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Kats B. - One of the best experts on this subject based on the ideXlab platform.
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The jump problem for certain Beltrami equation on Arcs
2020Co-Authors: Kats B., Mironova S., Pogodina A.Abstract:© 2017, Pleiades Publishing, Ltd.We consider the jump boundary problem on Jordan Arc for solutions of certain Beltrami equations
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The szegö function on a non-rectifiable Arc
2020Co-Authors: Kats B., Kats D.Abstract:Let γ be a simple Jordan Arc in the complex plane. The Szegö function, by definition, is a holomorphic in C\γ function with a prescribed product of its boundary values on γ. The problem of finding the Segö function in the case of piecewise smooth γ was solved earlier. In this paper we study this problem for non-rectifiable Arcs. The solution relies on properties of the Cauchy transform of certain distributions with the support on γ. © Allerton Press, Inc., 2012
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Metric dimensions, generalized integrations, cauchy transform, and riemann boundary-value problem on nonrectifiable Arcs
2020Co-Authors: Kats B.Abstract:We consider a nonrectifiable Jordan Arc Γ on the complex plane ℂ with endpoints a1 and a2. The Riemann boundary-value problem on this Arc is the problem of finding a function Φ(z) holomorphic in ℂ̄ \ Γ satisfying the equality, where Φ±(t) are the limit values of Φ(z) at a point t ∈ Γ \ {a1, a2} from the left and from the right, respectively. We introduce certain distributions with supports on nonrectifiable Arc Γ that generalize the operation of weighted integration along this Arc. Then we consider boundary behavior of the Cauchy transforms of these distributions, i.e., their convolutions with (2πiz)-1. As a result, we obtain a description of solutions of the Riemann boundary-value problem in terms of a new version of the metric dimension of the Arc Γ, the so-called approximation dimension. It characterizes the rate of best approximation of Γ by polygonal lines. © 2013 Springer Science+Business Media New York
Maxim Yattselev - One of the best experts on this subject based on the ideXlab platform.
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Convergent Interpolation to Cauchy Integrals over Analytic Arcs
Foundations of Computational Mathematics, 2009Co-Authors: Laurent Baratchart, Maxim YattselevAbstract:We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan Arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that Arc with Dini-smooth nonvanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the Arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the Arc.
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Convergent Interpolation to Cauchy Integrals over Analytic Arcs
Foundations of Computational Mathematics, 2009Co-Authors: Laurent Baratchart, Maxim YattselevAbstract:We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan Arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that Arc with Dini-smooth non-vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the Arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non-Hermitian orthogonal polynomials, for which we use classical properties of Hankel and Toeplitz operators on smooth curves. A construction of the appropriate interpolation schemes is explicit granted the parametrization of the Arc.
Masatoshi Fukushima - One of the best experts on this subject based on the ideXlab platform.
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stochastic komatu loewner evolutions and bmd domain constant
Stochastic Processes and their Applications, 2018Co-Authors: Zhenqing Chen, Masatoshi FukushimaAbstract:Let D = Hn [ N=1 Ck be a standard slit domain, where H is the upper half plane and Ck, 1 k N, are mutually disjoint horizontal line segments in H. Given a Jordan Arc D starting at @H; let gt be the unique conformal map from Dn [0;t] onto a standard slit domain Dt = Hn[ N=1 Ck(t) satisfying the hydrodynamic normalization at innity. It has been established recently that gt satises an ODE called a Komatu-Loewner equation in terms of the complex Poisson kernel of the Brownian motion with darning (BMD) for Dt, extending the classical chordal Loewner equation for the simply connected domain D = H: We randomize the Jordan Arc according to a system of probability measures on the family of equivalence classes of Jordan Arcs that enjoy a domain Markov property and a certain conformal invariance property. We show that the induced process ( (t); s(t)) satises a Markov type stochastic dierential equation, where (t) is a motion on @H and s(t) represents the motion of the endpoints of the slitsfCk(t); 1 k Ng: The diusion and drift coecients and b of (t) are homogeneous functions of degree 0 and 1, respectively, while s(t) has drift coecients only, determined by the BMD complex Poisson kernel that are known to be Lipschitz continuous. Conversely, given such functions and b with local Lipschitz continuity, the corresponding SDE admits a unique solution ( (t); s(t)). The latter produces random conformal maps gt(z) via the Komatu-Loewner equation. The resulting family of random growing hullsfFtg from the conformal mappings is called SKLE;b : We show that it enjoys a certain scaling property and a domain Markov property. Among other things, we further prove that SKLE;b has a locality property if = p 6 and b = bBMD, where bBMD is a BMD-domain constant that describes the discrepancy of a standard slit domain from H relative to BMD. AMS 2000 Mathematics Subject Classication : Primary 60J67, Secondary 30C20, 60H10, 60H30
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stochastic komatu loewner evolutions and bmd domain constant
