Orthogonal polynomials and rigged Hilbert space (Q1071277)

By functional analytic methods the classical question is investigated if a sequence of polynomials \((P_ n)\) satisfying the fundamental recurrence \[ zP_ n(z)=f_ nP_{n-1}(z)+g_ nP_ n(z)+h_ nP_{n+1}(z) \] with coefficients \(f_ n\), \(g_ n\), \(h_ n\) constitutes an orthogonal basis of \(L^ 2_{\mu}({\mathbb{R}})\) with respect to an appropriate positive measure \(\mu\). As a main result the question is answered positively in the symmetric case where \(0\neq h_ n=f^*_{n+1}\) (cf. Favard's theorem and the representation theorem). The analysis starts considering the associated Jacobi operator A in a separable Hilbert space H which acts on a fixed orthonormal basis \((\phi_ n)\) by \(A\phi_ n=f_ n\phi_{n-1}+g_ n\phi_ n+h_ n\phi_{n+1}\). The domain \(\Omega\) of A is chosen to consist of all \(\phi\in H\) for which the coefficients satisfy a certain set of decreasing conditions so as to form a stufen space (cf. definition) being dense in H. \(\Omega\) is supplied with an additional topology which is finer than the induced one and which turns \(\Omega\) into a complete, countable normed, nuclear space. It is shown that \(\Omega\) is invariant under A and that A is continuous with respect to the nuclear topology. So the situation of an operator in a rigged Hilbert space is achieved for A and (\(\Omega\),H). Now, in the case \(h_ n\neq 0\), the generalized eigenvectors \(F_ z\) for A corresponding to the eigenvalues \(z\in {\mathbb{C}}\) are found to satisfy the duality correspondence \(F_ z(\phi_ n)=P_ n(z).\) The remainder of the analysis consists in combining these results with spectral theory in the symmetric case.