Orthogonal polynomials with respect to a family of Sobolev inner products on the unit circle

Abstract:

The principal objective here is to look at some algebraic properties of the orthogonal polynomials $\Psi _n^{(b,s,t)}$ with respect to the Sobolev inner product on the unit circle \[ \langle f,g\rangle _{S^{(b,s,t)}} = (1-t) \langle f,g\rangle _{\mu ^{(b)}} + t \overline {f(1)} g(1) + s \langle f^{\prime },g^{\prime }\rangle _{\mu ^{(b+1)}}, \] where $\langle f,g\rangle _{\mu ^{(b)}} = \frac {\tau (b)}{2\pi } \int _{0}^{2\pi }\overline {f(e^{i\theta })} g(e^{i\theta }) (e^{\pi -\theta })^{\mathcal {I}m(b)} (\sin ^{2}(\theta /2))^{\mathcal {R}e(b)} d\theta .$ Here, $\mathcal {R}e(b) > -1/2$, $0 \leq t < 1$, $s > 0$ and $\tau (b)$ is taken to be such that $\langle 1,1\rangle _{\mu ^{(b)}} = 1$. We show that, for example, the monic Sobolev orthogonal polynomials $\Psi _n^{(b,s,t)}$ satisfy the recurrence $\Psi _n^{(b,s,t)}(z) - \beta _n^{(b,s,t)} \Psi _{n-1}^{(b,s,t)}(z) = \Phi _n^{(b,t)}(z),$ $n \geq 1$, where $\Phi _n^{(b,t)}$ are the monic orthogonal polynomials with respect to the inner product $\langle f,g\rangle _{\mu ^{(b,t)}} = (1-t) \langle f,g\rangle _{\mu ^{(b)}} + t \overline {f(1)} g(1)$. Some related bounds and asymptotic properties are also given.