On the Behavior of Algebraic Polynomial in Unbounded Regions with Piecewise Dini-Smooth Boundary

Abstract

Let G ⊂ ℂ be a finite region bounded by a Jordan curve L := ∂G, let $\Omega :=\mathrm{e}\mathrm{x}\mathrm{t}\overline{G}$ (with respect to $\overline{\mathbb{C}}$ ), let Δ := {w : |w| > 1}, and let w = Φ(z) be the univalent conformal mapping of Ω onto Δ normalized by Φ (∞) = ∞, Φ′(∞) > 0. Also let h(z) be a weight function and let A _p (h,G), p > 0 denote a class of functions f analytic in G and satisfying the condition $${\left\Vert f\right\Vert}_{A_p\left(h,G\right)}^p:={\displaystyle \int {\displaystyle \underset{G}{\int }h(z){\left|f(z)\right|}^pd{\sigma}_z<\infty, }}$$ where σ is a two-dimensional Lebesgue measure.

Moreover, let P _n (z) be an arbitrary algebraic polynomial of degree at most n ∈ ℕ. The well-known Bernstein–Walsh lemma states that * $$\begin{array}{cc}\hfill \left|{P}_n(z)\right|\le {\left|\varPhi (z)\right|}^n{\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)},\hfill & \hfill z\in \Omega .\hfill \end{array}$$

In this present work we continue the investigation of estimation (*) in which the norm ${\left\Vert {P}_n\right\Vert}_{C\left(\overline{G}\right)}$ is replaced by ${\left\Vert {P}_n\right\Vert}_{A_p\left(h,G\right)},p>0$ , for Jacobi-type weight function in regions with piecewise Dini-smooth boundary.