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On multiple extremal problems

Author: András Kroó
Journal: Proc. Amer. Math. Soc. 143 (2015), 3939-3949
MSC (2010): Primary 41A28, 41A50
Published electronically: March 6, 2015
MathSciNet review: 3359584
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Abstract: In linear space $ X$ consider arbitrary norms $ \Vert\cdot \Vert _j, j=1,2,...,d,$ and linear subspaces $ M_j\subset X$ of dim $ M_j=n_j, j=1,2,...,d,$ and set $ n=n_1+\dots +n_d$. Then given any subspace $ U\subset X$ of dimension $ n+1$ we shall verify the existence of $ u\in U\setminus \{0\}$ such that for every $ 1\leq j\leq d$ the element $ u$ is orthogonal to $ M_j$ in the $ \Vert\cdot \Vert _j$ norm, that is,

$\displaystyle \Vert u\Vert _j\leq \Vert u-m\Vert _j,\;\; \forall m\in M_j, \forall 1\leq j\leq d.$

In case of polynomial approximation with respect to distinct weighted uniform norms, the above extremal element is called a multiple Chebyshev polynomial. It will be shown that for weights whose ratios are monotone there always exists a unique multiple Chebyshev polynomial of maximal degree.

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Additional Information

András Kroó
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary — and — Department of Analysis, Budapest University of Technology and Economics, Budapest, Hungary

Keywords: Multiple orthogonal polynomials, multiple Chebyshev polynomials, uniqueness of normal multiple Chebyshev polynomial
Received by editor(s): January 31, 2014
Received by editor(s) in revised form: April 19, 2014, April 30, 2014, and May 13, 2014
Published electronically: March 6, 2015
Additional Notes: Supported by the OTKA Grant K111742.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society

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