A persistent local maximum of the pth power deviation on an interval, p < 1

T. S. Motzkin and J. L. Walsh

Vol. 24 (1968), No. 1, 133–142

DOI: 10.2140/pjm.1968.24.133

Abstract

The deviation of the polynomial p₀(x) ≡ c from the given function f(x) ≡|x|^1∕αsgx,p + α > 2,w(ixj) nonnegative, bounded, and integrable but not a null function, is defined as δ(c) ≡∫ ₋₁¹w(x)|c − f(x)|pdx, whence δ′′(0) < 0. Thus the error function c − f(x) has a strong oscillation in the interval [–1, 1], yet δ(c) has a local maximum at c = 0 provided δ′(0) = 0; this is true for every (allowable) choice of w(x). For suitably chosen w(x), the deviation δ(c) has a global maximum at c = 0,|c|≦ 1.

Mathematical Subject Classification

Primary: 41.30

Milestones

Received: 1 October 1965

Published: 1 January 1968