Optimal error estimates for chebyshev approximations of functions with limited regularity in fractional Sobolev-Type Spaces

In this paper, we introduce a new theoretical framework built upon fractional Sobolev-type spaces involving Riemann-Liouville fractional integrals/derivatives for optimal error estimates of Chebyshev polynomial approximations to functions with limited regularity. It naturally arises from exact representations of Chebyshev expansion coefficients. Here, the essential pieces of the puzzle for the error analysis include (i) fractional integration by parts (under the weakest possible conditions), and (ii) generalised Gegenbauer functions of fractional degree (GGF-Fs): a new family of special functions with notable fractional calculus properties. Under this framework, we are able to estimate the optimal decay rate of Chebyshev expansion coefficients for a large class of functions with interior and endpoint singularities, which are deemed suboptimal or complicated to characterise in existing literature. Then we can derive optimal error estimates for spectral expansions and the related Chebyshev interpolation and quadrature measured in various norms, and also improve available results in usual Sobolev spaces with integer regularity exponentials in several senses. As a byproduct, this study results in some analytically perspicuous formulas particularly on GGF-Fs, which are potentially useful in spectral algorithms. The idea and analysis techniques can be extended to general Jacobi polynomial approximations.