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Supremum norms for quadratic polynomials

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Abstract.

We define two norms on \({\Bbb R}^3\) as follows. For \((a,b,c) \in {\Bbb R}^3,\) we let \(\Vert (a,b,c)\Vert _{\Bbb R}\equiv \sup \, \{ |ax^2 + bx + c|\, :\, x \in [-1,1]\} \) and \(\Vert (a,b,c)\Vert _{\Bbb C}\equiv \sup \, \{ |az^2 + bz + c|:\) \(\, z\in {\Bbb C},\ |z| \leq 1\} .\) Geometric properties of these norms are investigated. In particular, we give explicit formulas for these norms, describe the extreme points of the corresponding unit balls, give a parametric description of the unit spheres, and plot their images.

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Received: 1.7.1999

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Aron, R., Klimek, M. Supremum norms for quadratic polynomials. Arch. Math. 76, 73–80 (2001). https://doi.org/10.1007/s000130050544

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  • DOI: https://doi.org/10.1007/s000130050544

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