A Family of Orthogonal Rational Functions and Other Orthogonal Systems with a skew-Hermitian Differentiation Matrix

Abstract

In this paper we explore orthogonal systems in L2(R) which give rise to a skew-Hermitian, tridiagonal differentiation matrix. Surprisingly, allowing the differentiation matrix to be complex leads to a particular family of rational orthogonal functions with favourable properties: they form an orthonormal basis for L2(R), have a simple explicit formulae as rational functions, can be manipulated easily and the expansion coefficients are equal to classical Fourier coefficients of a modified function, hence can be calculated rapidly. We show that this family of functions is essentially the only orthonormal basis possessing a differentiation matrix of the above form and whose coefficients are equal to classical Fourier coefficients of a modified function though a monotone, differentiable change of variables. Examples of other orthogonal bases with skew-Hermitian, tridiagonal differentiation matrices are discussed as well.

Fast Computation of Orthogonal Systems with a Skew‐Symmetric Differentiation Matrix

“…Which is preferable? As things stand, there is no clear answer (and things are complicated by the availability of yet another approach of this kind, using skew-Hermitian differentiation matrices, which is described in [21]). The full-range approximation has the virtue of simplicity, hence of easier implementation.…”

“…Some orthonormal bases (e.g., Hermite functions) have an essential singularity at infinity (viewed as the north pole of the Riemann sphere), as does, for example, the tanh map. Even when, like the Malmquist-Takenaka system [21], a basis is analytic in a strip surrounding R, our problems are not over because most analytic functions of interest are likely to have an essential singularity at infinity. As a striking example, while the Malmquist-Takenaka expansion coefficients of 1=.1 g x 2 / (which is analytic in a strip about R decay exponentially, as 3 jnj , the speed of decay for sin x=.1gx 2 / is O.jnj 5=4 /, barely better than linear [34]!…”

Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

“…Which is preferable? As things stand, there is no clear answer (and things are complicated by the availability of yet another approach of this kind, using skew-Hermitian differentiation matrices, which is described in [20]). The full-range approximation has the virtue of simplicity, hence of easier implementation.…”

“…Hermite functions) have an essential singularity at infinity (viewed as the North Pole of the Riemann sphere), as does, for example, the tanh map. Even when, like the Malmquist-Takenaka system [20], a basis is analytic in a strip surrounding R, our problems are not over because most analytic functions of interest are likely to have an essential singularity at infinity. As a striking example, while the Malmquist-Takenaka expansion coefficients of 1/(1 + x 2 ) (which is analytic ina strip about R decay exponentially, as 3 −|n| , the speed of decay for sin x/(1 + x 2 ) is O |n| −5/4 , barely better than linear [34]!…”

“…In Section 5 we list some conclusions of our work, and flesh out the basis for applications of these functions to computing the Fourier transform and to a fast Olver-Townsend-type spectral method on the real line [26]. This is the place to mention [20], a companion paper to the current one, where we explore complex-valued orthonormal systems with skew-Hermitian tridiagonal differentiation matrices. Inter alia we demonstrate there the existence of another system with coefficients that can be computed in O (N log 2 N) operations, this time using the Fast Fourier Transform.…”