Orthogonal Systems with a Skew-Symmetric Differentiation Matrix
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Springer Nature
https://doi.org/10.1007/s10208-019-09435-x
https://doi.org/10.1007/s10208-019-09435-x
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Funder: University of Manchester
In this paper, we explore orthogonal systems in L2(R)$$\mathrm {L}_2({\mathbb R})$$ 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 {φn}n∈Z+$$\{\varphi _n\}_{n\in {\mathbb Z}_+}$$ and a sequence of polynomials {pn}n∈Z+$$\{p_n\}_{n\in {\mathbb Z}_+}$$ orthonormal with respect to a symmetric probability measure dμ(ξ)=w(ξ)dξ$$\mathrm{d}\mu (\xi ) = w(\xi ){\mathrm {d}}\xi $$. If dμ$$\mathrm{d}\mu $$ is supported by the real line, this system is dense in L2(R)$$\mathrm {L}_2({\mathbb R})$$; otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from dμ$$\mathrm{d}\mu $$ to {φn}n∈Z+$$\{\varphi _n\}_{n\in {\mathbb Z}_+}$$ 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.
In this paper, we explore orthogonal systems in L2(R)$$\mathrm {L}_2({\mathbb R})$$ 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 {φn}n∈Z+$$\{\varphi _n\}_{n\in {\mathbb Z}_+}$$ and a sequence of polynomials {pn}n∈Z+$$\{p_n\}_{n\in {\mathbb Z}_+}$$ orthonormal with respect to a symmetric probability measure dμ(ξ)=w(ξ)dξ$$\mathrm{d}\mu (\xi ) = w(\xi ){\mathrm {d}}\xi $$. If dμ$$\mathrm{d}\mu $$ is supported by the real line, this system is dense in L2(R)$$\mathrm {L}_2({\mathbb R})$$; otherwise, it is dense in a Paley–Wiener space of band-limited functions. The path leading from dμ$$\mathrm{d}\mu $$ to {φn}n∈Z+$$\{\varphi _n\}_{n\in {\mathbb Z}_+}$$ 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.