Foundations of the Theory of Strongly Periodically Correlated Fields over Z2

The aim of this paper is to provide readers with basic concepts and techniques for analysis of strongly periodically correlated fields (SCF) over Z2. We show that every SCF over Z2 can be transformed into a coordinate-wise SCF (Fact 3.1) studied in [13]. The main result of the paper however is a specific decomposition of a strongly periodically correlated field (Theorem 4.1) which was not available for coordinate-wise SCFs. As consequences of the latter we obtain a description and an easy proof of existence of the spectral measures of an SCF (Theorem 5.1) as well as a functional description of an absolutely continuous SCF (Theorem 6.1). Most of the facts are explained in details and proved, with an exception of the proof of Theorem 6.1, which was too long for this publication and is left for a forthcoming paper.