Newman's Inequality for Müntz Polynomials on Positive Intervals Academic Article uri icon

abstract

  • The principal result of this paper is the following Markov-type inequality for Müntz polynomials. THEOREM (Newman's Inequality on [a, b] ⊂ (0, ∞)). Let A := (λj)∞j=0be an increasing sequence of nonnegative real numbers. Suppose λ0 = 0 and there exists a δ > 0 so that λj ≥ δj for each j. Suppose 0 < a < b. Then there exists a constant c(a, b, δ) depending only on a, b, and δ so that (equation presented) for every P ∈ Mn(Λ), where Mn(Λ) denotes the linear span of {xλ 0, xλ1, ..., xλn} over ℝ. When [a, b] = [0, 1] and with ∥P′∥[a, b] replaced with ∥xP′(x)∥[a, b] this was proved by Newman. Note that the interval [0, 1] plays a special role in the study of Müntz spaces Mn(Λ). A linear transformation y = αx + β does not preserve membership in Mn(Λ) in general (unless β = 0). So the analogue of Newman's Inequality on [a, b] for a > 0 does not seem to be obtainable in any straightforward fashion from the [0, b] case. © 1996 Academic Press, Inc.

author list (cited authors)

  • Borwein, P., & Erdélyi, T.

citation count

  • 4

publication date

  • May 1996