Geodesic
Exact interpolation, spurious poles, and uniform convergence of multipoint Pad\'e approximants
Sbornik. Mathematics, Tome 209 (2018) no. 3, pp. 432-448

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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: Padé approximation, multipoint Padé approximants, spurious poles. 
@article{SM_2018_209_3_a6,
     author = {D. S. Lubinsky},
     title = {Exact interpolation, spurious poles, and uniform convergence of multipoint {Pad\'e} approximants},
     journal = {Sbornik. Mathematics},
     pages = {432--448},
     publisher = {mathdoc},
     volume = {209},
     number = {3},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_3_a6/}
}
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D. S. Lubinsky. Exact interpolation, spurious poles, and uniform convergence of multipoint Pad\'e approximants. Sbornik. Mathematics, Tome 209 (2018) no. 3, pp. 432-448. http://geodesic.mathdoc.fr/item/SM_2018_209_3_a6/