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In the context of orthogonal polynomials in the plane we introduce the notion of a polynomially small (PS) perturbation of a measure. In such a case we establish relative asymptotic results for the two sequences of the associated orthonormal polynomials. We also provide results dealing with the behaviour of the zeros of PS perturbations of area orthogonal (Bergman) polynomials., Comment: 21 pages, 4 figures

By exploiting a link between Bergman orthogonal polynomials and the Grunsky matrix, probably first observed by K{\"u}hnau in 1985, we improve some recent results on strong asymptotics of Bergman polynomials outside the domain G of orthogonality, and on entries of the Bergman shift operator. In our proofs we suggest a new matrix approach involving the Grunsky matrix, and use well-established results in the literature relating properties of the Grunsky matrix to the regularity of the boundary of G, and the associated conformal maps. For quasiconformal boundaries, this approach allows for new insights for Bergman polynomials.

There is an ongoing debate on the therapeutic potential of vaso-modulatory interventions against glioma invasion. Prominent vasculature-targeting therapies involve functional tumour-associated blood vessel deterioration and normalisation. The former aims at tumour infarction and nutrient deprivation medi- ated by vascular targeting agents that induce occlusion/collapse of tumour blood vessels. In contrast, the therapeutic intention of normalising the abnormal structure and function of tumour vascular net- works, e.g. via alleviating stress-induced vaso-occlusion, is to improve chemo-, immuno- and radiation therapy efficacy. Although both strategies have shown therapeutic potential, it remains unclear why they often fail to control glioma invasion into the surrounding healthy brain tissue. To shed light on this issue, we propose a mathematical model of glioma invasion focusing on the interplay between the mi- gration/proliferation dichotomy (Go-or-Grow) of glioma cells and modulations of the functional tumour vasculature. Vaso-modulatory interventions are modelled by varying the degree of vaso-occlusion. We discovered the existence of a critical cell proliferation/diffusion ratio that separates glioma invasion re- sponses to vaso-modulatory interventions into two distinct regimes. While for tumours, belonging to one regime, vascular modulations reduce the tumour front speed and increase the infiltration width, for those in the other regime the invasion speed increases and infiltration width decreases. We show how these in silico findings can be used to guide individualised approaches of vaso-modulatory treatment strategies and thereby improve success rates.

Let $E$ be a compact set of positive logarithmic capacity in the complex plane and let $\{P_n(z)\}_{1}^{\infty}$ be a sequence of asymptotically extremal monic polynomials for $E$ in the sense that \begin{equation*}%\label{} \limsup_{n\to\infty}\|P_n\|_E^{1/n}\le\mathrm{cap}(E). \end{equation*} The purpose of this note is to provide sufficient geometric conditions on $E$ under which the (full) sequence of normalized counting measures of the zeros of $\{P_n\}$ converges in the weak-star topology to the equilibrium measure on $E$, as $n\to\infty.$ Utilizing an argument of Gardiner and Pommerenke dealing with the balayage of measures, we show that this is true, for example, if the interior of the polynomial convex hull of $E$ has a single component and the boundary of this component has an "inward corner" (more generally, a "non-convex singularity"). This simple fact has thus far not been sufficiently emphasized in the literature. As applications we mention improvements of some known results on the distribution of zeros of some special polynomial sequences., Comment: 14 pages, 4 figures

Let $G$ be a finite union of disjoint and bounded Jordan domains in the complex plane, let $\mathcal{K}$ be a compact subset of $G$ and consider the set $G^\star$ obtained from $G$ by removing $\mathcal{K}$; i.e., $G^\star:=G\setminus \mathcal{K}$. We refer to $G$ as an archipelago and $G^\star$ as an archipelago with lakes. Denote by $\{p_n(G,z)\}_{n=0}^\infty$ and $\{p_n(G^\star,z)\}_{n=0}^\infty$, the sequences of the Bergman polynomials associated with $G$ and $G^\star$, respectively; that is, the orthonormal polynomials with respect to the area measure on $G$ and $G^\star$. The purpose of the paper is to show that $p_n(G,z)$ and $p_n(G^\star,z)$ have comparable asymptotic properties, thereby demonstrating that the asymptotic properties of the Bergman polynomials for $G^\star$ are determined by the boundary of $G$. As a consequence we can analyze certain asymptotic properties of $p_n(G^\star,z)$ by using the corresponding results for $p_n(G,z)$, which were obtained in a recent work by B. Gustafsson, M. Putinar, and two of the present authors. The results lead to a reconstruction algorithm for recovering the shape of an archipelago with lakes from a partial set of its complex moments., Comment: 24 pages, 9 figures

Let G be a bounded Jordan domain in the complex plane and consider the infinite upper Hessenberg matrix M associated with the Bergman orthogonal polynomials of G. This matrix represents the Bergman shift operator of G. The main purpose of the paper is to describe and analyze a close relation between M and the Toeplitz matrix with symbol the normalized conformal map of the exterior of the unit circle onto the complement of the closure of G. Our results are based on the strong asymptotics of the Bergman polynomials. As an application, we describe and analyze an algorithm for recovering the shape of G from its area moments., Comment: 21 pages, 3 figures

We establish the strong asymptotics for Bergman orthogonal polynomials defined over Jordan domains with corners. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for domains bounded by analytic curves, and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries. In order to do so, we use a new approach based on tools from quasiconformal mapping theory. The impact of the resulting theory is demonstrated in a number of applications, varying from coefficient estimates in the well-known class Sigma of univalent functions and a connection with operator theory, to the computation of capacities and a reconstruction algorithm from moments., Comment: 40 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:0910.1788

Let G be a bounded Jordan domain in the complex plane with piecewise analytic boundary. We present theoretical estimates and numerical evidence for certain phenomena, regarding the application of the Bergman kernel method with algebraic and pole singular basis functions, for approximating the conformal mapping of G onto the normalized disk. In this way, we complete the task of providing full theoretical justification of this method., Comment: 37 pages, 1 figure

The aim of the paper is to establish the strong asymptotics for the Bergman orthogonal polynomials defined over non-smooth domains in the complex plane. This complements an investigation started in 1923 by T. Carleman, who derived the strong asymptotics for domains with analytic boundaries and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries., Comment: 23 pages, 1 figure; corrected typos

We consider the series of the Bergman orthogonal polynomials associated with a bounded simply-connected domain in the complex plane, whose boundary is a Jordan curve. These are the polynomials that are orthonormal with respect to the area measure on the domain. The purpose of this note is to report on recent results regarding the fine asymptotic behaviour of the the leading coefficients and the polynomials in the complement of the domain, in cases when the boundary includes corners. These results complement an investigation started in the 1920's by T. Carleman, who obtained the fine asymptotics for domains with analytic boundaries and carried over by P.K. Suetin in the 1960's, who established them for domains with smooth boundaries., Comment: 4 pages

Growth estimates of complex orthogonal polynomials with respect to the area measure supported by a disjoint union of planar Jordan domains (called, in short, an archipelago) are obtained by a combination of methods of potential theory and rational approximation theory. The study of the asymptotic behavior of the roots of these polynomials reveals a surprisingly rich geometry, which reflects three characteristics: the relative position of an island in the archipelago, the analytic continuation picture of the Schwarz function of every individual boundary and the singular points of the exterior Green function. By way of explicit example, fine asymptotics are obtained for the lemniscate archipelago $|z^m-1|