S. P. Suetin

S. P. Suetin
Russian Academy of Sciences | RAS · Steklov Mathematical Institute

PhD (Russian Candidate), DrSci

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133
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Publications

Publications (133)
Preprint
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Let $[f_0,\dots,f_m]$ be a tuple of series in nonnegative powers of $1/z$, $f_j(\infty)\neq0$. It is supposed that the tuple is in "general position". We give a construction of type I and type II Hermite--Pad\'e polynomials to the given tuple of degrees $\leq{n}$ and $\leq{mn}$ respectively and the corresponding $(m+1)$-multi-indexes with the follo...
Preprint
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In the paper, we discuss how it would be possible to succeed in Stahl's novel approach, 1987--1988, to explore Hermite--Pad\'e polynomials based on Riemann surface properties. In particular, we explore the limit zero distribution of type I Hermite--Pad\'e polynomials $Q_{n,0},Q_{n,1},Q_{n,2}$, $\operatorname{deg}{Q_{n,j}}\leq{n}$, for a collection...
Article
We propose and justify an algorithm for producing Hermite- Padé polynomials of type I for an arbitrary tuple of formal power series , , about the point ( ) under the assumption that the series have a certain (‘general position’) nondegeneracy property. This algorithm is a straightforward extension of the classical Viskovatov algorithm for construct...
Preprint
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Under the assumption of the existence of Stahl's $S$-compact set we give a short proof of the limit zeros distribution of Pad\'e polynomials and convergence in capacity of diagonal Pad\'e approximants for a generic class of algebraic functions. The proof is direct but not from the opposite as Stahl's original proof is. The generic class means in pa...
Preprint
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We discuss the relation between the linear Tschebyshev-Pad\'e approximations to analytic function $f$ and the diagonal type I Hermite-Pad\'e polynomials for the tuple of functions $[1,f_1,f_2]$ where the pair of functions $f_1,f_2$ forms certain Nikishin system. An approach is proposed of how to extend the seminal Stahl's Theory for Pad\'e approxim...
Preprint
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In the paper we represent two examples which are based on the properties of discrete measures. In the first part of the paper we prove that for each probability measure $\mu$, $\operatorname{supp}{\mu}=[-1,1]$, which logarithmic potential is a continuous function on $[-1,1]$ there exists a (discrete) measure $\sigma=\sigma(\mu)$, $\operatorname{sup...
Preprint
The structure of a Nuttall partition into sheets of some class of four-sheeted Riemann surfaces is studied. The corresponding class of multivalued analytic functions is a special class of algebraic functions of fourth order generated by the function inverse to the Zhukovskii function. We show that in this class of four-sheeted Riemann surfaces, the...
Preprint
We propose an algorithm for producing Hermite-Pad\'e polynomials of type I for an arbitrary tuple of $m+1$ formal power series $[f_0,\dots,f_m]$, $m\geq1$, about $z=0$ ($f_j\in{\mathbb C}[[z]]$) under the assumption that the series have a certain (`general position') nondegeneracy property. This algorithm is a straightforward extension of the class...
Article
Using the scalar equilibrium problem posed on the two-sheeted Riemann surface, we prove the existence of a limit distribution of the zeros of Hermite-Padé polynomials of type II for a pair of functions forming a Nikishin system. We discuss the relation of the results obtained here to some results of H. Stahl (1988) and present results of numerical...
Preprint
The existence of the limit distribution of the zeros of Hermite-Pad\'e polynomials of type II for a pair of functions forming a Nikishin system is proved using the scalar equilibrium problem posed on the two-sheeted Riemann surface. The relation of the results obtained here to some results of H. Stahl (1988) is discussed. Results of numerical exper...
Article
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We prove the equivalence of the vector and scalar equilibrium problems which arise naturally in the study of the limit zeros distribution of type I Hermite-Padé polynomials for a pair of functions forming a Nikishin system. Bibliography: 22 titles.
Preprint
Full-text available
We prove the equivalence of the vector and scalar equilibrium problems which arise naturally in the study of the limit zeros distribution of type I Hermite--Pad\'e polynomials for a pair of functions forming a Nikishin system.
