Stepanov almost periodically correlated and almost periodically unitary processes

This paper extends the structure and properties of almost periodically correlated (APC) and almost periodically unitary (APU) processes, which were defined in the sense of Bohr, to a larger class of processes for which the sense of almost periodicity is that of Stepanov. These processes are not necessarily continuous in quadratic mean, as are the Bohr APC and APU processes, yet exhibit a sense of almost periodicity. For example, processes formed by amplitude modulation $f(t)X(t)$ or time-scale modulation $X(t+f(t))$ of a wide sense stationary process $X(t)$ by a scalar Stepanov AP function $f(t)$, which need not be continuous, are Stepanov APU and APC. The principal results on APC and APU processes are extended to the new class. We extend Gladyshev's characterization of APC correlation functions to Stepanov APC processes and show that their correlation functions are completely represented by a Fourier series having a countable number of coefficient functions that are Fourier transforms of complex measures. We show that Stepanov APU processes are also Stepanov APC and are given by $X(t)=U(t)[P(t)]$, where $\{U(t),t\in\mathbb R\}$ is a strongly continuous group of unitary operators and $P(t)$ is a vector-valued Stepanov almost periodic function. As in the case of Bohr APU processes, the preceding fact leads to representations of $X(t)$ based on the spectral theory for unitary operators and for Stepanov almost periodic functions.