发表机构
University of Göttingen; Chemnitz University of Technology; University of Rostock（哥廷根大学; 开姆尼茨工业大学; 罗斯托克大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究了DCT-I和DST-I用于参数数值积分时的误差，给出了显式误差估计，并发现有理函数和亚纯函数分别导致几何衰减与代数平台，以及临界几何速率。
AI 中文摘要
快速傅里叶变换（FFT）及其实数对应物——离散余弦变换（DCT）和离散正弦变换（DST）的快速算法——常被用于数值评估傅里叶余弦变换和傅里叶正弦变换，从而作为基于给定连续函数有限样本的参数化求积规则。但所得结果的精度如何？本文研究了将第一类DCT（DCT-I）和第一类DST（DST-I）应用于计算$\int_0^{\infty} f(x)\cos(2\pi xv)\d x$和$\int_0^{\infty} f(x)\sin(2\pi xv)\d x$（其中$f \in L^1(\Rp) \cap C(\Rp)$，$v \in \Rp$）时产生的误差，以及使用DCT-I计算函数$h \in C(I)$在区间$I:= [-1,1]$上的切比雪夫系数$\frac{2}{\pi}\int_0^{\pi} h(\cos\theta)\\,\cos(n\theta)\d\theta$（$n \in \Np$）时产生的误差。我们在多项式衰减、指数衰减和混合衰减条件下给出了实用且显式的误差估计。对于傅里叶余弦和正弦变换，有理函数导致频率参数$P$的几何衰减，并结合采样参数$L$的$L^{-3/2}$阶代数平台；而对于诸如$\operatorname{sech}(\pi\\,\cdot)$的亚纯函数，在两个参数中均达到临界几何速率。对于在$\C \setminus I$中具有简单极点的有理函数的切比雪夫逼近，求积误差具有临界几何速率$\rho_0^{-P}$，并带有显式常数，其中$\rho_0 > 1$是最大解析性伯恩斯坦椭圆的参数。多个数值示例说明了该理论。
英文摘要
The fast Fourier transform (FFT) and its real counterparts, the fast algorithms of the discrete cosine transform (DCT) and the discrete sine transform (DST), are frequently employed for the numerical evaluation of the Fourier cosine and Fourier sine transform, thereby acting as parametric quadrature rules based on finitely many samples of a given continuous function. But how accurate are the obtained results? In this paper we study the error occurring if the DCT of type I (DCT-I) and the DST of type I (DST-I) are applied to compute $\int_0^{\infty} f(x)\cos(2πxv)\d x$ and $\int_0^{\infty} f(x)\sin(2πxv)\d x$ for $f \in L^1(\Rp) \cap C(\Rp)$ and $v \in \Rp$, as well as the error occurring if the Chebyshev coefficients $\frac{2}π\int_0^π h(\cosθ)\,\cos(nθ)\dθ$, $n \in \Np$, of a function $h \in C(I)$ on $I := [-1,1]$ are computed by the DCT-I. We present practicable, explicit error estimates under polynomial, exponential, and mixed decay conditions. For the Fourier cosine and sine transform, rational functions lead to a geometric decay in the frequency parameter $P$ combined with an algebraic plateau of order $L^{-3/2}$ in the sampling parameter $L$, while for meromorphic functions such as $\operatorname{sech}(π\,\cdot)$ the critical geometric rate is attained in both parameters. For Chebyshev approximation of a rational function with simple poles in $\C \setminus I$, the quadrature error has the critical geometric rate $ρ_0^{-P}$ with an explicit constant, where $ρ_0 > 1$ is the parameter of the largest Bernstein ellipse of analyticity. Several numerical examples illustrate the theory.
