Bernstein–Walsh Inequalities in Higher Dimensions over Exponential Curves

Shirali Kadyrov, Mark Lawrence

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Let x= (x1, ⋯, xd) ∈ [- 1 , 1] d be linearly independent over Z, and set K={(ez,ex1z,ex2z⋯,exdz):|z|≤1}. We prove sharp estimates for the growth of a polynomial of degree n, in terms of En(x):=sup{‖P‖Δd+1:P∈Pn(d+1),‖P‖K≤1},where Δ d+1 is the unit polydisk. For all x∈ [- 1 , 1] d with linearly independent entries, we have the lower estimate logEn(x)≥nd+1(d-1)!(d+1)logn-O(nd+1);for Diophantine x, we have logEn(x)≤nd+1(d-1)!(d+1)logn+O(nd+1).In particular, this estimate holds for almost all x with respect to Lebesgue measure. The results here generalize those of Coman and Poletsky [6] for d= 1 without relying on estimates for best approximants of rational numbers which do not hold in the vector-valued setting.

Original languageEnglish
Pages (from-to)327-338
Number of pages12
JournalConstructive Approximation
Volume44
Issue number3
DOIs
Publication statusPublished - Dec 1 2016

Keywords

  • Bernstein–Walsh inequalities
  • Hausdorff dimension
  • Liouville vectors
  • Polynomial inequalities

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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