Fast Hankel Transforms Using Related and Lagged Convolutions

Abstract

A heuristic algorithm is presented for fast and accurate evaluation of complex Hankel transforms of orders 0 and 1. Concepts using linear digital convolution are introduced, where Bessel function evaluations are not required. Related convolution is defined over a set of transforms whose integrands are related algebraically. Given any transform argument range, fast lagged convolution is performed m place within a predeslgned digital filter By arranging related and lagged convolutions m matrix form, a whole matrix of complex Hankel transforms is evaluated with a minimum of integrand function calls. For each point in the matrix, an adaptive convolution algorithm (based on function behavior and a given truncation tolerance) further reduce unnecessary evaluations along the decreasing digital-filter tails. The class of mtegrands (excluding the Bessel factor) must be continuous monotonic decreasing complex (or real) functions of a real variable defined in (0, oo). Higher integerorder Hankel transforms may be expressed in terms of orders 0 and 1 by recursion, and can be rapidly evaluated by related convolution.