On the angle between two fourier subspaces

Examining an approximation inspired by equalization theory, we consider the minimum angle ΩN between the subspaces of Hilbert space generated by the sequences {e^ikω}^N-N and {e^ikω}|k| > N. Here ω ∊ [-π,π] and the inner product for the Hilbert space involve a positive, bounded weight function r(ω). The finite Toeplitz matrices R and Γ generated by r(ω) and l/r(ω), respectively, play a crucial role, and, in fact, sin²ΩN is the reciprocal of the largest eigenvalue of RΓ. In general, sin²ΩN is shown to be bounded away from unity as N becomes large. The geometry of the problem enables us to give some results concerning the product matrix RΓ, which, out of the present context, may seem surprising.