Markov-type inequalities on certain irrational arcs and domains Academic Article uri icon

abstract

  • Let Pnd denote the set of real algebraic polynomials of d variables and of total degree at most n. For a compact set K ⊂ Rd set ∥P∥K = supx∈K P(x) . Then the Markov factors on K are defined by Mn(K): = max {∥ DωP∥K: P ∈ Pnd, ∥P∥K ≤ 1, ω ∈ Sd-1}. (Here, as usual, Sd-1 stands for the Euclidean unit sphere in Rd.) Furthermore, given a smooth curve Γ ⊂ Rd, we denote by DTP the tangential derivative of P along Γ (T is the unit tangent to Γ). Correspondingly, consider the tangential Markov factor of Γ given by MnT (Γ): = max {∥DTP∥Γ : P ∈ Pnd, ∥P∥ Γ ≤ 1}. Let Γα:= {(x, xα) : 0≤x≤1}. We prove that for every irrational number α > 0 there are constants A, B > 1 depending only on α such that An ≤ MnT(Γα) ≤ Bn for every sufficiently large n. Our second result presents some new bounds for Mn(Ωα), where Ωα:={(x, y) ∈ ℝ2 : 0≥x≥1; 1/2xα≤y≤2xα} (d=2, α > 1). We show that for every α > 1 there exists a constant c > 0 depending only on α such that Mn (Ωα)≤nclogn. © 2004 Elsevier Inc. All rights reserved.

author list (cited authors)

  • Erdélyi, T., & Kroó, A.

citation count

  • 4

publication date

  • October 2004