Optimal Error Estimates for Chebyshev Approximations of Functions with Endpoint Singularities in Fractional Spaces

In this paper, we introduce some new definitions and more general results of fractional spaces in order to deal with functions with endpoint singularities. Based on this theoretical framework, we derive optimal decay rates for Chebyshev expansion coefficients by applying the uniform upper bounds of generalized Gegenbauer functions of fractional degree (GGF-Fs). This enables us to further present the optimal L^∞ -estimates and L² -estimates of the Chebyshev polynomial approximations. In particular, we provide point-wise error estimates and the precise upper and lower bounds for u(x) = (1 + x) ^α , α> 0 on Ω¯ = [- 1 , 1] in L^∞ -norm. Moreover, we also discuss the extension of our main results to optimal error estimates of the related Chebyshev interpolation and quadrature measured in various norms at Chebyshev–Gauss points. Numerical results demonstrate the perfect coincidence with the error estimates. Indeed, the analysis techniques can enrich the theoretical foundation of p and hp methods for singular problems.