## Abstract

In this paper we explore orthogonal systems in 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, as 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 or-

thonormal system {f'n}nεZ+ and a sequence of polynomials {pn}nεZ+ orthonor-

mal with respect to a symmetric probability measure dμ(ξ) = w(ξ)dξ. If dμ is

supported by the real line this system is dense in L2(R), otherwise it is dense in

a Paley{Wiener space of band-limited functions. The path leading from dµ to

{f'n}nεZ+ 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.

skew-symmetric, tridiagonal, irreducible differentiation matrix. Such systems are

important since they are stable by design and, as 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 or-

thonormal system {f'n}nεZ+ and a sequence of polynomials {pn}nεZ+ orthonor-

mal with respect to a symmetric probability measure dμ(ξ) = w(ξ)dξ. If dμ is

supported by the real line this system is dense in L2(R), otherwise it is dense in

a Paley{Wiener space of band-limited functions. The path leading from dµ to

{f'n}nεZ+ 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.

Original language | English |
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Journal | Foundations of Computational Mathematics |

Early online date | 10 Oct 2019 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Differentiation matrix
- special functions
- orthogonal polynomials
- energy preserving numerical methods