Explicit representations for Banach subspaces of Lizorkin distributions

TL;DR

This paper explores Banach subspaces within Lizorkin distributions, establishing their density in continuous functions and applying these findings to enhance results in fractional splines and neural networks.

Contribution

It introduces Banach subspaces of Lizorkin distributions with continuous representation operators and develops a variational framework for applications.

Findings

Lizorkin space is dense in C_0(R^d)

Constructed Banach subspaces facilitate new analysis methods

Strengthened results for fractional splines and ReLU networks

Abstract

The Lizorkin space is well-suited for studying various operators; e.g., fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. However, we can show that it is dense in $C_0(\mathbb R^d)$, a property that is shared by the larger Schwartz space and that turns out to be useful for applications. Based on this result, we investigate subspaces of Lizorkin distributions that are Banach spaces and for which a continuous representation operator exists. Then, we introduce a variational framework involving these spaces and that makes use of the constructed operator. By investigating two particular cases of this framework, we are able to strengthen existing results for fractional splines and 2-layer ReLU networks.