FAST FOURIER METHODS IN COMPUTATIONAL COMPLEX ANALYSIS
FAST FOURIER METHODS IN COMPUTATIONAL COMPLEX ANALYSIS
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DOI：
10.1137/1021093
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发表时间：
1979-01-01
期刊：
影响因子：
10.2
通讯作者：
HENRICI, P
中科院分区：
文献类型：
--
作者：
HENRICI, P
本文讨论了离散傅里叶变换，并指出了它在复分析中的一些计算问题。我们开始通过描述变换的基本性质及其在一维和多维情况下的有效实现，通过Cooley，刘易斯和Welch的约化公式（IBM Res，paper，1967），讨论了下列应用：用衰减因子计算傅里叶系数，数值保角变换中Symm积分方程的解，数值保角变换中Symm积分方程的解，数值保角变换中Symm积分方程的解，数值保角变换中Symm积分方程的解。三角插值;共轭周期函数的确定及其在单连通区域和双连通区域保角映射的Theodorsen积分方程中的应用; Laurent系数的确定及其在数值微分、生成函数和拉普拉斯变换的数值反演中的应用;确定高次多项式零点的“密度”。然后，我们讨论卷积及其应用的时间序列分析，乘法多项式和大整数，快速泊松求解器。本文最后叙述了布伦特和Kung（美国梅隆大学，1975年，1976年）关于快速算法操纵的权力系列
In this paper we discuss the discrete Fourier transform and point out some computational problems in (mainly) complex analysis where it can be fruitfully applied. We begin by describing the elementary properties of the transform and its efficient implementation, both in the one-dimensional and in the multi-dimensional case, by the reduction formulas of Cooley, Lewis, and Welch (IBM Res, paper, 1967).The following applications are then discussed: Calculation of Fourier coefficients using attenuation factors; solution of Symm’s integral equation in numerical conformal mapping; trigonometric interpolation; determination of conjugate periodic functions and their application to Theodorsen’s integral equation for the conformal mapping of simply and of doubly connected regions; determination of Laurent coefficients with applications to numerical differentiation, generating functions, and the numerical inversion of Laplace transforms; determination of the “density” of the zeros of high degree polynomials. We then discuss convolution and its application to time series analysis, to the multiplication of polynomials and of large integers, and to fast Poisson solvers. The paper concludes with an account of some recent results of Brent and Kung (Carnegie-Mellon Univ., 1975, 1976) concerning fast algorithms for manipulating power series