No abstract uploaded!

Given a fixed$p$≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $ , provided that the dimension$d$is sufficiently large. The method also yields new$L$^{$q$}space-time regularity results for solutions of the wave equation in high dimensions.

No abstract uploaded!

We show that Q-Fano varieties of fixed dimension with anti-canonical degrees and alpha-invariants bounded from below form a bounded family. As a corollary, K-semistable Q-Fano varieties of fixed dimension with anti-canonical degrees bounded from below form a bounded family.

In its simplest form, convolution neural networks (CNNs) consist of a fully connected layer g composed with a sequence of convolution layers T . Although g is known to have the universal approximation property, it is not known if CNNs, which has the form g ◦ T inherits this property, especially when the kernel size in T is small. In this paper, we show that under suitable conditions, CNNs does inherit the universal approximation property and its sample complexity can be characterized. In addition, we discuss concretely how the nonlinearity of T can improve the approximation power. Finally, we show that when the target function class has a certain compositional form, convolutional networks are far more advantageous compared with fully connected networks, in terms of number of parameters needed to achieve a desired accuracy.