Nikos Stylianopoulos
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Research Interests
Nikos Stylianopoulos has a total of 31 co-authors. Here are the top co-authors by number of shared publications:
N. Papamichael
Department of Mathematics and Statistics, University of Cyprus. PO Box 20537, CY-1678 Nicosia, Cyprus, CY
10 shared publications- Edward Barry Saff
Center for Constructive Approximation, Vanderbilt University, Nashville, USA
8 shared publications - Mihai Putinar
University of California at Santa Barbara
6 shared publications - Bernhard Beckermann
Université de Lille, Lille, France
2 shared publications Björn Gustafsson
Department of Mathematics, The Royal Institute of Technology, S-10044, Stockholm, Sweden
2 shared publicationsEdward B. Saff
Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, 37240 Nashville, USA
2 shared publications
Latest Publications
- 20 April 2020
- Published by Springer Nature in Foundations of Computational Mathematics
- Vol. 21 (1) , 71-124
Perturbations of Christoffel–Darboux Kernels: Detection of Outliers
- 20 April 2020
- Published by Springer Nature in Foundations of Computational Mathematics
- Vol. 21 (1) , 71-124
Perturbations of Christoffel–Darboux Kernels: Detection of Outliers
- Open Access
Perturbations of Christoffel-Darboux kernels. I: detection of outliers
Two central objects in constructive approximation, the Christoffel-Darboux kernel and the Christoffel function, are encoding ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel-Darboux kernel in C^d, with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri)potential theory are widely used in the derivation of our perturbation formulas.
Open AccessPerturbations of Christoffel-Darboux kernels. I: detection of outliers
Two central objects in constructive approximation, the Christoffel-Darboux kernel and the Christoffel function, are encoding ample information about the associated moment data and ultimately about the possible generating measures. We develop a multivariate theory of the Christoffel-Darboux kernel in C^d, with emphasis on the perturbation of Christoffel functions and their level sets with respect to perturbations of small norm or low rank. The statistical notion of leverage score provides a quantitative criterion for the detection of outliers in large data. Using the refined theory of Bergman orthogonal polynomials, we illustrate the main results, including some numerical simulations, in the case of finite atomic perturbations of area measure of a 2D region. Methods of function theory of a complex variable and (pluri)potential theory are widely used in the derivation of our perturbation formulas.
- 26 February 2018
- Published by Steklov Mathematical Institute
- Vol. 209 (3) , 168-188
Относительная асимптотика ортогональных многочленов для возмущенных мер
В работе дается обзор и доказывается ряд новых результатов о поведении ортогональных многочленов на плоскости под воздействием малых возмущений меры ортогональности. Вводится понятие полиномиально малого возмущения меры. Именно, если $\{p_n(\mu_j,z)\}_{n=0}^\infty$, $j=0,1$, - последовательность многочленов, ортогональных по этой мере, то мера $\mu_0 \geqslant \mu_1$ называется полиномиально малым возмущением меры $\mu_1$, если $\|p_n(\mu_1, \cdot )\|_{L_2(\mu_0-\mu_1)}\to 0$ при $n\to\infty$. Устанавливаются результаты об асимптотическом поведении двух последовательностей ортонормированных многочленов. Также доказываются результаты о поведении нулей полиномиально малых возмущения многочленов, ортогональных относительно плоской меры (многочленов Бергмана). Библиография: 35 названий.- 26 February 2018
- Published by Steklov Mathematical Institute
- Vol. 209 (3) , 168-188
Относительная асимптотика ортогональных многочленов для возмущенных мер
В работе дается обзор и доказывается ряд новых результатов о поведении ортогональных многочленов на плоскости под воздействием малых возмущений меры ортогональности. Вводится понятие полиномиально малого возмущения меры. Именно, если $\{p_n(\mu_j,z)\}_{n=0}^\infty$, $j=0,1$, - последовательность многочленов, ортогональных по этой мере, то мера $\mu_0 \geqslant \mu_1$ называется полиномиально малым возмущением меры $\mu_1$, если $\|p_n(\mu_1, \cdot )\|_{L_2(\mu_0-\mu_1)}\to 0$ при $n\to\infty$. Устанавливаются результаты об асимптотическом поведении двух последовательностей ортонормированных многочленов. Также доказываются результаты о поведении нулей полиномиально малых возмущения многочленов, ортогональных относительно плоской меры (многочленов Бергмана). Библиография: 35 названий.- 5 June 2017
- Published by Springer Nature in Constructive Approximation
- Vol. 47 (2) , 211-235
Bergman Orthogonal Polynomials and the Grunsky Matrix
- 5 June 2017
- Published by Springer Nature in Constructive Approximation
- Vol. 47 (2) , 211-235
Bergman Orthogonal Polynomials and the Grunsky Matrix
- Open Access
Relative Asymptotics of Orthogonal Polynomials for Perturbed Measures
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.