Article
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An example of a Markov function f = const + \(\hat \sigma \) such that the three functions f, f², and f³ constitute a Nikishin systemis given. It is conjectured that there exists aMarkov function f such that, for each n ∈ N, the system of f, f²,..., fⁿ is a Nikishin system.
Preprint
Full-text available
The paper puts forward an example of a~Markov function $f=\operatorname{const}+\widehat{\sigma}$ such that the three functions $f,f^2$ and $f^3$ form a Nikishin system. A conjecture is proposed that there exists a~Markov function $f$ such that, for each $n\in\mathbb N$, the system $f,f^2,\dots,f^n$ constitutes a~Nikishin system. Bibliography:~20~ti...
Preprint
Full-text available
We discuss a new approach to realization of the well-known Weierstrass's programme on efficient continuation of an analytic element corresponding to a~multivalued analytic function with finite number of branch points. Our approach is based on the use of Hermite--Pad\'e polynomials.
Preprint
Full-text available
A new approach to the problem of the zero distribution of Hermite-Pad\'e polynomials of type I for a pair of functions $f_1,f_2$ forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field, which is posed on a two-sheeted Riemann surface.
Article
Full-text available
A new approach to the problem of the zero distribution of type I Hermite—Padé polynomials for a pair of functions f1, f2 forming a Nikishin system is discussed. Unlike the traditional vector approach, we give an answer in terms of a scalar equilibrium problem with harmonic external field which is posed on a two-sheeted Riemann surface.
Article
An analog of Pólya’s theorem on the estimate of the transfinite diameter for a class of multivalued analytic functions with finitely many branch points and of the corresponding class of admissible compact sets located on the associated (with this function) two-sheeted Stahl–Riemann surface is obtained.
Article
Full-text available
Part II of this paper elaborates on the unique capability of the proposed power flow analysis framework to obtain the true solution corresponding to the stable operating point of a network. It explains the significance of obtaining the true solution for an accurate assessment of the voltage collapse margin. This feature distinguishes the framework...
Article
Full-text available
Part I of this paper embeds the AC power flow problem with voltage control and exponential load model in the complex plane. Modeling the action of network controllers that regulate the magnitude of voltage phasors is a challenging task in the complex plane as it has to preserve the framework of holomorphicity for obtention of these complex variable...
Article
Full-text available
In the paper, we propose two new conjectures about the convergence of Hermite Approximants of multivalued analytic functions of Laguerre class ${\mathscr L}$. The conjectures are based in part on the numerical experiments, made recently by the authors in [26] and [27].
Chapter
Type I Hermite–Padé polynomials for a set of functions f 0 , f 1 ,. .. , f s at infinity, Q n,0 , Q n,1 ,. .. , Q n,s , is defined by the asymptotic condition R n (z) := Q n,0 f 0 +Q n,1 f 1 +Q n,2 f 2 +· · ·+Q n,s f s (z) = O 1 z sn+s , z → ∞, with the degree of all Q n,k ≤ n. We describe an approach for finding the asymptotic zero distribution of...
Article
Full-text available
The problem of analytic continuation of a multivalued analytic function with finitely many branch points on the Riemann sphere is discussed. The focus is on Pad'e approximants: classical (one-point) Pad'e approximants, multipoint Pad'e approximants, and Hermite-Pad'e approximants. The main result is a theorem on the distribution of zeros and the co...
Article
Full-text available
We study two potential-theory equilibrium problems that arise naturally in the theory of the limit distribution of zeros of the Hermite–Padé polynomials. We analyze the relationship between these problems and prove that the equilibrium measure for one of the problems is the balayage of the equilibrium measure for the other problem.
Article
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In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Pad\'e polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider t...
Article
For an interval $E=[a,b]$ on the real line, let $\mu$ be either the equilibrium measure, or the normalized Lebesgue measure of $E$, and let $V^{\mu}$ denote the associated logarithmic potential. In the present paper, we construct a function $f$ which is analytic on $E$ and possesses four branch points of second order outside of $E$ such that the fa...
