Orthogonal Systems with a Skew-Symmetric Differentiation Matrix

Funder: University of ManchesterAbstract 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.</jats:p

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

Funder: University of ManchesterAbstractIn this paper we explore orthogonal systems in $$\mathrm {L}_2(\mathbb {R})$$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 $$\mathrm {L}_2(\mathbb {R})$$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.</jats:p

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

Funder: University of ManchesterAbstractIn this paper we explore orthogonal systems in $$\mathrm {L}_2(\mathbb {R})$$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 $$\mathrm {L}_2(\mathbb {R})$$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.</jats:p