differentiation matrix
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2020 ◽  
Vol 76 ◽  
pp. 100651 ◽  
Author(s):  
Charles-Andrew Vande Catsyne ◽  
Sufyan Ali Sayyed ◽  
Patricia Molina-Ortiz ◽  
Bastien Moes ◽  
David Communi ◽  
...  

2019 ◽  
Vol 19 (6) ◽  
pp. 1191-1221 ◽  
Author(s):  
Arieh Iserles ◽  
Marcus Webb

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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