Let w be a nonnegative
integrable weight function on the real line R such that (logw)∕(1 + x2) is also
integrable. Let FT and PT denote, respectively, the closed linear spans in L2(R,wdx)
of {eiax: a ≧ T} and {eiax: a ≦ T}. Let 𝜃(T) denote the angle between P0 and FT.
The problem considered here is that of describing those weights w for which
𝜃(T) → π∕2 as T tends to infinity (such weights arise as the spectral densities of
strongly mixing stationary Caussian processes). Some necessary conditions on
w are given for 𝜃(T) → π∕2, and a construction is given to show that w
may have arbitrarily wild oscillatory discontinuities even if 𝜃(T) → π∕2.
Another measure of the interdependence of P0 and FT is introduced: let
𝜃 ∗ (T) denote the angle between PT⊖ (PT∩ F0) and F0⊖ (PT∩ F0). A
complete structural characterization is given of those weights w for which both
𝜃(T) and 𝜃 ∗ (T) tend to π∕2. Moreover, it is shown that if either 𝜃(T) or
𝜃 ∗ (T) is eventually positive and the other tends to π∕2, then they both
do.
