TY - JOUR
T1 - Bernstein–Walsh Inequalities in Higher Dimensions over Exponential Curves
AU - Kadyrov, Shirali
AU - Lawrence, Mark
N1 - Publisher Copyright:
© 2015, Springer Science+Business Media New York.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.
PY - 2016/12/1
Y1 - 2016/12/1
N2 - 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.
AB - 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.
KW - Bernstein–Walsh inequalities
KW - Hausdorff dimension
KW - Liouville vectors
KW - Polynomial inequalities
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U2 - 10.1007/s00365-015-9314-2
DO - 10.1007/s00365-015-9314-2
M3 - Article
AN - SCOPUS:84945301611
VL - 44
SP - 327
EP - 338
JO - Constructive Approximation
JF - Constructive Approximation
SN - 0176-4276
IS - 3
ER -