Construction of Lighthill's unitary functions: The imbricate series of unity

Abstract

The Poisson Sum Theorem implies that every spatially periodic function f( x) can be written as a so-called “imbricate” series, that is, as the superposition of an infinite number of evenly space copies of a pattern function A( x). The pattern is the Fourier Transform of a function a( n) that specifies the usual Fourier coefficients. Because there are many a( n) which coincide with a given set of Fourier coefficients aj when n is an integer, the pattern function A( x) is not unique. In this work, we show how to construct pattern functions of the form A(x) = f(x)Ω(x) where f( x) is the periodic function itself (the imbrication of A( x)) and where Ω(x) is what Lighthill called a “unitary” function. We derive an explicit formula to construct all possible unitary functions and tabulate eight examples. We also prove that all unitary functions are pattern functions of unity, that is, for the imbricate series that sums to f( x) ≡ 1∀ x.