On spline collocation and the Hilbert transform

In this paper we examine a relationship between the spline collocation projection operator pi(n) and the Hilbert singular integral operator H-0. We use Fourier analysis to prove that under certain conditions, a commutator property holds between the two operators. More specifically, we show that for u is an element of H-t, parallel to(pi H-n(0) - H-0 pi(n))u parallel to t <= Ch(lambda)parallel to u parallel to(s) (where h = 1/n), for some t,s and lambda is an element of R.