The theory of multi-dimensional polynomial approximation |
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Authors: | Moshe Dubiner |
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Institution: | (1) School of Mathematical Sciences, Tel aviv University, Ramat Aviv, 69978 Tel Aviv, Israel |
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Abstract: | We consider the problem of polynomial approximation to a real valued functionf defined on a compact set
. An approximation theorem is proven in terms of the newly defined modulus of approximation. It is shown to imply a multidimensional
Jackson type theorem which is stronger than previously known results even for the interval −1, 1]. A strong multidimensional
Bernstein type inverse theorem is also proven. We allow quite general approximation quasi-norms including
for 0<q≤∞.
We have found that the space of polynomials ℙ on a compact setX induces a semimetric
which encapsulates the local structure of ℙ. Any semimetric ρ equivalent to
suffices for the rough theory presented here. Many examples of sets
and their metrics are presented. |
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Keywords: | |
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