从非等距离散样本近似傅里叶变换
Approximating the Fourier Transform from Non-equispaced Discrete Samples
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中文总结 AI 辅助
该研究针对函数傅里叶变换的近似问题，采用概率密度逆累积分布函数变换生成的非等距节点采样，推导了确定性误差界，优化密度可恢复等距采样收敛速率并降低预渐近误差，数值实验验证了方法有效性。
中文摘要 AI 辅助
我们研究从有限个样本近似函数的傅里叶变换。脱离等距设置，我们通过概率密度的逆累积分布函数变换[0,1]上的等距点，得到确定性非等距节点来对函数采样。这种变量变换紧致化了实直线，因此无需截断空间域。结合中点法则的精确混叠恒等式与变换后振荡积分的平稳相位估计，我们推导了每个频率及L_p空间下的确定性误差界。对于空间和频率具有多项式衰减的函数，显式优化的密度可恢复等距采样的收敛速率，而额外的方差参数可大幅降低预渐近误差常数。对于亚指数衰减的函数，多项式衰减的密度可产生亚指数速率。数值实验验证了该理论，并证明其与最优缩放的等距采样相比，预渐近误差更小。
英文摘要
We study the approximation of the Fourier transform of a function from finitely many samples. Departing from the equispaced setting, we sample the function at deterministic non-equispaced nodes obtained by transforming equispaced points on $[0,1]$ through the inverse cumulative distribution function of a probability density. This change of variables compactifies the real line, so that no truncation of the space domain is necessary. Combining an exact aliasing identity for the midpoint rule with stationary-phase estimates for the transformed oscillatory integrals, we derive deterministic error bounds for every frequency and in $L_p$. For functions with polynomial decay in space and frequency, an explicitly optimized density recovers the equispaced convergence rate, while an additional variance parameter substantially reduces the pre-asymptotic error constants. For (sub-)exponentially decaying functions a polynomially decaying density yields (sub-)exponential rates. Numerical experiments confirm the theory and demonstrate the reduced pre-asymptotic error compared with optimally scaled equispaced sampling.
