Approximation with Neural Networks in Variable Lebesgue Spaces
This addresses a theoretical gap in neural network approximation for variable Lebesgue spaces, which is incremental as it extends known results to more general function spaces.
The paper tackles the problem of approximating functions with neural networks in variable Lebesgue spaces, showing that shallow networks can approximate any function with any desired accuracy when the exponent function is bounded, and characterizing approximable subspaces when it is unbounded.
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.