A counterexample in \(H^{\infty}+BUC\)
Let BUC be the space of bounded uniformly continuous functions on \({\mathbb{R}}\), \(QBUC=(H^{\infty}+BUC)\cap \overline{(H^{\infty}+BUC)}.\) The main result of the article under review is an example of a function H which is an outer function in \(H^ 1\) in the half-plane and such that \(\bar H/| H| \in H^{\infty}+BUC\) but \(\bar H/| H|\) cannot be represented as the product of an inner function and a function in QBUC. Thus the situation is very different from that in the case \(H^{\infty}+C\). The counterexample mentioned permits one to construct a counterexample to a problem in prediction theory. It is proved that there exists a strongly mixing stationary Gaussian process with continuous time whose density cannot be represented as \(| P|^ 2 e^{u+\tilde v}\) where P is an entire function of exponential type in \(L^ 2\) and u,v\(\in BUC\).
strongly mixing stationary Gaussian process
scientific article; zbMATH DE number 3877516