Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation

Abstract

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

Error analysis of Fourier–Legendre and Fourier–Hermite spectral-Galerkin methods for the Vlasov–Poisson system

“…To test the convergence rate on direction of 𝑣, we take 𝜖 = 𝜔 = 1, 𝜏 = 0.005, Ω 𝑥 = (0, 2𝜋), Ω 𝑣 = (−8, 8) and fix 2𝑀 + 1 = 13 Fourier modes, but run up 𝑇 = 0.1. In Figure 2, it shows that the convergence orders in 𝐿 2 norm, 𝐿 ∞ norm are 𝑠 + 1 2 and 𝑠, respectively, and the convergence behaviours are consistent with the Legendre approximation results for such functions with limited regularity (see e.g., [20]).…”

Are best approximations really better than Chebyshev?

“…The error estimate of other orthogonal projections such as Legendre, Gegenbauer and Jacobi projections has attracted attention in recent years (see, e.g., [1,16,29,30,32,33]). In [29, Figure 3] it has been observed that the error curves of Legendre projections illustrate similar character as that of Chebyshev projections.…”

Optimal rates of convergence and error localization of Gegenbauer projections

“…It is clear that the test functions in (3.4) also belong to this space and f 4 ∈ H 4 , f 5 ∈ H 5 and f 6 ∈ H 3 . This space is preferable when developing error estimates for various orthogonal polynomial approximations to differentiable function (see, e.g., [16,17,27,30,33]). Comparing the space H m and the assumption of piecewise analytic functions, which one is better in the sense that optimal rate of convergence of S λ n (f ) can be predicted?…”

“…Moreover, to the best of the author's knowledge, the sharpness of the derived error estimates has not been addressed. For the latter approach, a remarkable advantage is that some computable error bounds of S n (f ) can be established (see, e.g., [1,16,17,27,28,30,31,32,33,35]). However, as shown in [30,31], the convergence rate predicted by this approach may be overestimated for differentiable functions.…”