Uniform estimates for polynomial approximation in domains with corners

AbstractLet G⊂C be a domain with a Jordan boundary ∂G, consisting of l smooth curves Γj, such that {zj}≔Γj-1∩Γj≠∅, j=1,…,l, where Γ0≔Γl. Denote by αjπ, 0<αj⩽2, the angles at zj's between the curves Γj-1 and Γj, exterior with respect to G. Let Φ be a conformal mapping of the exterior C⧹G¯ of G¯=G∪∂G onto the exterior of the unit disk, normed by Φ′(∞)>0. We assume that there is a neighborhood U of G¯, such that 0<c(G)⩽ϕ(z)Φ′(z)⩽C(G),z∈U⧹G¯, whereϕ(z)≔∏j=1lz-zj1-1αj,z∈C,z≠zj if αj⩽1. Set gG≔supg(z):z∈G. Then we prove Theorem. Let r∈N and 0⩽β⩽r. If a function f is analytic in G and f(r)ϕβG<+∞, then for each n⩾lr there is an algebraic polynomial Pn of degree