# Rational interpolants with preassigned poles, theoretical aspects

Studia Mathematica (1999)

• Volume: 132, Issue: 1, page 1-14
• ISSN: 0039-3223

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## Abstract

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Let ⨍ be an analytic function on a compact subset K of the complex plane ℂ, and let ${r}_{n}\left(z\right)$ denote the rational function of degree n with poles at the points ${{b}_{ni}}_{i=1}^{n}$ and interpolating ⨍ at the points ${{a}_{ni}}_{i=0}^{n}$. We investigate how these points should be chosen to guarantee the convergence of ${r}_{n}$ to ⨍ as n → ∞ for all functions ⨍ analytic on K. When K has no “holes” (see [8] and [3]), it is possible to choose the poles ${{b}_{ni}}_{i,n}$ without limit points on K. In this paper we study the case of general compact sets K, when such a separation is not always possible. This fact causes changes both in the results and in the methods of proofs. We consider also the case of functions analytic in open domains. It turns out that in our general setting there is no longer a “duality” ([8], Section 8.3, Corollary 2) between the poles and the interpolation points.

## How to cite

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Ambroladze, Amiran, and Wallin, Hans. "Rational interpolants with preassigned poles, theoretical aspects." Studia Mathematica 132.1 (1999): 1-14. <http://eudml.org/doc/216583>.

@article{Ambroladze1999,
abstract = {Let ⨍ be an analytic function on a compact subset K of the complex plane ℂ, and let $r_n(z)$ denote the rational function of degree n with poles at the points $\{b_\{ni\}\}^\{n\}_\{i=1\}$ and interpolating ⨍ at the points $\{a_\{ni\}\}^\{n\}_\{i=0\}$. We investigate how these points should be chosen to guarantee the convergence of $r_n$ to ⨍ as n → ∞ for all functions ⨍ analytic on K. When K has no “holes” (see [8] and [3]), it is possible to choose the poles $\{b_\{ni\}\}_\{i,n\}$ without limit points on K. In this paper we study the case of general compact sets K, when such a separation is not always possible. This fact causes changes both in the results and in the methods of proofs. We consider also the case of functions analytic in open domains. It turns out that in our general setting there is no longer a “duality” ([8], Section 8.3, Corollary 2) between the poles and the interpolation points.},
author = {Ambroladze, Amiran, Wallin, Hans},
journal = {Studia Mathematica},
language = {eng},
number = {1},
pages = {1-14},
title = {Rational interpolants with preassigned poles, theoretical aspects},
url = {http://eudml.org/doc/216583},
volume = {132},
year = {1999},
}

TY - JOUR
AU - Ambroladze, Amiran
AU - Wallin, Hans
TI - Rational interpolants with preassigned poles, theoretical aspects
JO - Studia Mathematica
PY - 1999
VL - 132
IS - 1
SP - 1
EP - 14
AB - Let ⨍ be an analytic function on a compact subset K of the complex plane ℂ, and let $r_n(z)$ denote the rational function of degree n with poles at the points ${b_{ni}}^{n}_{i=1}$ and interpolating ⨍ at the points ${a_{ni}}^{n}_{i=0}$. We investigate how these points should be chosen to guarantee the convergence of $r_n$ to ⨍ as n → ∞ for all functions ⨍ analytic on K. When K has no “holes” (see [8] and [3]), it is possible to choose the poles ${b_{ni}}_{i,n}$ without limit points on K. In this paper we study the case of general compact sets K, when such a separation is not always possible. This fact causes changes both in the results and in the methods of proofs. We consider also the case of functions analytic in open domains. It turns out that in our general setting there is no longer a “duality” ([8], Section 8.3, Corollary 2) between the poles and the interpolation points.
LA - eng
UR - http://eudml.org/doc/216583
ER -

## References

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1. [1] A. Ambroladze and H. Wallin, Convergence of rational interpolants with preassigned poles, J. Approx. Theory 89 (1997), 238-256. Zbl0872.41007
2. [2] A. Ambroladze and H. Wallin, Rational interpolants with preassigned poles, theory and practice, Complex Variables Theory Appl. 34 (1997), 399-413. Zbl0892.30032
3. [3] T. Bagby, Rational interpolation with restricted poles, J. Approx. Theory 7 (1973), 1-7. Zbl0252.30039
4. [4] N. S. Landkof, Foundations of Modern Potential Theory, Grundlehren Math. Wiss. 190, Springer, New York, 1972. Zbl0253.31001
5. [5] T. Ransford, Potential Theory in the Complex Plane, London Math. Soc. Student Texts 28, Cambridge Univ. Press, Cambridge, 1995. Zbl0828.31001
6. [6] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987.
7. [7] H. Stahl and V. Totik, General Orthogonal Polynomials, Encyclopedia Math. Appl., Cambridge Univ. Press, Cambridge, 1992. Zbl0791.33009
8. [8] J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, 4th ed., Amer. Math. Soc. Colloq. Publ. 20, Amer. Math. Soc., Providence, R.I., 1965. Zbl0146.29902
9. [9] A. E. Wegert and L. N. Trefethen, From the Buffon needle problem to the Kreiss matrix theorem, Amer. Math. Monthly 101 (1994), 132-139. Zbl0799.30002

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