Coefficients of Polynomials of Restricted Growth on the Real Line

Letφ:(-∞, ∞)→(0, ∞) be a given continuous even function and letmbe a positive integer. We show that, with some additional restrictions onφ, there exist decreasing sequencesx₁, ..., x_mandy₁, ..., y_m-1of symmetrically located points on (-∞, ∞) and corresponding polynomialsPandQof degreesm-1 andm, respectively, satisfyingP(x)≤φ(x)^m,Q(x)≤φ(x) ^m,-∞<x<∞,where equality holds with alternating signs at the corresponding sequence of points (and also at ±∞ forQ). Moreover, for any polynomialpof degree at mostm, (a)if p(x_j)≤φ(x_j)^mforj=1, ..., m, then p^(k)(0)≤P^(k)(0) wheneverkandmhave opposite parity and 0≤k<m; (b)if p(y_j)≤φ(y_j)^mforj=1, ..., m-1 and if limsup_y→∞p(y)/φ(y)^m≤1, then p^(k)(0)≤Q^(k)(0) wheneverkandmhave the same parity and 0≤k≤m. We give two computational methods for determining these sequences of points and thusPandQ.