Strong Divergence for System Approximations

TL;DR

This paper proves that certain stable linear systems, like the Hilbert transform, can be strongly divergent when approximated via sampling series, with divergence occurring for all subsequences and analyzing divergence speed.

Contribution

It establishes the phenomenon of strong divergence in system approximations, extending previous weak divergence results, and explores divergence behavior at Nyquist and oversampling rates.

Findings

Strong divergence occurs for all subsequences in certain system approximations.

Divergence speed can be characterized when divergence occurs.

Connections between divergence phenomena and the Banach-Steinhaus theorem are discussed.

Abstract

In this paper we analyze the approximation of stable linear time-invariant systems, like the Hilbert transform, by sampling series for bandlimited functions in the Paley-Wiener space $\mathcal{PW}_{\pi}^{1}$. It is known that there exist systems and functions such that the approximation process is weakly divergent, i.e., divergent for certain subsequences. Here we strengthen this result by proving strong divergence, i.e., divergence for all subsequences. Further, in case of divergence, we give the divergence speed. We consider sampling at Nyquist rate as well as oversampling with adaptive choice of the kernel. Finally, connections between strong divergence and the Banach-Steinhaus theorem, which is not powerful enough to prove strong divergence, are discussed.