AbstractThis 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 $$\text {C}^1(a,b)$$ C 1 ( a , b ) weight function such that $$w(a)=w(b)=0$$ w ( a ) = w ( b ) = 0 , we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case $$a=-\infty $$ a = - ∞ , $$b=+\infty $$ b = + ∞ , 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 $$x^\alpha \textrm{e}^{-x}$$ x α e - x for $$x>0$$ x > 0 and $$\alpha >0$$ α > 0 and the ultraspherical weight function $$(1-x^2)^\alpha $$ ( 1 - x 2 ) α , $$x\in (-1,1)$$ x ∈ ( - 1 , 1 ) , $$\alpha >0$$ α > 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 $$\alpha $$ α , and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.