On the Angle Between Two Fourier Subspaces

On the Angle Between Two Fourier Subspaces By J. E. MAZO (Manuscript received September 3, 1976) Examining an approximation inspired by equalization theory, we consider the minimum angle tl^ between the subspaces of Hilbert space generated by the sequences elkwN~N and {e,/?a,}|fc| > N. Here a> (E [--7r,7r] and the inner product for the Hilbert space involve a positive, bounded weight function r(co). The finite Toeplitz matrices R and r generated by r(co) and l/r(co), respectively, play a crucial role, and, in fact, sin2Qiv is the reciprocal of the largest eigenvalue of RT. In general, sin2$lN 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 RT, which, out of the present context, may seem surprising.