On the optimal rates of convergence of Gegenbauer projections
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In this paper we present a comprehensive convergence rate analysis of Gegenbauer projections. We show that, for analytic functions, the convergence rate of the Gegenbauer projection of degree n is the same as that of the best approximation of the same degree when λ≤0 and the former is slower than the latter by a factor of n^λ when λ>0, where λ is the parameter in Gegenbauer polynomials. For piecewise analytic functions, we demonstrate that the convergence rate of the Gegenbauer projection of degree n is the same as that of the best approximation of the same degree when λ≤1 and the former is slower than the latter by a factor of n^λ-1 when λ>1. The extension to functions of fractional smoothness is also discussed. Our theoretical findings are illustrated by numerical experiments.
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