Abstract:
We survey and present some new results that are related to the behaviour of orthogonal polynomials in the plane under small perturbations of the measure of orthogonality. More precisely, we introduce the notion of a polynomially small (PS) perturbation of a measure. Namely, if $\mu_0 \geqslant \mu_1$ and $\{p_n(\mu_j,z)\}_{n=0}^\infty$, $j=0,1$, are the associated orthonormal polynomial sequences, then $\mu_0$ is a PS perturbation of $\mu_1$ if $\|p_n(\mu_1,\,\cdot\,)\|_{L_2(\mu_0-\mu_1)}\to 0$, as $n\to\infty$. In such a case we establish relative asymptotic results for the two sequences of orthonormal polynomials. We also provide results dealing with the behaviour of the zeros of PS perturbations of area orthogonal (Bergman) polynomials.
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