Markoff type inequalities for curved majorants Academic Article uri icon

abstract

  • AbstractLet pn(x) be a real algebraic polynomial of degree n, and consider the Lp norms on I = [−1, 1]. A classical result of A. A. Markoff states that if ‖pn‖. ∞ ≤ 1, then ‖P′n‖∞ ≤ n2. A generalization of Markoff's problem, first suggested by P. Turán, is to find upper bounds for ‖pn(J)p if ∣pn(x)∣≤ ψ(x)xI. Here ψ(x) is a given function, a curved majorant. In this paper we study extremal properties of ‖p′n2 and ‖p″n2 if pn(x) has the parabolic majorant ∣p(x)∣≤ 1 − x2, xI. We also consider the problem, motivated by a well-known result of S. Bernstein, of maximising ‖(1 − x2)

publication date

  • February 1995