Alpan, Gökalp Widom factors for generalized Jacobi measures. (English) Zbl 1495.41001 J. Math. Anal. Appl. 511, No. 2, Article ID 126098, 12 p. (2022). Summary: We study optimal lower and upper bounds for Widom factors \(W_{\infty, n}(K, w)\) associated with Chebyshev polynomials for the weights \(w(x) = \sqrt{1+x}\) and \(w(x) = \sqrt{1-x}\) on compact subsets of \([-1, 1]\). We show which sets saturate these bounds. We consider Widom factors \(W_{2,n}(\mu)\) for \(L_2 (\mu)\) extremal polynomials for measures of the form \(d\mu (x) = (1-x)^{\alpha} (1+x)^{\beta} d\mu_K (x)\) where \(\alpha + \beta \geq 1, \alpha, \beta \in \mathbb{N} \cup \{ 0\}\) and \(\mu_K\) is the equilibrium measure of a compact regular set \(K\) in \([-1,1]\) with \(\pm 1 \in K\). We show that for such measures the improved lower bound (which was first studied in [4]) \([W_{2,n} (\mu)]^2 \geq 2S(\mu)\) holds. For the special cases \(d\mu (x) = (1 - x^2) d\mu_K (x), d\mu(x) = (1 - x) d\mu_K (x), d\mu(x) = (1 + x) d\mu_K (x)\) we determine which sets saturate this lower bound and discuss how saturated lower bounds for \([W_{2,n} (\mu)]^2\) and \(W_{\infty,n} (K, w)\) are related. MSC: 41A10 Approximation by polynomials 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) Keywords:Widom factors; Chebyshev polynomials; orthogonal polynomials; Jacobi polynomials; extremal polynomials PDFBibTeX XMLCite \textit{G. Alpan}, J. Math. Anal. Appl. 511, No. 2, Article ID 126098, 12 p. 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