This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any $\mathrm{C}^1(a,b)$ weight function such that $w(a)=w(b)=0$, we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case $a=-\infty$, $b=+\infty$, only a limited number of powers of that matrix is bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function $x^\alpha \mathrm{e}^{-x}$ for $x>0$ and $\alpha>0$ and the ultraspherical weight function $(1-x^2)^\alpha$, $x\in(-1,1)$, $\alpha>0$, and establish their properties. Both weights share a most welcome feature of {\em separability,\/} which allows for fast computation. The quality of approximation is highly sensitive to the choice of $\alpha$ and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.

翻译：本文关注于给定零狄利克雷边界条件的实区间上的标准正交系统。具体而言，我们感兴趣的是具有反对称微分矩阵的正交系统（此类系统不包含标准正交多项式）。我们考虑此类系统的一种简单构造方法，并探讨其衍生性质。一般地，给定任意满足$w(a)=w(b)=0$的$\mathrm{C}^1(a,b)$权函数，均可生成具有反对称微分矩阵的标准正交系统。除$a=-\infty$、$b=+\infty$的情形外，该矩阵仅有有限次幂有界，我们建立了权函数性质与有界性之间的关联。特别地，我们详细研究了两种权函数：定义在$x>0$且$\alpha>0$时的拉盖尔权函数$x^\alpha \mathrm{e}^{-x}$，以及定义在$x\in(-1,1)$且$\alpha>0$时的超球权函数$(1-x^2)^\alpha$，并建立了它们的基本性质。这两种权函数均具有一个备受青睐的特征——**可分性**，这为实现快速计算提供了可能。近似质量对参数$\alpha$的选择高度敏感，我们讨论了如何根据零边界条件的数量最优选取该参数。