neural networks in variable Lebesgue spaces. We show that, whenever the exponent function of the space is bounded, every function can be approximated with shallow neural networks with any desired accuracy. This result subsequently leads to determine the universality of the approximation depending on the boundedness of the exponent function. Furthermore, whenever the exponent is unbounded, we obtain some characterization results for the subspace of functions that can be approximated.