  In this paper, we introduce a new theoretical framework built upon fractional
Sobolev-type spaces involving Riemann-Liouville (RL) fractional
integrals/derivatives, which is naturally arisen from exact representations of
Chebyshev expansion coefficients, for optimal error estimates of Chebyshev
approximations to functions with limited regularity. 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 characterize in existing literature. We can then
derive optimal error estimates for spectral expansions and the related
Chebyshev interpolation and quadrature measured in various norms, and also
improve the available results in usual Sobolev spaces of integer regularity
exponentials in several senses. As a by-product, 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 spectral approximations.
