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Pointwise and Uniform Convergence of Fourier Extensions
Constructive Approximation ( IF 2.7 ) Pub Date : 2019-11-19 , DOI: 10.1007/s00365-019-09486-x
Marcus Webb , Vincent Coppé , Daan Huybrechs

Fourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

中文翻译:

傅里叶扩展的逐点收敛和一致收敛

区间上连续但非周期函数的傅立叶级数近似会受到吉布斯现象的影响,这意味着在端点附近存在永久振荡超调。傅里叶扩展通过使用在较大间隔上周期性的傅里叶级数来逼近函数来规避这个问题。先前关于傅立叶扩展收敛的结果集中在 $$L^2$$L 2 范数中的误差,但在本文中,我们分析了傅立叶扩展的逐点和一致收敛(公式化为 $$L 中的最佳近似) ^2$$ L 2 规范)。我们表明傅里叶扩展的逐点收敛比经典傅里叶级数更类似于勒让德级数。特别是,与经典傅立叶级数不同,傅立叶扩展在区间的端点处产生逐点收敛。与勒让德级数类似,端点处的逐点收敛比内部慢了一半的代数阶。证明是通过分析相关的 Lebesgue 函数以及用于傅立叶扩展的 Jackson 和 Bernstein 型定理来进行的。提供了数值实验。我们以关于傅立叶扩展的正则化和过采样最小二乘插值版本的开放性问题来结束本文。
更新日期:2019-11-19
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