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 \(\text {C}^1(a,b)\) 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 few powers of that matrix are 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 \(x^\alpha \textrm{e}^{-x}\) 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 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.

本文关注的是实数区间内的正交系统，给定零狄利克雷边界条件。更具体地说，我们感兴趣的是具有斜对称微分矩阵的系统（这不包括正交多项式）。我们考虑此类系统的简单构建并探究其后果。一般来说，给定任何\(\text {C}^1(a,b)\)权重函数，使得\(w(a)=w(b)=0\)，我们可以生成具有倾斜的正交系统-对称微分矩阵。除了\(a=-\infty \)和\(b=+\infty \)的情况外，该矩阵只有少数幂是有界的，我们在权重函数的属性和有界性之间建立了联系。特别是，我们详细研究了两个权重函数：\(x>0\)和\(\alpha >0\)的拉盖尔权重函数\(x^\alpha \textrm{e}^{-x}\ )和超球权重函数\((1-x^2)^\alpha \)、\(x\in (-1,1)\)、\(\alpha >0\)，并建立它们的属性。这两个权重都有一个最受欢迎的可分离性特征，可以实现快速计算。近似的质量对\(\alpha \)的选择高度敏感，我们讨论如何根据零边界条件的数量来最佳选择该参数。