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Explicit Representations for Banach Subspaces of Lizorkin Distributions

S. Neumayer, M. Unser

Analysis and Applications, vol. 21, no. 5, pp. 1223–1250, September 2023.

The Lizorkin space is well suited to the study of operators like fractional Laplacians and the Radon transform. In this paper, we show that the space is unfortunately not complemented in the Schwartz space. In return, we show that it is dense in C0(ℝd), 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 that involves 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. 

@ARTICLE(http://bigwww.epfl.ch/publications/neumayer2302.html,
AUTHOR="Neumayer, S. and Unser, M.",
TITLE="Explicit Representations for {B}anach Subspaces of {L}izorkin
	Distributions",
JOURNAL="Analysis and Applications",
YEAR="2023",
volume="21",
number="5",
pages="1223--1250",
month="September",
note="")

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