2019
Том 71
№ 11

Özkartepe P.

Articles: 2
Article (English)

On the interference of the weight and boundary contour for algebraic polynomials in weighted Lebesgue spaces. II

Ukr. Mat. Zh. - 2017. - 69, № 12. - pp. 1633-1651

We continue to study the estimation of the modulus of algebraic polynomials on the boundary contour with weight function, when the contour and the weight function have certain singularities with respect to the their quasinorm in the weighted Lebesgue space. In particular, the exact estimates were obtained for polynomials orthonormal on the curve with respect to the weight function with zeros on the same curve.

Article (English)

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

Ukr. Mat. Zh. - 2014. - 66, № 5. - pp. 579–597

Let G ⊂  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.