Semiorthogonal functions and orthogonal polynomials on the unit circle
Let \(\mathcal P=\mathbb R[x]\) be the linear space of polynomials with real coefficients, \(\Sigma=\mathcal P[\sqrt{1-x^2}]\), and \(L\) be the space of Laurent polynomials with complex coefficients. To each Hermitian functional \(\mathcal L\) on \(L\) corresponds a functional \(\mathcal M_{\mathcal L}\) on \(\Sigma\), establishing a bijection. The authors construct a basis of \(\Sigma\) satisfying orthogonality conditions with respect to \(\mathcal M_{\mathcal L}\), and show its connection with \(2 \times 2\) matrix polynomials and polynomials orthogonal on the unit circle. The result is applied to the study of the rational modifications of Hermite functionals \(\mathcal L\) of the type \[ \widetilde{\mathcal L}=\left(\tfrac 12 (x+x^{-1})-a\right)\mathcal L , \quad a \in \mathbb R . \]
scientific article; zbMATH DE number 1326351