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.

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The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by r G (λ) ν (x) (right GGF-Fs) and l G (λ) ν (x) (left GGF-Fs) with x ∈ (−1, 1), λ > −1/2 and real ν ≥ 0, are special functions (usually non-polynomials), which are defined upon the hypergeometric representation of the classical Gegenbauer polynomial by allowing integer degree to be real fractional degree. Remarkably, the GGF-Fs become indispensable for optimal error estimates of polynomial approximation to singular functions, and have intimate relations with several families of nonstandard basis functions recently introduced for solving fractional differential equations. However, some properties of GGF-Fs, which are important pieces for the analysis and applications, are unknown or under explored. The purposes of this paper are twofold. The first is to show that for λ, ν > 0 and x = cos θ with θ ∈ (0, π),and derive the precise expression of the "residual" term R (λ) ν (θ). With this at our disposal, we obtain the bounds of GGF-Fs uniform in ν. Under an appropriate weight function, the bounds are uniform for θ ∈ [0, π] as well. Moreover, we can study the asymptotics of GGF-Fs with large fractional degree ν. The second is to present miscellaneous properties of GGF-Fs for better understanding of this family of useful special functions.2010 Mathematics Subject Classification. 30E15, 41A10, 41A25, 41A60, 65G50. (λ) ν (x) can be viewed as special g-Jacobi functions (see Mirevski et al [15]), defined by replacing the integer-order derivative in the Rodrigues' formula of the Jacobi polynomials by the RL fractional derivative. However, both the definition and derivation of some properties in [15] have flaws (see Remark 4.1). 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. [10,8]) recently introduced in developing efficient spectral methods for fractional differential equations. It is also noteworthy that the seminal work of Gui and Babuška [9] on hpestimates of Legendre approximation of singular functions essentially relied on some non-classical Jacobi polynomials with the parameter α or β < −1, which turned out closely related to GGF-Fs. In a nutshell, the GGF-Fs (and more generally the generalised Jacobi functions of fractional degree) can be of great value for numerical analysis and computational algorithms, but many of their properties are still under explored.It is known that the study of asymptotics has been a longstanding subject of special functions and their far reaching applications (see, e.g., [16,23,17]). Most of the asymptotic results of classical orthogonal polynomials can be found in the books [22,17], and are reported in the review papers [14,26,27] in more general senses. We highlight that the asymptotic formulas of the hypergeometric function: 2 F 1 (a − µ, b + µ; c; (1 − z)/2) in terms of Bessel functions for lar...

This paper concerns optimal error estimates for Legendre polynomial expansions of singular functions whose regularities are naturally characterised by a certain fractional Sobolev-type space introduced in [34, Math. Comput., 2019]. The regularity is quantified as the Riemann-Liouville (RL) fractional integration (of order 1 − s ∈ (0, 1)) of the highest possible integer-order derivative (of order m) of the underlying singular function that is of bounded variation. Different from Chebyshev approximation, the usual L ∞ -estimate is non-optimal for functions with interior singularities. However, we show that the optimality can be achieved in a certain weighted L ∞ -sense. We also provide point-wise error estimates that can answer some open questions posed in several recent literature. Here, our results are valid for all polynomial orders, as in most applications the polynomial orders are relatively small compared to those in the asymptotic range. KeywordsApproximation by Legendre polynomials • Optimal estimates • Singular functions with interior and endpoint singularities • Functions of bounded variation Mathematics Subject Classification (2010) 41A10 • 41A25 • 41A50 • 65N35 • 65M60

We propose a spectral collocation method for the numerical solution of the time-dependent Schrödinger equation, where the newly developed nonpolynomial functions in a previous study are used as basis functions. Equipped with the new basis functions, various boundary conditions can be imposed exactly. The preferable semi-implicit time marching schemes are employed for temporal discretization. Moreover, the new basis functions build in a free parameter intrinsically, which can be chosen properly so that the semi-implicit scheme collapses to an explicit scheme. The method is further applied to linear Schrödinger equation set in unbounded domain. The transparent boundary conditions are constructed for time semidiscrete scheme of the linear Schrödinger equation. We employ spectral collocation method using the new basis functions for the spatial discretization, which allows for the exact imposition of the transparent boundary conditions. Comprehensive numerical tests both in bounded and unbounded domain are performed to demonstrate the attractive features of the proposed method. KEYWORDS collocation method, nonpolynomial basis functions, Schrödinger equation, transparent boundary conditions 186

In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

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