Logarithmic Barron Spectrum Spaces and Their Applications

In this paper, the authors give the definition of Barron spectrum space of logarithmic smoothness, and clarify the embedding relationship between Barron spectrum space of logarithmic smoothness and Besov space of logarithmic smoothness. In addition, the decaying behavior of radial functions in the Barron spectrum space of logarithmic smoothness have been studied. Finally, authors provide an application of Barron spectrum space of logarithmic smoothness in neural network approximation. All of these are generalizations of the corresponding results in the classical case.