Chebyshev and Fast Decreasing Polynomials
Document Type
Article
Publication Date
2015
Digital Object Identifier (DOI)
https://doi.org/10.1112/plms/pdv014
Abstract
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.
Was this content written or created while at USF?
Yes
Citation / Publisher Attribution
Proceedings of the London Mathematical Society, v. 110, issue 5, p. 1057-1098
Scholar Commons Citation
Totik, Vilmos and Varga, Tamás, "Chebyshev and Fast Decreasing Polynomials" (2015). Mathematics and Statistics Faculty Publications. 120.
https://digitalcommons.usf.edu/mth_facpub/120