Article
Full-text available
This paper proposes a method to embed the AC power flow problem with voltage magnitude constraints in the complex plane. Modeling the action of network controllers that regulate the magnitude of voltage phasors is a challenging task in the complex plane as it has to preserve the framework of holomorphicity for obtention of these complex variables w...
Article
Full-text available
Type I Hermite--Pad\'e polynomials for a set of functions $f_0, f_1, ..., f_s$ at infinity, $Q_{n,0}$, $Q_{n,1}$, ..., $Q_{n,s}$, is defined by the asymptotic condition $$ R_n(z):=\bigl(Q_{n,0}f_0+Q_{n,1}f_1+Q_{n,2}f_2+...+Q_{n,s}f_s\bigr)(z) =\mathcal O (\frac1{z^{s n+s}}), \quad z\to\infty, $$ with the degree of all $Q_{n,k}\leq n$. We describe a...
Article
Full-text available
We introduce and analyze some numerical results obtained by the authors experimentally. These experiments are related to the well known problem about the distribution of the zeros of Hermite--Pad\'e polynomials for a collection of three functions $[f_0 \equiv 1,f_1,f_2]$. The numerical results refer to two cases: a pair of functions $f_1,f_2$ forms...
Article
We obtain Nuttall's integral equation provided that the corresponding complex-valued function σ(x) does not vanish and belongs to the Dini-Lipschitz class. Using this equation, we obtain a complex analogue of Bernshtein's classical asymptotic formulae for polynomials orthogonal on the closed unit interval Δ = [- 1,1] with respect to a complex-value...
Article
Full-text available
The well-known approach of J. Nuttall to the derivation of strong asymptotic formulas for the Hermite-Padé polynomials for a set of m multivalued functions is based on the conjecture that there exists a canonical (in the sense of decomposition into sheets) m-sheeted Riemann surface possessing certain properties. In this paper, for m = 3, we introdu...
Article
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The paper presents some heuristic results about the distribution of zeros of Hermite-Pade polynomials of first kind for the case of three functions $1,f,f^2$, where $f$ has the form $f(z): = \prod\limits_ {j = 1 } ^3 (z-a_j) ^ {\alpha_j} $, $\alpha_j \in \mathbb C\setminus \mathbb Z $, $ \sum \limits_ {j = 1 } ^ 3 \alpha_j = 0 $, $ f (\infty) = 1 $...
Article
Full-text available
The variation of equilibrium energy is analyzed for three different functionals that naturally arise in solving a number of problems in the theory of constructive rational approximation of multivalued analytic functions. The variational approach is based on the relationship between the variation of the equilibrium energy and the equilibrium measure...
Article
The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem 1) that the zeros have a limiting dis...
Article
This paper gives a proof of the theorem announced by the authors in the preceding paper with the same title. The theorem states that asymptotically the behaviour of the polynomials which are orthonormal with respect to the varying weight coincides with the asymptotic behaviour of the Nuttall psi-function, which solves a special boundary-value probl...
Article
Padé-Chebyshev approximants are considered for multivalued analytic functions that are real-valued on the unit interval . The focus is mainly on non-linear Padé-Chebyshev approximants. For such rational approximations an analogue is found of Stahl's theorem on convergence in capacity of the Padé approximants in the maximal domain of holomorphy of t...
Article
We obtain a sufficiently general variational formula for a Green’s function, which, in particular, implies the classic variational formulas of Hadamard and Schiffer.
Article
We obtain a strong asymptotic formula for the leading coefficient α n (n) of a degree n polynomial q n (z;n) orthonormal on a system of intervals on the real line with respect to a varying weight. The weight depends on n as e -2nQ(x) , where Q(x) is a polynomial and corresponds to the “hard-edge-case”. The formula in Theorem 1 is quite similar to W...
Article
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This paper studies a variation of the equilibrium energy for a certain fairly general functional which appears naturally in the solution of many rational approximation problems of multi-valued analytic functions. The main result of this work states that for the energy functional under consideration and a certain class of admissible compact sets, re...
Article
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This is a survey of results constituting the foundations of the modern convergence theory of Padé approximants.Bibliography: 204 titles.
