非光滑函数的分数阶插值公式

本文基于局部分数阶Taylor展开式构造非光滑函数的分数阶插值公式,证明了插值公式的存在和唯一性,给出了分数阶插值的Lagrange表示形式及其误差余项,讨论了一种混合型的分段分数阶插值和整数阶插值的收敛阶.数值算例验证了对于非光滑函数分数阶插值明显优于通常的多项式插值,并说明在实际计算中采用分段混合分数阶和整数阶插值可以使得插值误差在区间上分布均匀,能够极大地提高插值精度.

A FRACTIONAL INTERPOLATION FORMULA FOR NON-SMOOTH FUNCTIONS

This paper constructs a fractional interpolation formula for non-smooth functions based on the local fractional Taylor's expansion. The existence and uniqueness of the fractional interpolation formula are proved. The formula with Lagrange basis and its error remainder are provided. The convergence order of a hybrid pattern for piecewise fractional order interpolation and integer-order interpolation is also discussed. Numerical examples demonstrate that the fractional interpolation is obviously superior to the traditional polynomial interpolation for non-smooth functions. They also show that the interpolating error can be uniformly distributed on the interval by using the piecewise hybrid interpolation, which can exceedingly improve the interpolation accuracy.