Convergence of Fourier series of stationary Gaussian processes (Q1100806)

Let \(\xi\) (t), \(t\in [-\pi,\pi]\), be a restriction on [-\(\pi\),\(\pi\) ] of a real-valued stationary Gaussian process. If the process \(\xi\) (t) is continuous a.e. then \[ \| \xi -D_ n*\xi \|_{\infty}\to 0,\quad n\to \infty, \] in probability, where [D*f](t), \(n\geq 1\), are the partial Fourier sums for \(f\in L_ 1[-\pi,\pi]\) and \(\| \cdot \|_{\infty}\) is the uniform norm on the interval [-a,a]\(\subset [- \pi,\pi]\). The paper contains some analogous statements for the convergence \(\xi -D_ n*\xi\) in the norms generated by continuity modules of a sufficiently wide class. The main result is the following theorem: Let \(\sigma\) (x) be some continuity module and the separable Banach space \(H^ o_{\sigma}[a,b]\) consist of such functions \(\phi\in C[a,b]\) that \[ \| \phi \|^{(\sigma)}_{[a,b]}=\sup_{t\in [a,b]}| \phi (t)| +\sup_{t\neq s}| \phi (t)-\phi (s)| /\sigma (| t- s|)<\infty \] \[ and\quad \sup_{| t-s| <h}| \phi (t)- \phi (s)| =o(\sigma (h)). \] If \(\xi \in H^ o_{\sigma}[-\pi,\pi]\) a.e., then \(\| \xi -D_ n*\xi \|^{(\sigma)}_{[-a,a]}\to 0\) in probability for every interval [-a,a]\(\subset [-\pi,\pi]\).