Stationary fields with positive angle (Q1091037)

Let \({\mathcal H}\) be a complex Hilbert space and let \({\mathcal M}\) and \({\mathcal N}\) be closed subspaces of \({\mathcal H}\), let \[ \rho ({\mathcal M},{\mathcal N})=\sup \{| (x,y)|:| x| =| y| =1,\quad x\in {\mathcal M},y\in {\mathcal N}\}. \] The subspaces \({\mathcal M}\) and \({\mathcal N}\) are said to be at positive angle if \(\rho\) (\({\mathcal M},{\mathcal N})<1\). Let \(X=X_{mn}\), m,n\(\in {\mathbb{Z}}\), be a stationary field, \({\mathcal M}(U^ v)\) and \({\mathcal M}(\bar U^ v)\) be a vertical past and future of the field X, respectively, where \(U^ v=\{(m,n)\); \(m\leq -1\}\cup \{(m,n)\); \(m=0,n\leq -1\}\), \(\bar U{}^ v={\mathbb{Z}}^ 2\setminus U^ v\), let \(\gamma^ v=\rho ({\mathcal M}(U^ v)\), \({\mathcal M}(\bar U^ v)).\) An analytic characterization of stationary fields with a positive angle between the vertical past and future is given. The connection between the positivity of the angle and the convergence of the Fourier series for certain functions is studied. An example of a stationary field for which the vertical angle is positive, but whose horizontal angle is zero, is given.