Widom factors for generalized Jacobi measures

2021/07/28 by Alpan, Gökalp · 1 citation
#33C45 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 41A17 #Secondary 41A44

paper · doi:10.48550/arxiv.2107.13245

Abstract

We study optimal lower and upper bounds for Widom factors W_∞,n(K,w) associated with Chebyshev polynomials for the weights w(x)=√(1+x) and w(x)=√(1-x) on compact subsets of [-1,1]. We show which sets saturate these bounds. We consider Widom factors W_2,n(μ) for L₂(μ) extremal polynomials for measures of the form dμ(x)=(1-x)^α(1+x)^βdμ_K(x) where α+β≥ 1, α,β∈ℕ∪ \0\ and μ_K is the equilibrium measure of a compact regular set K in [-1,1] with ± 1∈ K. We show that for such measures the improved lower bound [W_2,n(μ)]²≥ 2S(μ) holds. For the special cases dμ(x)=(1-x²)dμ_K(x), dμ(x)=(1-x)dμ_K(x), dμ(x)=(1+x)dμ_K(x) we determine which sets saturate this lower bound and discuss how saturated lower bounds for [W_2,n(μ)]² and W_∞,n(K,w) are related.

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