On an Example of the Nikishin System

Suetin, Sergey Pavlovich

doi:10.1134/s0001434618110342

Cited by 16 publications

(16 citation statements)

References 16 publications

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“…All that was said above, was discussed in [31] on an example of multivalued functions f from the class introduced earlier by the author of the present paper (see [29]) -this being the class of functions of the form…”

Section: Introduction and The Statement Of The Main Resultsmentioning

confidence: 97%

Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

Ikonomov¹,

Suetin²

2021

Preprint

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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 boundary between the second and third sheets of the Nuttall partition of the Riemann surface, is completely characterized in terms of an extremal problem posed on the two-sheeted Riemann surface of the function w defined by the equation w 2 = z 2 − 1. In particular, we show that in this class of functions the boundary between the second and third sheets does not intersect both the boundary between the first and second sheets and the boundary between the third and fourth sheets.Bibliography: [34] titles.

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Section: Introduction and The Statement Of The Main Resultsmentioning

confidence: 97%

Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

Ikonomov¹,

Suetin²

2021

Preprint

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“…This Program is based on a generalization of the classical Viskovatov algorithm [22]. Here the zeros of Padé polynomials P n,0 (z), P n,1 (z) of order n = 100 for the function f given by (42) are plotted (blue points for P n,0 and red points for P n,1 ). These zeros simulate the segment ∆ = [−1, 1] which is the Stahl compact set for f .…”

Section: One Numerical Examplementioning

confidence: 99%

“…Let now take into account all that was said before about the connection between type I Hermite-Padé polynomials and Tschebyshev-Padé approximations and about max-min problem which was conjectured to describe the limit zeros distribution of Hermite-Padé polynomials. Then based on that information we might suppose that the limit zero distribution of the corresponding HP-polynomials for both tuples [1, f, f 2 ] and [1, 1/(z 2 − 1) 1/2 , f ], where f is from (42), should be just the same. The numerical results represented on the Figure 1 and Figure 2 are in a good accordance with that statement.…”

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confidence: 99%

Tschebyshev-Padé approximations for multivalued functions

Rakhmanov¹,

Suetin²

2021

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We discuss the relation between the linear Tschebyshev-Padé approximations to analytic function f and the diagonal type I Hermite-Padé 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 Padé approximations for multivalued analytic functions to the Tschebyshev-Padé approximations. The approach is based on the relation between Tschebyshev-Padé approximations and Hermite-Padé polynomials and also on a connection of Hermite-Padé polynomials and multipoint Padé approximants.Bibliography: [47] titles. Contents 1. Tschebyshev-Padé Approximations 1 2. Padé approximants at infinity 4 3. Hermite-Padé polynomials. 6 4. General interpolation problem 8 5. GRS theorem 10 6. Conjecture on Tschebyshev-Padé approximants 13 7. One numerical example 14 References 18

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“…We emphasize once again that it is supposed that the parameters A j and B j satisfy the above conditions. Notice that for m = 1 the systems w, w 2 and w, w 2 , w 3 form Nikishin systems (see [40]).…”

mentioning

confidence: 99%

“…Note that since f ∈ C(z, w) in general is a complex-valued function on the real line, then the powerful methods developed in the papers [13], [14] and [2] are not applicable to prove the relations ( 38)- (40) (in this connection see also [27], [5], [32], [34], [4]). If (39) would be proven, then from that and (36) it would be followed that on the base of type I and type II HP-polynomials a multi-valued analytic function f ∈ C(z, w) is constructively recovered on the two sheets of the RS R 3 (w ∞ ).…”

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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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On an Example of the Nikishin System

Cited by 16 publications

References 16 publications

Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

Structure of the Nuttall partition for some class of four-sheeted Riemann surfaces

Tschebyshev-Padé approximations for multivalued functions

Maximum Principle and Asymptotic Properties of Hermite--Padé Polynomials

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