Abstract
In this work we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with interior zero and exterior non zero angles.
Highlights
Let C denote the complex plane and C: = C ∪ {∞}; G ⊂ C be a bounded Jordan region with boundary L: = ∂G such that 0 ∈ G; Let {zj}l j=1 be the fixed system of distinct points on the curve L.We consider generalizedJacobi weight function h(z) which is defined as follows: h(z): = ∏lj=1 |z − zj|γj, z ∈ C, (1)where γj > −2, for all j = 1,2, . . . , l.Let ℘n denotes the class of all algebraic polynomials Pn(z) of degree at most n ∈ N
We study the pointwise estimations for the derivative |Pn′(z)| in unbounded region Ω with zero angles as the following type
We give the definitions of regions with a piecewise smooth curve, which we present our main result and some notation that will be used later in the text
Summary
Jacobi weight function h(z) which is defined as follows: h(z): = ∏lj=1 |z − zj|γj , z ∈ C,. Let ℘n denotes the class of all algebraic polynomials Pn(z) of degree at most n ∈ N. For the Jordan region G, we introduce:. ∞, and Ap(1, G) ≡ Ap(G), where σ be the two-dimensional Lebesgue measure. When L is rectifiable, for any p > 0, let. ‖Pn‖L∞(1,L): = mz∈aLx|Pn(z)|, p = ∞, and Lp(1, L) ≡ Lp(L). Let us set Ω: = C \G = extL; Δ(w, R): = {w: |w| > R, R > 1}, Δ: = Δ(0,1) and let w = Φ(z) be the univalent conformal
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
