TITLE:
On the divergence of polynomial interpolation
AUTHORS:
Angel Jorba and Joan Carles Tatjer
Departament de Matematica Aplicada i Analisi,
Universitat de Barcelona,
Gran Via 585, 08007 Barcelona, Spain
E-mails: angel@maia.ub.es, jcarles@maia.ub.es
ABSTRACT:
Consider a triangular interpolation scheme on a piecewise $C^1$ curve
of the complex plane, and let $\Gamma$ be the closure of this
triangular scheme. Given a meromorphic function $f$ with no
singularities on $\Gamma$, we are interested in the region of
convergence of the sequence of interpolating polynomials to the
function $f$. In particular, we focus on the case in which $\Gamma$ is
not fully contained in the interior of the region of convergence
defined by the standard logarithmic potential. Let us call
$\Gamma_{out}$ the subset of $\Gamma$ outside of the convergence
region.
In the paper we show that the sequence of interpolating polynomials,
$\{P_n\}_n$, is divergent on all the points of $\Gamma_{out}$, except
on a set of zero Lebesgue measure. Moreover, the structure of the set
of divergence is also discussed: the subset of values $z$ for which
there exists a partial sequence of $\{P_n(z)\}_n$ that converges to
$f(z)$ has zero Hausdorff dimension (so it also has zero Lebesgue
measure), while the subset of values for which all the partials are
divergent has full Lebesgue measure.
The classical Runge example is also considered. In this case we show
that, for all $z$ in the part of the interval $(-5,5)$ outside the
region of convergence, the sequence $\{P_n(z)\}_n$ is divergent.