Coefficients of Polynomials of Restricted Growth on the Real Line

Lawrence A. Harris

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

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, ..., xmandy1, ..., ym-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(xj)≤φ(xj)mforj=1, ..., m, then p(k)(0)≤P(k)(0) wheneverkandmhave opposite parity and 0≤k<m; (b)if p(yj)≤φ(yj)mforj=1, ..., m-1 and if limsupy→∞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.

Original languageEnglish
Pages (from-to)293-312
Number of pages20
JournalJournal of Approximation Theory
Volume93
Issue number2
DOIs
StatePublished - May 1998

ASJC Scopus subject areas

  • Analysis
  • Numerical Analysis
  • Mathematics (all)
  • Applied Mathematics

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