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 L²-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.

中文翻译：

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具有有界变化的分数阶导数的奇异函数的勒让德展开的最优误差估计

我们为奇异函数提出了一个新的分数泰勒公式，其 Caputo 分数阶导数是有界变化的。它使用两个连续的整数阶来桥接和“插入”通常的泰勒公式。这使我们能够获得此类奇异函数的勒让德展开系数的类似公式，并进一步推导出勒让德多项式近似的最优（加权）\(L^{\infty }\) -估计和L ^{2 -}估计. 这组结果可以丰富关于奇异问题的p和hp方法的现有理论，并回答一些最近文献中提出的一些开放性问题。