Extremal polynomials on the n-grid

01/04/2023

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by   Arno B. J. Kuijlaars, et al.

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The n-grid E_n consists of n equally spaced points in [-1,1] including the endpoints ± 1. The extremal polynomial p_n^* is the polynomial that maximizes the uniform norm p _[-1,1] among polynomials p of degree ≤α n that are bounded by one on E_n. For every α∈ (0,1), we determine the limit of 1/nlog p_n^*_[-1,1] as n →∞. The interest in this limit comes from a connection with an impossibility theorem on stable approximation on the n-grid.