On the distribution of extremal points of general Chebyshev polynomials

Kroó, András; Peherstorfer, Franz

doi:10.1090/S0002-9947-1992-1012514-4

On the distribution of extremal points of general Chebyshev polynomials
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by András Kroó and Franz Peherstorfer PDF

Trans. Amer. Math. Soc. 329 (1992), 117-130 Request permission

Abstract:

For a linear subspace ${\mathcal {U}_n} = {\operatorname {span}}[{\varphi _1}, \ldots ,{\varphi _n}]$ in $C[a,b]$ we introduce general Chebyshev polynomials as solutions of the minimization problem ${\operatorname {min}_{{a_i}}}{\left \| {{\varphi _n} - \sum \nolimits _{i = 1}^{n - 1} {{a_i}{\varphi _i}} } \right \|_C}$. For such a Chebyshev polynomial we study the distribution of its extremal points (maximum and minimum points) in terms of structural and approximative properties of ${\mathcal {U}_n}$.

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