Exact interpolation, spurious poles, and uniform convergence of multipoint Padé approximants

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.
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