TY - JOUR

T1 - A family of orthogonal rational functions and other orthogonal systems with a skew-Hermitian differentiation matrix

AU - Iserles, Arieh

AU - Webb, Marcus

PY - 2020/1/23

Y1 - 2020/1/23

N2 - 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 formulæ 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.

AB - 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 formulæ 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.

KW - Orthogonal systems

KW - orthogonal rational functions

KW - spectral methods

KW - Fast Fourier Transform

KW - Malmquist–Takenaka system

U2 - 10.1007/s00041-019-09718-5

DO - 10.1007/s00041-019-09718-5

M3 - Article

JO - Journal of Fourier Analysis and Applications

JF - Journal of Fourier Analysis and Applications

SN - 1531-5851

ER -