Optimal error estimates for Chebyshev approximations of functions with limited regularity in fractional Sobolev-type spaces

Abstract

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.2000 Mathematics Subject Classification. 41A10, 41A25, 41A50, 65N35, 65M60.

Asymptotics of the generalized Gegenbauer functions of fractional degree

“…On the other hand, the Handbook [17, (15.9.15)] listed r G (λ)ν (x) but without presented any of their properties. Interestingly, as pointed out in [13], the GGF-Fs have a direct bearing on Jacobi polyfractonomial (cf. [28]) and generalised Jacobi functions (cf.…”

“…(2.2)) under the left RL fractional derivative (cf. [13]). As a generalization of Gegenbauer polynomials, the GGF-Fs satisfy the following fractional Rodrigues' formula.…”

“…For completeness, we quote the following estimates, which were very useful in the error analysis in [13]. For λ ≥ 1 and real ν ≥ 0, we have…”

“…Typically, such results are established in Jacobi-weighted Sobolev spaces with integral-order regularity exponentials (see, e.g., [21]), or weighted Besov spaces with fractional regularity exponentials using the notion of space interpolation (see, e.g., [5,6,7]). In a very recent work [13], we introduced a new framework of fractional Sobolev-type spaces involving Riemann-Liouville (RL) fractional integrals and derivatives in the study of polynomial approximation to singular functions. Such spaces are naturally arisen from exact representations of orthogonal polynomial expansion coefficients, and could best characterize the fractional differentiability/regularity, leading to optimal error estimates.…”

Calculation of fractional integrals using partial sums of Fourier series for structural mechanics problems

“…This is due to a number of reasons. First, the development of the theory of fractional integration and differentiation as such, such as the works [1,2], the review work [3], which presents the experience of M. Jrbashyan's research related to the modern theory of fractional calculus. Second, extensive applications of this mathematical apparatus in various fields of science and industry [4], especially in fields related to nanotechnology, diffusion problems, as well as in creating structures that take into account the state of matter.…”