On spline collocation and the Hilbert transform

Micula, Sanda



In this paper we examine a relationship between the spline collocation projection operator \pi_n and the Hilbert singular
integral operator {\mathcal{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 \in H^t, ||(\pi_n {\mathcal{H}_0} - {\mathcal{H}_0} \pi_n)u||_t \le C h^{\la} ||u||_s (where h=1/n), for some t, s and \la \in \R.

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Micula, Sandra