Spectrum of periodically correlated fields

TL;DR

This paper explores the structure and spectral properties of periodically correlated Hilbert space valued fields over locally compact Abelian groups, extending the concept of PC processes to multi-parameter fields and analyzing their covariance and spectrum.

Contribution

It introduces a framework for analyzing periodically correlated fields over general groups, extending existing theory from processes to multi-parameter fields and providing detailed spectral analysis.

Findings

Characterization of covariance functions for PC fields

Development of an analogue of the spectrum for these fields

Detailed examination of a weakly PC field over Z^2

Abstract

The paper deals with Hilbert space valued fields over any locally compact Abelian group G, in particular over G = Z^n x R^m, which are periodically correlated (PC) with respect to a closed subgroup of G. PC fields can be regarded as multi-parameter extensions of PC processes. We study structure, covariance function, and an analogue of the spectrum for such fields. As an example a weakly PC field over Z^2 is thoroughly examined.