The Experts below are selected from a list of 10530 Experts worldwide ranked by ideXlab platform
Zhou X. - One of the best experts on this subject based on the ideXlab platform.
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Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Pad\'e Approximants and Asymptotics
2019Co-Authors: Vartanian A., Zhou X.Abstract:Given $K$ arbitrary poles, which are neither necessarily distinct nor bounded, on the Extended Real Line, a corresponding ordered base of rational functions orthogonal with respect to varying exponential weights is constructed: this gives rise to a $K$-fold family of orthogonal rational functions (ORFs). The ORF problem is characterised as a family of $K$ matrix Riemann-Hilbert problems (RHPs) on the Extended Real Line, and a corresponding family of $K$ energy minimisation (variational) problems containing external fields with singular points is formulated, and the existence, uniqueness, and regularity properties of the associated family of equilibrium measures is established. The family of $K$ equilibrium measures is used to derive a family of $K$ model matrix RHPs on the Extended Real Line that are amenable to asymptotic analysis via the Deift-Zhou non-Linear steepest-descent method: this is used to derive uniform asymptotics, in a certain double-scaling limit, of the ORFs and their leading coefficients, as well as related, important objects, in the entire complex plane. A family of $K$ multi-point Pad\'e approximants (MPAs) for the Markov-Stieltjes transform is also presented, and uniform asymptotics, in a certain double-scaling limit, are obtained for the corresponding MPAs and their associated errors in approximation (MPA error terms) in the entire complex plane.Comment: 343 pages, 15 figure
Laiyi Zhu - One of the best experts on this subject based on the ideXlab platform.
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the convergence properties of orthogonal rational functions on the Extended Real Line and analytic on the upper half plane
International Journal of Wavelets Multiresolution and Information Processing, 2020Co-Authors: Laiyi ZhuAbstract:We study the orthogonal rational functions with a given sequence of poles on the half plane. We give the weak-star convergence results for the rational measures which relate to the Poisson kernel a...
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orthogonal rational functions on the Extended Real Line and analytic on the upper half plane
Rocky Mountain Journal of Mathematics, 2018Co-Authors: Laiyi ZhuAbstract:Let $\{\alpha _k\}_{k=1}^\infty$ be an arbitrary sequence of complex numbers in the upper half plane. We generalize the orthogonal rational functions $\phi _n$ based upon those points and obtain the Nevanlinna measure, together with the Riesz and Poisson kernels, for Caratheodory functions $F(z)$ on the upper half plane. Then, we study the relation between ORFs and their functions of the second kind as well as their interpolation properties. Further, by using a Linear transformation, we generate a new class of rational functions and state the necessary conditions for guaranteeing their orthogonality.
X Zhou - One of the best experts on this subject based on the ideXlab platform.
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riemann hilbert characterisation of rational functions with a general distribution of poles on the Extended Real Line orthogonal with respect to varying exponential weights multi point pad e approximants and asymptotics
arXiv: Classical Analysis and ODEs, 2019Co-Authors: A Vartanian, X ZhouAbstract:Given $K$ arbitrary poles, which are neither necessarily distinct nor bounded, on the Extended Real Line, a corresponding ordered base of rational functions orthogonal with respect to varying exponential weights is constructed: this gives rise to a $K$-fold family of orthogonal rational functions (ORFs). The ORF problem is characterised as a family of $K$ matrix Riemann-Hilbert problems (RHPs) on the Extended Real Line, and a corresponding family of $K$ energy minimisation (variational) problems containing external fields with singular points is formulated, and the existence, uniqueness, and regularity properties of the associated family of equilibrium measures is established. The family of $K$ equilibrium measures is used to derive a family of $K$ model matrix RHPs on the Extended Real Line that are amenable to asymptotic analysis via the Deift-Zhou non-Linear steepest-descent method: this is used to derive uniform asymptotics, in a certain double-scaling limit, of the ORFs and their leading coefficients, as well as related, important objects, in the entire complex plane. A family of $K$ multi-point Pade approximants (MPAs) for the Markov-Stieltjes transform is also presented, and uniform asymptotics, in a certain double-scaling limit, are obtained for the corresponding MPAs and their associated errors in approximation (MPA error terms) in the entire complex plane.
Vartanian A. - One of the best experts on this subject based on the ideXlab platform.
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Riemann-Hilbert Characterisation of Rational Functions with a General Distribution of Poles on the Extended Real Line Orthogonal with Respect to Varying Exponential Weights: Multi-Point Pad\'e Approximants and Asymptotics
2019Co-Authors: Vartanian A., Zhou X.Abstract:Given $K$ arbitrary poles, which are neither necessarily distinct nor bounded, on the Extended Real Line, a corresponding ordered base of rational functions orthogonal with respect to varying exponential weights is constructed: this gives rise to a $K$-fold family of orthogonal rational functions (ORFs). The ORF problem is characterised as a family of $K$ matrix Riemann-Hilbert problems (RHPs) on the Extended Real Line, and a corresponding family of $K$ energy minimisation (variational) problems containing external fields with singular points is formulated, and the existence, uniqueness, and regularity properties of the associated family of equilibrium measures is established. The family of $K$ equilibrium measures is used to derive a family of $K$ model matrix RHPs on the Extended Real Line that are amenable to asymptotic analysis via the Deift-Zhou non-Linear steepest-descent method: this is used to derive uniform asymptotics, in a certain double-scaling limit, of the ORFs and their leading coefficients, as well as related, important objects, in the entire complex plane. A family of $K$ multi-point Pad\'e approximants (MPAs) for the Markov-Stieltjes transform is also presented, and uniform asymptotics, in a certain double-scaling limit, are obtained for the corresponding MPAs and their associated errors in approximation (MPA error terms) in the entire complex plane.Comment: 343 pages, 15 figure
A Vartanian - One of the best experts on this subject based on the ideXlab platform.
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riemann hilbert characterisation of rational functions with a general distribution of poles on the Extended Real Line orthogonal with respect to varying exponential weights multi point pad e approximants and asymptotics
arXiv: Classical Analysis and ODEs, 2019Co-Authors: A Vartanian, X ZhouAbstract:Given $K$ arbitrary poles, which are neither necessarily distinct nor bounded, on the Extended Real Line, a corresponding ordered base of rational functions orthogonal with respect to varying exponential weights is constructed: this gives rise to a $K$-fold family of orthogonal rational functions (ORFs). The ORF problem is characterised as a family of $K$ matrix Riemann-Hilbert problems (RHPs) on the Extended Real Line, and a corresponding family of $K$ energy minimisation (variational) problems containing external fields with singular points is formulated, and the existence, uniqueness, and regularity properties of the associated family of equilibrium measures is established. The family of $K$ equilibrium measures is used to derive a family of $K$ model matrix RHPs on the Extended Real Line that are amenable to asymptotic analysis via the Deift-Zhou non-Linear steepest-descent method: this is used to derive uniform asymptotics, in a certain double-scaling limit, of the ORFs and their leading coefficients, as well as related, important objects, in the entire complex plane. A family of $K$ multi-point Pade approximants (MPAs) for the Markov-Stieltjes transform is also presented, and uniform asymptotics, in a certain double-scaling limit, are obtained for the corresponding MPAs and their associated errors in approximation (MPA error terms) in the entire complex plane.
