一种带自适应频率检测的局部核心-尾部傅里叶-拉盖尔框架方法用于无界域问题
A Localized Core-Tail Fourier-Laguerre Frame Method with Adaptive Frequency Detection for Unbounded-Domain Problems
- School of Mathematics and Statistics, Shandong University of Technology（山东理工大学数学与统计学院）
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AI总结：
本文提出一种局部核心-尾部傅里叶-拉盖尔框架方法，将实线域分解为核心与半无限尾部，分别用傅里叶延拓和调制拉盖尔框架处理，并引入自适应频率检测，实现无界域问题的高精度逼近。
AI中文摘要：
我们提出了一种用于实线上逼近和模型问题的局部核心-尾部傅里叶-拉盖尔框架方法。计算域被分解为一个有限核心和两个半无限尾部。在核心区域使用局部傅里叶延拓来解析非周期、振荡和局部非光滑结构，而在尾部使用调制拉盖尔框架，使得拉盖尔函数仅逼近缓慢变化的衰减包络。局部傅里叶延拓分量还提供了两种数据驱动机制：用于内部边缘检测的系数能量指标和用于选择尾部调制中心的局部频率指标。我们推导了误差估计，将核心逼近误差、调制拉盖尔包络误差、频率失配效应和有限尾部截断误差分开。分析表明，尾部复杂度主要受调制后的残余相位控制，而非原始载波频率。数值实验展示了该方法对振荡、多频率和导数不连续函数的高精度，一个衰减模型问题说明了该表示可通过精确界面约束与微分算子结合。
英文摘要：
We propose a localized core--tail Fourier--Laguerre frame method for approximation and model problems on the real line. The domain is decomposed into a finite core and two semi-infinite tails. Local Fourier extension is used in the core to resolve nonperiodic, oscillatory, and locally nonsmooth structures, while modulated Laguerre frames are used in the tails so that the Laguerre functions approximate only slowly varying decaying envelopes. The local Fourier extension component also provides two data-driven mechanisms: coefficient-energy indicators for internal edge detection and local frequency indicators for selecting tail modulation centers. We derive error estimates that separate the core approximation error, the modulated Laguerre envelope error, the effect of frequency mismatch, and the finite-tail truncation error. The analysis shows that the tail complexity is governed mainly by the residual phase after modulation rather than by the original carrier frequency. Numerical experiments demonstrate high accuracy for oscillatory, multi-frequency, and derivative-discontinuous functions, and a decaying model problem illustrates that the representation can be combined with differential operators through exact interface constraints.
