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

Arieh Iserles; Marcus Webb

doi:10.1002/cpa.21973

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

Communications on Pure and Applied Mathematics ◽

10.1002/cpa.21973 ◽

2021 ◽

Vol 74 (3) ◽

pp. 478-506

Author(s):

Arieh Iserles ◽

Marcus Webb

Keyword(s):

Fast Computation ◽

Differentiation Matrix ◽

Orthogonal Systems

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

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-019-09718-5 ◽

2020 ◽

Vol 26 (1) ◽

Author(s):

Arieh Iserles ◽

Marcus Webb

Keyword(s):

Rational Functions ◽

Orthogonal Rational Functions ◽

Differentiation Matrix ◽

Orthogonal Systems

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Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

Foundations of Computational Mathematics ◽

10.1007/s10208-019-09435-x ◽

2019 ◽

Vol 19 (6) ◽

pp. 1191-1221 ◽

Cited By ~ 3

Author(s):

Arieh Iserles ◽

Marcus Webb

Keyword(s):

Probability Measure ◽

Weight Function ◽

Real Line ◽

Orthonormal System ◽

Hermite Functions ◽

Time Dependent ◽

Polynomial Sequence ◽

Differentiation Matrix ◽

Band Limited ◽

Orthogonal Systems

Abstract In this paper, we explore orthogonal systems in $$\mathrm {L}_2({\mathbb R})$$L2(R) which give rise to a real skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are important since they are stable by design and, if necessary, preserve Euclidean energy for a variety of time-dependent partial differential equations. We prove that there is a one-to-one correspondence between such an orthonormal system $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ and a sequence of polynomials $$\{p_n\}_{n\in {\mathbb Z}_+}$${pn}n∈Z+ orthonormal with respect to a symmetric probability measure $$\mathrm{d}\mu (\xi ) = w(\xi ){\mathrm {d}}\xi $$dμ(ξ)=w(ξ)dξ. If $$\mathrm{d}\mu $$dμ is supported by the real line, this system is dense in $$\mathrm {L}_2({\mathbb R})$$L2(R); otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from $$\mathrm{d}\mu $$dμ to $$\{\varphi _n\}_{n\in {\mathbb Z}_+}$${φn}n∈Z+ is constructive, and we provide detailed algorithms to this end. We also prove that the only such orthogonal system consisting of a polynomial sequence multiplied by a weight function is the Hermite functions. The paper is accompanied by a number of examples illustrating our argument.

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