广义FC-Gram逼近框架及其分析与应用
A Generalized FC-Gram Approximation Framework with Analysis and Applications
AI总结：
提出广义FC-Gram框架，通过截断函数控制Gram多项式延拓，建立收敛定理，误差为O(n^{-min(r+β,d)})，并应用于线性ODE和抛物型PDE的高阶求解。
摘要：
FC-Gram算法通过将非周期函数周期延拓到更大区间来构造高阶三角逼近，延拓区间上Gram多项式的混合延续质量直接决定逼近精度。我们提出GenFC，一个广义FC-Gram框架，其中每个Gram多项式的延续由满足指定边界平坦条件的截断函数塑造。我们建立了一个收敛定理，表明对于任何这样的族，GenFC逼近误差在原区间的上确界范数下满足$O(n^{-\min(r+\beta,\,d)})$，其中$f \in C^r([0,1])$具有可积的$(r+1)$阶导数，$d$是Gram多项式的数量，$\beta \in [0,1]$是$f^{(r+1)}$的傅里叶衰减指数。作者最近引入的改进FC-Gram算法作为特例被恢复，并且文中构造了几个满足这些条件的显式族。在光滑、有限正则性和快速振荡测试案例上的数值实验证实了理论预测。该框架进一步应用于通过后向微分公式(BDF)时间步进的线性ODE和抛物型PDE的高阶求解器，展示了全程的高阶精度。
英文摘要：
The FC-Gram algorithm constructs high-order trigonometric approximations of nonperiodic functions by periodically extending them to a larger interval, with the quality of the blending continuation of Gram polynomials over the extension interval directly governing the approximation accuracy. We introduce GenFC, a generalized FC-Gram framework in which the continuation of each Gram polynomial is shaped by a cutoff function satisfying prescribed boundary flatness conditions. We establish a convergence theorem showing that for any such family the GenFC approximation error satisfies $O(n^{-\min(r+β,\,d)})$ in the supremum norm on the original interval, where $f \in C^r([0,1])$ has an integrable $(r+1)$th derivative, $d$ is the number of Gram polynomials, and $β\in [0,1]$ is the Fourier decay exponent of $f^{(r+1)}$. The modified FC-Gram algorithm, recently introduced by the authors, is recovered as a special case, and several explicit families satisfying these conditions are constructed in the paper. Numerical experiments across smooth, limited-regularity, and rapidly oscillating test cases confirm the theoretical predictions. The framework is further applied to high-order solvers for linear ODEs and parabolic PDEs via backward differentiation formulae (BDF) time-stepping, demonstrating high-order accuracy throughout.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
