  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.
