Exact convergence rates for best L _P rational approximation to the signum function and for optimal quadrature in H ^P

Abstract

For the signum function s on [−1, 1], the exact rate of convergence (up to a multiplicative constant) for best L _p approximation to s by real rational functions of degree n was obtained by Vjacheslavov for 1≤p<∞. Here we show that the same rates hold for complex rational approximation and we obtain extensions to L _p for 0<p<∞. Exact L _p convergence rates are also obtained for best symmetric rational approximation on the unit circle to the function H _ω(z) that equals one on the arc z=e ^iθ, −ω<θ<ω, and zero elsewhere. These results yield a new proof for the exact convergence rate for optimal quadrature in the Hardy space H ^p

Research conducted while visiting the University of South Florida.

Research supported in part by the National Science Foundation under grant DMS-881-4026.