Chebyshev and Fast Decreasing Polynomials

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Extending a classical result of Widom from 1969, polynomials with small supremum norms are constructed for a large family of compact setsΓ: their norm is at most a constant times the theoretical lower limitcap(Γ)n, wherecap(Γ)denotes logarithmic capacity. The construction is based on a discretization of the equilibrium measure, and the polynomials have the additional property that outside the given setΓthey increase as fast as possible, namely ascap(Γ)nexp⁡(ngC¯∖Γ(z)), with the Green's function with pole at infinity in the exponent. This latter fact allows us to use these polynomials as building blocks in constructing Dirac delta-type polynomials around corners: if a compact setKhas a corner at some pointz0, then Dirac delta-type polynomials (fast decreasing polynomials) peaking atz0are polynomialsPn(z)withPn(z0)=1that decrease as|Pn(z)|≺exp⁡(−nβ|z−z0|γ)on the setKaszmoves away fromz0. The possible(β,γ)pairs are completely described in turn of the angleαπatz0(β<1andγ⩾β/(2−α)orβ=1andγ>β/(2−α)). As application of these fast decreasing polynomials sharp Nikolskii- and Markov-type inequalities are proved for Jordan domains with corners. The paper uses distortion properties of conformal maps, potential theoretic techniques as well as the theory of weighted logarithmic potentials.

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Proceedings of the London Mathematical Society, v. 110, issue 5, p. 1057-1098