Open AccessRelative Asymptotics of Orthogonal Polynomials for Perturbed Measures
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.
- 1 January 2016
- Published by American Mathematical Society (AMS)
- Vol. 661, 187-198
Boundary Estimates for Bergman Polynomials in Domains with Corners
Let G G be a bounded simply-connected domain in the complex plane C \mathbb {C} , whose boundary Γ := ∂ G \Gamma :=\partial G is a Jordan curve, and let { p n } n = 0 ∞ \{p_n\}_{n=0}^{\infty } denote the sequence of Bergman polynomials of G G . This is defined as the unique sequence of polynomials { p n ( z ) } n = 0 ∞ \{p_n(z)\}_{n=0}^\infty , with positive leading coefficient, that are orthonormal with respect to the area measure on G G . The asymptotic behaviour of p n ( z ) p_n(z) in the exterior of Γ \Gamma , in cases when Γ \Gamma is a piecewise analytic Jordan curve have been established recently in \cite{St-CA13}. The purpose of this note is to derive, for the same class of curves, estimates for the asymptotics of p n ( z ) p_n(z) on Γ \Gamma .
- 1 January 2016
- Published by American Mathematical Society (AMS)
- Vol. 661, 187-198
Boundary Estimates for Bergman Polynomials in Domains with Corners
Let G G be a bounded simply-connected domain in the complex plane C \mathbb {C} , whose boundary Γ := ∂ G \Gamma :=\partial G is a Jordan curve, and let { p n } n = 0 ∞ \{p_n\}_{n=0}^{\infty } denote the sequence of Bergman polynomials of G G . This is defined as the unique sequence of polynomials { p n ( z ) } n = 0 ∞ \{p_n(z)\}_{n=0}^\infty , with positive leading coefficient, that are orthonormal with respect to the area measure on G G . The asymptotic behaviour of p n ( z ) p_n(z) in the exterior of Γ \Gamma , in cases when Γ \Gamma is a piecewise analytic Jordan curve have been established recently in \cite{St-CA13}. The purpose of this note is to derive, for the same class of curves, estimates for the asymptotics of p n ( z ) p_n(z) on Γ \Gamma .
- 1 March 2015
- Published by Elsevier in Journal of Approximation Theory
- Vol. 191, 118-127
On the zeros of asymptotically extremal polynomial sequences in the plane
- 1 March 2015
- Published by Elsevier in Journal of Approximation Theory
- Vol. 191, 118-127
On the zeros of asymptotically extremal polynomial sequences in the plane
- 1 January 2015
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 47 (3) , 2442-2463
Orthogonal Polynomials for Area-Type Measures and Image Recovery
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...- 1 January 2015
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 47 (3) , 2442-2463
Orthogonal Polynomials for Area-Type Measures and Image Recovery
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...- Open Access
On the zeros of asymptotically extremal polynomial sequences in the plane
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.Open AccessOn the zeros of asymptotically extremal polynomial sequences in the plane
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. - Open Access
Orthogonal polynomials for area-type measures and image recovery
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.Open AccessOrthogonal polynomials for area-type measures and image recovery
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.