Potentials on a Compact Riemann Surface

Чирка, Евгений Михайлович

doi:10.1134/s0081543818040211

Cited by 24 publications

(18 citation statements)

References 8 publications

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“…The existence of bipolar Green functions on any compact Riemann surface (which is well known), and all their properties we need are proved in [5] (see also [8]). The functions g( q, p; z) are still defined up to an additive constant.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…. , m, are connected, the values of f on all Nuttall sheets, except the 'last' R (m) and except of the set π −1 (F ) (see (8)), are constructively reconstructed with the help of the polynomial Hermite-Padé m-system. It is worth pointing out that the sheet D is not related in any way to the Nuttall sheets of R (j) .…”

Section: Statements Of the Main Resultsmentioning

confidence: 99%

“…The function ϕ on S is called δsubharmonic if it can be locally represented as the difference of two subharmonic functions. We denote by δ-sh(S) the space of all δ-subharmonic functions on S. It is well known (see, for example, [8]) that δ-sh(S) consists precisely of the potentials of all neutral signed measures on S and δ-sh(S) ∈ L p (S) for any p ∈ [1, ∞). To get rid of the ambiguity of the operator (dd c ) −1 on the space of neutral signed measures, which is related to the possibility of adding a constant, one fixes a continuous linear functional φ, say, on the space L 1 (S) satisfying the condition φ(1) = 0, and instead of the space δ-sh(S) considers the space Pot φ (S) := {v ∈ δ-sh(S) : φ(v) = 0}.…”

Section: Auxiliary Resultsmentioning

confidence: 99%

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Hermite-Padé approximants for meromorphic functions on a compact Riemann surface

Komlov¹,

Palvelev²,

Suetin³

et al. 2017

Russ. Math. Surv.

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For an arbitrary tuple of m + 1 germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Padé m-system (of order n, n ∈ N), which consists of m tuples of polynomials; these tuples, which are indexed by a natural number k ∈ [1, . . . , m], are called the kth polynomials of the Hermite-Padé m-system. We study the weak asymptotics of the polynomials of the Hermite-Padé m-system constructed at the point ∞ from the tuple of germs [1, f 1,∞ , . . . , f m,∞ ] of the functions 1, f 1 , . . . , f m that are meromorphic on some (m + 1)-sheeted branched covering π : R → C of the Riemann sphere C of a compact Riemann surface R. In particular, under some additional condition on π, we find the limit distribution of the zeros and the asymptotics of the ratios of the kth polynomials for all k ∈ [1, . . . , m]. It turns out that in the case, where f j = f j for some meromorphic function f on R, the ratios of some kth polynomials of such Hermite-Padé m-system converge to the sum of the values of the function f on the first k sheets of the Nuttall partition of the Riemann surface R into sheets. Contents 1 Introduction 2 Statements of the main results 3 Reconstruction of the values of a function meromorphic on R from its germ via the polynomial Hermite-Padé m-system 4 The Riemann surface R [k] and the definition of the kth polynomials of the Hermite-Padé m-system in terms of this surface 5 Proofs of Theorems 1 and 2 6 The condition of connectedness of the Riemann surface R [k]

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Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Statements Of the Main Resultsmentioning

confidence: 99%

Section: Auxiliary Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Hermite-Padé approximants for meromorphic functions on a compact Riemann surface

Komlov¹,

Palvelev²,

Suetin³

et al. 2017

Russ. Math. Surv.

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“…Then according to the descendence principle (see [5, Chapter I, § 3, Theorem 1.3], [10], [2] and [3]), we have that…”

Section: For Each ρ ∈ (1 R) We Denote By γmentioning

confidence: 99%

A direct proof of Stahl's theorem for a generic class of algebraic functions

Suetin¹

2021

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 Padé polynomials and convergence in capacity of diagonal Padé 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 particular that all branch points of the multi-sheeted Riemann surface of the algebraic function are of the first order (i.e., we assume the surface is such that all branch points are of square root type).We do not use the relations of orthogonality at all. The proof is based on the maximum principle only.Bibliography: [13] titles.

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“…Finally we notice that the asymptotic properties of Hermite-Padé polynomials are still of unabated interest (see [1], [5], [22], [34], [23], [34], [4] and the bibliography therein). An accessible presentation of the potential theory on Riemann surface is given in the papers [9]- [11]. β n means that for each compact set K ⊂ G and n = 1, 2, .…”

mentioning

confidence: 99%

Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials

Suetin

2021

Preprint

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In the paper, we discuss how it would be possible to succeed in Stahl's novel approach, 1987Stahl's novel approach, -1988, to explore Hermite-Padé polynomials based on Riemann surface properties.In particular, we explore the limit zero distribution of type I Hermite-Padé polynomials Qn,0, Qn,1, Qn,2, deg Qn,j n, for a collection of three analytic elements [1, f∞, f 2 ∞ ]. The element f∞ is an element of a function f from the class C(z, w) where w is supposed to be from the class Z ±1/2 ([−1, 1]) of multivalued analytic functions generated by the inverse Zhukovskii function with the exponents from the set {±1/2}. The Riemann surface corresponding to f ∈ C(z, w) is a four-sheeted Riemann surface R4(w) and all branch points of f are of the first order (i.e., all branch points are of square root type).It is proved that there exits the similar limit zero distribution for all three polynomials Qn,j. The answer is done in terms of Nuttall's condenser which was introduced by E. Rakhmanov and the author in 2013. The corresponding limit measure is supported on the second plate of Nuttall's condenser which coincides with the projection of the boundary between the first and the second sheets of the three-sheeted Riemann surface R3(w∞) associated by Nuttall with the element w∞. The limit measure is a unique solution of a mixed Green-logarithmic theoretical-potential equilibrium problem. In general the surface R3(w∞) can be of any genus.Since the algebraic function f ∈ C(z, w) is of fourth order and we consider the triple of the analytic elements [1, f∞, f 2 ∞ ] but not the quadruple [1, f∞, f 2 ∞ , f 3 ∞ ] ones, the result is new and does not follow from the known results.As in previous paper arXiv: 2108.00339 and following to Stahl's ideas, 1987-1988, we do not use the orthogonality relations at all. The proof is based on the maximum principle only.Bibliography: [45] titles.

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Potentials on a Compact Riemann Surface

Cited by 24 publications

References 8 publications

Hermite-Padé approximants for meromorphic functions on a compact Riemann surface

Hermite-Padé approximants for meromorphic functions on a compact Riemann surface

A direct proof of Stahl's theorem for a generic class of algebraic functions

Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials

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