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%DIFDELCMD < ]%%% . The regularity is quantified as the Riemann-Liouville (RL) fractional integration (of order 1-s\in (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^\infty-estimate is non-optimal for functions with interior singularities. However, we show that the optimality can be achieved in a certain weightedL^\infty-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 .
We present a new fractional Taylor formula for singular functions whose %DIFDELCMD < ]%%% Caputo fractional derivatives are of bounded variation. It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) L^\infty-estimates and L^2-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for p and hp methods for singular problems, and answer some open questions posed in some recent literature .
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This paper concerns optimal error estimates for Legendre polynomial expansions of
We present a new fractional Taylor formula for
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regularities are naturally characterised by a certain fractional Sobolev-type space introduced in
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34, Math. Comput., 2019
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. The regularity is quantified as the Riemann-Liouville (RL) fractional integration (of order 1-s\in (0,1)) of the highest possible integer-order derivative (of order m) of the underlying singular function that is of
Caputo fractional derivatives are of
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Different from Chebyshev approximation, the usual L^\infty-estimate is non-optimal for functions with interior singularities. However, we show that the optimality can be achieved in a certain weightedL^\infty-sense. We also provide point-wise error estimates that can
It bridges and ``interpolates" the usual Taylor formulas with two consecutive integer orders. This enables us to obtain an analogous formula for the Legendre expansion coefficient of this type of singular functions, and further derive the optimal (weighted) L^\infty-estimates and L^2-estimates of the Legendre polynomial approximations. This set of results can enrich the existing theory for p and hp methods for singular problems, and
R
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
some recent literature