On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights

Details On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights

2009-10-22

arxiv.org/abs/0910.4223v1

Mathematics

Classical Analysis and ODEs

Scientific paper

For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \leq p \leq \infty$. It is shown that eventually all but a uniformly bounded number of the roots of the $L^p$-optimal polynomials lie within a small neighborhood of the support of a certain equilibrium measure; asymptotics for the $n$th roots of the $L^p$ norms are also provided. The case $p=\infty$ is well known and corresponds to weighted Chebyshev polynomials; the case $p=2$ corresponding to orthogonal polynomials as well as any other $1\leq p <\infty$ is our contribution.

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