Approximating convolution products better than the DFT while keeping the FFT

Abstract

In recent years there has been considerable interest in the use of the Fast Fourier Transform Algorithm (FFT) to calculate the Discrete Fourier Transform (DFT), allowing—in particular—for the fast computation of convolution products of finite sequences of numbers. Generalizations of the DFT and FFT to dimensions n = 2, 3,… are immediate, but their use in dimensions n > 1 to (approximately) calculate convolution integrals appears quite limited, even though integral equations involving multi-dimensional convolutions are common in physics. Most likely this situation is due to the fact that the quadrature formulas for approximating multi-dimensional convolution integrals obtained via the DFT are quite poor if n > 1. It is shown how the FFT can be used to calculate each of a whole class of newly defined transforms, the LPT or Lattice Point Transforms (hence, each LPT has a “fast algorithm” implementation). In a manner analogous to the n-dimensional DFT, each n-dimensional LPT allows one to (approximately) compute n-dimensional convolution integrals. Some of the quadrature formulas so obtained are exceptionally good. Such quadrature formulas correspond to LPTs generated by “good lattice points”. The cataloguing of “good lattice points” represents an area of research in present day multi-dimensional integration theory. Where N 1 denotes the number of points of functional evaluation used, the expected error of the quadrature formulas arising through the use of the DFT is O(N 1 −1 n ) , while the expected error of the quadrature formulas arising through the use of LPTs generated by “good lattice points” is only slightly larger than O( N 1 −1). Applications to integral equations are discussed.