基于Legendre多项式的数值微分: 加权$L^2$空间中的收敛性分析

Legendre Polynomials-Based Numerical Differentiation: A Convergence Analysis in a Weighted $L^2$ Space

中文关键词:  Legendre多项式  数值微分  Jacobi多项式  加权$L^2$空间

英文关键词:Legendre polynomials  numerical differentiation  Jacobi polynomials  weighted $L^2$ space

中文摘要:

      我们研究利用$f^{\delta}$的Fourier-Legendre级数部分和的导数近似函数$f$的导数问题, 其中$f^{\delta}$表示噪音版本的$f$.我们在一个加权$L^2$空间中执行重建过程, 而不是在观测$L^2$空间. 这充分地利用了Legendre多项式的性质进而导致了收敛阶的轻微改进.最后, 我们用几个数值例子来说明所提方法的效率.

英文摘要:

      We consider the problem of estimating the derivative of a function $f$ from its noisy version $f^{\delta}$ by using the derivatives of the partial sums of Fourier-Legendre series of $f^{\delta}$. Instead of the observation $L^2$ space, we perform the reconstruction of the derivative in a weighted $L^2$ space. This takes full advantage of the properties of Legendre polynomials and results in a slight improvement on the convergence order. Finally, we provide several numerical examples to demonstrate the efficiency of the proposed method.