arXiv: Probability, 2014Co-Authors: Zhenqing Chen, Masatoshi FukushimaAbstract:Let $D={\mathbb H} \setminus \cup_{k=1}^N C_k$ be a standard slit domain, where ${\mathbb H}$ is the upper half plane and $C_k$, $1\leq k\leq N$, are mutually disjoint horizontal line segments in $H$. Given a Jordan Arc $\gamma\subset D$ starting at $\partial H$, let $g_t$ be the unique conformal map from $D\setminus\gamma[0,t]$ onto a standard slit domain $D_t={\mathbb H} \setminus \cup_{k=1}^N C_k(t)$ satisfying the hydrodynamic normalization at infinity. It has been established recently that $g_t$ satisfies an ODE called a Komatu-Loewner equation in terms of the complex Poisson kernel of the Brownian motion with darning (BMD) for $D_t$. We randomize the Jordan Arc $\gamma$ according to a system of probability measures on the family of equivalence classes of Jordan Arcs that enjoy a domain Markov property and a certain conformal invariance property. We show that the induced process $(\xi(t), {\bf s}(t))$ satisfies a Markov type stochastic differential equation, where $\xi(t)$ is a motion on $\partial {\mathbb H}$ and ${\bf} s(t)$ represents the motion of the endpoints of the slits $\{C_k(t),\; 1\le k\le N \}.$ Conversely, given such functions $\alpha$ and $b$ with local Lipschitz continuity, the corresponding SDE admits a unique solution $(\xi(t), {\bf s}(t))$. The latter produces random conformal maps $g_t(z)$ via the Komatu-Loewner equation. The resulting family of random growing hulls $\{F_t\}$ from the conformal mappings is called ${\rm SKLE}_{\alpha,b}.$ We show that it enjoys a certain scaling property and a domain Markov property. Among other things, we further prove that ${\rm SKLE}_{\alpha,-b_{\rm BMD}}$ for a constant $\alpha >0$ has a locality property if and only if $\alpha = \sqrt{6}$, where $b_{\rm BMD}$ is a BMD-domain constant that describes the discrepancy of a standard slit domain from ${\mathbb H}$ relative to BMD.
Emilio Torrano - One of the best experts on this subject based on the ideXlab platform.
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The Hessenberg matrix and the Riemann mapping function
Advances in Computational Mathematics, 2013Co-Authors: Carmen Escribano, Antonio Giraldo, M. Asunción Sastre, Emilio TorranoAbstract:We consider a Jordan Arc Γ in the complex plane ${\mathbb C}$ and a regular measure μ whose support is Γ. We denote by D the upper Hessenberg matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with μ. We show in this work that, if the Hessenberg matrix D is uniformly asymptotically Toeplitz, then the symbol of the limit operator is the restriction to the unit circle of the Riemann mapping function ?(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure μ. We use this result to show how to approximate the Riemann mapping function for the support of μ from the entries of the Hessenberg matrix D.
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The Hessenberg matrix and the Riemann mapping
arXiv: Spectral Theory, 2011Co-Authors: Carmen Escribano, Antonio Giraldo, M. Asunción Sastre, Emilio TorranoAbstract:We consider a Jordan Arc \Gamma in the complex plane \mathbb{C} and a regular measure \mu whose support is \Gamma . We denote by D the upper Hessenberg matrix of the multiplication by z operator with respect to the orthonormal polynomial basis associated with \mu . We show in this work that, if the Hessenberg matrix D is uniformly asymptotically Toeplitz, then the symbol of the limit operator is the restriction to the unit circle of the Riemann mapping function \phi(z) which maps conformally the exterior of the unit disk onto the exterior of the support of the measure \mu . We use this result to show how to approximate the Riemann mapping function for the support of \mu from the entries of the Hessenberg matrix D.
B. A. Kats - One of the best experts on this subject based on the ideXlab platform.
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Metric dimensions, generalized integrations, cauchy transform, and riemann boundary-value problem on nonrectifiable Arcs
Journal of Mathematical Sciences, 2013Co-Authors: B. A. KatsAbstract:We consider a nonrectifiable Jordan Arc Γ on the complex plane $$ \mathbb{C} $$ with endpoints a _1 and a _2. The Riemann boundary-value problem on this Arc is the problem of finding a function Φ( z ) holomorphic in $$ \bar{\mathbb{C}} $$ \ Γ satisfying the equality $$ {\varPhi^{+}}(t)=G(t){\varPhi^{-}}(t)+g(t),\,\,\,\,\,\,t\in \varGamma \backslash \left\{ {{a_1},{a_2}} \right\}, $$ where Φ ± ( t ) are the limit values of Φ( z ) at a point t ∈ Γ \ { a _1 , a _2} from the left and from the right, respectively. We introduce certain distributions with supports on nonrectifiable Arc Γ that generalize the operation of weighted integration along this Arc. Then we consider boundary behavior of the Cauchy transforms of these distributions, i.e., their convolutions with (2 πiz )^ − 1. As a result, we obtain a description of solutions of the Riemann boundary-value problem in terms of a new version of the metric dimension of the Arc Γ, the so-called approximation dimension. It characterizes the rate of best approximation of Γ by polygonal lines.
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The Szegö function on a non-rectifiable Arc
Russian Mathematics, 2012Co-Authors: B. A. Kats, D. B. KatsAbstract:Let Γ be a simple Jordan Arc in the complex plane. The Szegö function, by definition, is a holomorphic in ℂ \ Γ function with a prescribed product of its boundary values on Γ. The problem of finding the Segö function in the case of piecewise smooth Γ was solved earlier. In this paper we study this problem for non-rectifiable Arcs. The solution relies on properties of the Cauchy transform of certain distributions with the support on Γ.
