Generalized Toeplitz kernels, stationarity and harmonizability
Given a Hilbert space \({\mathfrak M}\), an \({\mathfrak M}\)-process is a function X:\({\mathbb{Z}}\to {\mathfrak M}\) with its covariance kernel \(K(m,n)=(X(m),X(n))\). X is called generalized stationary process if its kernel K(m,n) is generalized Toeplitz kernel i.e. satisfies \(K(m,n)=K(m+1\), \(n+1)\) for \(m\neq -1\), \(n\neq -1\). An \({\mathfrak M}\)-valued process is said to be generalized harmonizable one if it admits representation \(X(n)=\hat V_ 1(n)\), \(n\geq 0\), \(X(n)=\hat V_ 2(n)\), \(n<0\) where \(V_ 1,V_ 2\) are bounded \({\mathfrak M}\)-valued measures on \({\mathbb{T}}\). The authors establish connection between these two notions and discuss some equivalent definitions of generalized harmonizable process. It is proved that for certain operations \(\Lambda\) the equation \(\Lambda X=Y\), Y being generalized stationary process, has a generalized stationary solution if and only if it has a generalized harmonizable one.
scientific article; zbMATH DE number 3927591