论文标题
具有有限变化的分数导数的单数函数的Legendre扩展的最佳误差估计值
Optimal error estimates for Legendre expansions of singular functions with fractional derivatives of bounded variation
论文作者
论文摘要
我们提出了一个新的泰勒公式，用于奇异函数，其Caputo分数衍生物具有有界变化。 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 $ p $和$ hp $的现有理论用于单数问题，并回答了一些最近的文献中提出的一些开放问题。
We present a new fractional Taylor formula for singular functions whose 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.
