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Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants
Sbornik. Mathematics, Tome 209 (2018) no. 3, pp. 432-448 Cet article a éte moissonné depuis la source Math-Net.Ru

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We introduce the concept of an exact interpolation index $n$ associated with a function $f$ and open set $\mathscr{L}$: all rational interpolants ${R=p/q}$ of type $(n,n)$ to $f$, with interpolation points in $\mathscr{L}$, interpolate exactly in the sense that $fq-p$ has exactly $2n+1$ zeros in $\mathscr{L}$. We show that in the absence of exact interpolation, there are interpolants with interpolation points in $\mathscr{L}$ and spurious poles. Conversely, for sequences of integers that are associated with exact interpolation to an entire function, there is at least a subsequence with no spurious poles, and consequently, there is uniform convergence. Bibliography: 22 titles.
Keywords: spurious poles.
Mots-clés : Padé approximation, multipoint Padé approximants 
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D. S. Lubinsky. Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants. Sbornik. Mathematics, Tome 209 (2018) no. 3, pp. 432-448. http://geodesic.mathdoc.fr/item/SM_2018_209_3_a6/

[1] G. A. Baker, Jr., P. Graves-Morris, Padé approximants, v. 59, Encyclopedia Math. Appl., 2nd ed., Addison-Wesley Publishing Co., Reading, MA, 1996, xiv+746 pp. | DOI | MR | MR | Zbl | Zbl

[2] A. I. Aptekarev, M. L. Yattselev, “Padé approximants for functions with branch points – strong asymptotics of Nuttall–Stahl polynomials”, Acta Math., 215:2 (2015), 217–280 | DOI | MR | Zbl

[3] B. Beckermann, G. Labahn, A. C. Matos, “On rational functions without Froissart doublets”, Numer. Math., first online 2017, 1–19 ; arXiv: 1605.00506 | DOI

[4] V. I. Buslaev, “On the Baker–Gammel–Wills conjecture in the theory of Padé approximants”, Sb. Math., 193:6 (2002), 811–823 | DOI | DOI | MR | Zbl

[5] V. I. Buslaev, “Convergence of the Rogers–Ramanujan continued fraction”, Sb. Math., 194:6 (2003), 833–856 | DOI | DOI | MR | Zbl

[6] V. I. Buslaev, A. A. Gonchar, S. P. Suetin, “On convergence of subsequences of the $m$th row of a Padé table”, Math. USSR-Sb., 48:2 (1984), 535–540 | DOI | MR | Zbl

[7] D. Coman, E. A. Poletsky, “Overinterpolation”, J. Math. Anal. Appl., 335:1 (2007), 184–197 | DOI | MR | Zbl

[8] J. Gilewicz, Y. Kryakin, “Froissart doublets in Padé approximation in the case of polynomial noise”, J. Comput. Appl. Math., 153:1-2 (2003), 235–242 | DOI | MR | Zbl

[9] A. A. Gončar, “Estimates of the growth of rational functions and some of their applications”, Math. USSR-Sb., 1:3 (1967), 445–456 | DOI | MR | Zbl

[10] A. A. Gonchar, “On uniform convergence of diagonal Padé approximants”, Math. USSR-Sb., 46:4 (1983), 539–559 | DOI | MR | Zbl

[11] A. A. Gonchar, L. D. Grigoryan, “On estimates of the norm of the holomorphic component of a meromorphic function”, Math. USSR-Sb., 28:4 (1976), 571–575 | DOI | MR | Zbl

[12] L. D. Grigorjan, “Estimates of the norm of the holomorphic components of functions meromorphic in domains with a smooth boundary”, Math. USSR-Sb., 29:1 (1976), 139–146 | DOI | MR | Zbl

[13] D. V. Khristoforov, “On the phenomenon of spurious interpolation of elliptic functions by diagonal Padé approximants”, Math. Notes, 87:4 (2010), 564–574 | DOI | DOI | MR | Zbl

[14] D. S. Lubinsky, “Distribution of poles of diagonal rational approximants to functions of fast rational approximability”, Constr. Approx., 7 (1991), 501–519 | DOI | MR | Zbl

[15] D. S. Lubinsky, “Spurious poles in diagonal rational approximation”, Progress in approximation theory (Tampa, FL, 1990), Springer Ser. Comput. Math., 19, Springer, New York, 1992, 191–213 | DOI | MR | Zbl

[16] D. S. Lubinsky, “Rogers–Ramanujan and the Baker–Gammel–Wills (Padé) conjecture”, Ann. of Math. (2), 157:3 (2003), 847–889 | DOI | MR | Zbl

[17] E. A. Rakhmanov, “On the convergence of Padé approximants in classes of holomorphic functions”, Math. USSR-Sb., 40:2 (1981), 149–155 | DOI | MR | Zbl

[18] T. Ransford, Potential theory in the complex plane, London Math. Soc. Stud. Texts, 28, Cambridge Univ. Press, Cambridge, 1995, x+232 pp. | DOI | MR | Zbl

[19] H. Stahl, “Spurious poles in Padé approximation”, J. Comput. Appl. Math., 99:1-2 (1998), 511–527 | DOI | MR | Zbl

[20] S. P. Suetin, “Distribution of the zeros of Padé polynomials and analytic continuation”, Russian Math. Surveys, 70:5 (2015), 901–951 | DOI | DOI | MR | Zbl

[21] F. Wielonsky, “Riemann–Hilbert analysis and uniform convergence of rational interpolants to the exponential function”, J. Approx. Theory, 131:1 (2004), 100–148 | DOI | MR | Zbl

[22] M. Yattselev, Meromorphic approximation: symmetric contours and wandering poles, manuscript, 2012, 19 pp. https://math.iupui.edu/~maxyatts/research.html