Article
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In 1986 J. Nuttall published in Constructive Approximation the paper "Asymptotics of generalized Jacobi polynomials", where with his usual insight he studied the behavior of the denominators ("generalized Jacobi polynomials") and the remainders of the Pade approximants to a special class of algebraic functions with 3 branch points. 25 years later w...
Article
The paper is devoted to the asymptotic properties of diagonal Padé approximants for Markov-type meromorphic functions. The main result is strong asymptotic formulas for the denominators of diagonal Padé approximants for Markov-type meromorphic functions f = $ \hat \sigma $ \hat \sigma + r under additional constraints on the measure σ (r is a rati...
Article
Properties of Jacobi operators generated by Markov functions are studied. The main results refer to the case where the support of the corresponding spectral measure µ consists of several intervals of the real line. In this class of operators, a comparative asymptotic formula for two solutions of the corresponding difference equation, polynomials or...
Article
Full-text available
Some new results on the convergence of nonlinear diagonal Pad\'e--Chebyshev approximations to multivalued analytic function given on the segment $[-1,1]$, are proved. We show that these approximations converge to the given function in the "maxmimal" domain of its meromorphity and that the boundary of this maxmimal domain is an $S$-curve. The result...
Article
Full-text available
We announce some new results on the convergence of Chebyshev--Pad\'e approximations to real-valued algebraic function given on the segment $[-1,1]$. The rate of convergence on the segment and in the corresponding maximal domain of meromorphity of a given function is charactirized in terms of a theoretical potential equilibrium problem.
Article
We present examples of two functions that are analytic on the interval [−1, 1] and satisfy the condition that, for any n = 2, 3, …, the first of them does not have nonlinear Padé-Chebyshev approximations of type (n, 2) and the second function does not have nonlinear Padé-Chebyshev approximations of type (n, n) (i.e., does not have diagonal approxim...
Article
For polynomials orthogonal with respect to a complex-valued weight on the closed interval a strong asymptotic formula in a neighbourhood of is obtained. In particular, for the 'trigonometric' weight , , this formula yields a description of the asymptotic behaviour of each of the zeros of the th orthogonal polynomial as . This strong asymptotic form...
Article
The uniform convergence of Padé diagonal approximants is studied for functions in some class that is a natural generalization of hyperelliptic functions. The study is based on Nuttall's approach, which consists in the analysis of a certain Riemann boundary-value problem on the corresponding hyperelliptic Riemann surface. In terms of the solution of...
Article
A non-linear system of differential equations ("generalized Dubrovin system") is obtained to describe the behaviour of the zeros of polynomials orthogonal on several intervals that lie in lacunae between the intervals. The same system is shown to describe the dynamical behaviour of zeros of this kind for more general orthogonal polynomials: the den...
Article
The class of Jacobi operators generated by unit Borel measures with support formed by finitely many intervals of the real line and finitely many points in lying outside the convex hull of these intervals is investigated. An asymptotic formula for the diagonal Green's function in this class is obtained as well as the trace formulae for sequences cor...
Article
Dumas's classical theorem on the behaviour of the Chebyshëv continued fraction corresponding to an elliptic function holomorphic at is extended to a fairly general class of elliptic functions. The behaviour of the Chebyshëv continued fractions corresponding to functions in that class is characterized in terms relating to the mutual position of the...
Article
We establish analogs of the theorem of Montessus de Ballore for rational approximations (of the type of the Padé approximants) to series of orthogonal polynomials and of Faber polynomials. Bibliography: 11 titles.
Article
The question is considered of the existence of a subsequence of the th row of the Padé table of a function that converges uniformly on compact subsets of the disk ( the radius of -meromorphy of ) which do not contain poles of this function.Bibliography: 8 titles.
Article
The author considers the connection between the asymptotic behavior of the poles of the mth row of the Padé table of a function given by a power series and the singular points of this function on the boundary of its mth disc of meromorphicity.Bibliography: 9 titles.
Article
The connection is considered between the asymptotic behavior of the poles of the mth row of the Padé table of a function given by a Taylor series and the singular points of this function on the boundary of its disk of m-meromorphy.Bibliography: 6 titles.

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