Fast Fourier Methods in Computational Complex Analysis

Abstract

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 * The John von Neumann Lecture delivered at the SIAM National Meeting, May, 1978.

Interpolation on Uniform Meshes by the Translates of One Function and Related Attenuation Factors

“…Criteria for the unique solvability of this system may be obtained with the aid of discrete Fourier analysis. (Here and in the sequel we use the notation and results from the paper of Henrici on fast Fourier algorithms [7].) If we denote by a, s, and / the sequences a = i .…”

Numerical solution of boundary integral equations by means of attenuation factors

“…Up until 1975, this splitting was performed in a way that destroyed the periodicity, namely as log|z(t) − z(s)| = log|t − s| + log z(t) − z(s) t − s (see Hsiao et al, 1980). Then, Henrici suggested using the kernel for the circle as the denominator, giving log|z(t) − z(s)| = log|e it − e is | + log z(t) − z(s) e it − e is (1.4) (see Henrici, 1979;Reichel, 1984Reichel, , 1986, independently came up with the same idea), and used it to analytically solve the equation for the ellipse. In 1975 also, the second author of the present article (Berrut, 1976) and Reichel independently suggested using this splitting in the numerical solution of (1.1) with trigonometric polynomials for general curves (see Henrici, 1986).…”

Fast solvers of integral and pseudodifferential equations on closed curves

“…Assume (4) - (6). Due to Lemmas 3.1 and 3.2 we can represent A = B −1 (I − T ), and the Fredholm properties of…”

“…are known and satisfy the conditions (6) where W (b j ) = 1 2π arg b j (t)| t=1 t=0 is the winding number [15] of the function b j , j = 1, 2.…”