Optimal error estimates for Legendre approximation of singular functions with limited regularity

06/01/2020 ∙ by Wenjie Liu, et al. ∙ 0

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.



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