Uniform Convergence of the Generalized Bieberbach Polynomials in Regions with Zero Angles

Uniform convergence of the p-Bieberbach polynomials in domains with zero angles

“…The proof of this lemma is given similar to those of [11,Lemma 15], [7] and [5]. Therefore, by taking α n from Lemmas 6.1-6.4, β n from Corollary 4.2 and in addition, these combine for G ∈ P Q(K, α, β) in cases α = 0 or β = 0, we prove Theorems 1.4-1.15.…”

“…The approximation properties in the C-norm of π n (z) on G were originally observed by Keldych [16] for the domains with sufficiently smooth boundaries. Considerable progress in this area was achieved by Mergelyan [20], Simonenko [23], Suetin [25], Andrievskii [9,11], Gaier [14], Abdullayev [2,4,6], Israfilov [15], and others.…”

“…Lemma 5.1 (See [5]). Let a measurable function f (z) be given on the arc γ and there exist a monotone increasing function ν(t), ν(0) = 0, such that |f (ζ)| ≺ ν(|ζ − z 0 |) for all ζ ∈ γ.…”

On Bernstein–Walsh-type lemmas in regions of the complex plane

“…Definition 2.1 [12]. We say that G 2 C Â if L WD @G has a continuous tangent Â.z/ WD Â.z.s// at all points z.s/:…”

Uniform estimates for polynomial approximation in domains with corners

“…To prove them one can use, say, the arguments in [9,Chapter IX.4]. If in addition all j = 2, then Lemma 3 follows, say, from Lemma 2.2 in [1].…”