TY - JOUR
T1 - Lebesgue-Type Inequalities for Quasi-greedy Bases
AU - Garrigós, Gustavo
AU - Hernández, Eugenio
AU - Oikhberg, Timur
N1 - Funding Information:
First author partially supported by Grants MTM2010-16518 and MTM2011-25377 (Spain). Second author supported by Grant MTM2010-16518 (Spain). Third author supported by Simons Foundation Travel Grant 210060, and by a COR grant from University of California System.
PY - 2013/12
Y1 - 2013/12
N2 - We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x-G N x∥ and the best N-term approximation σ N(x) is controlled by max{μ(N),k N}, where μ(N) and k N are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.
AB - We show that, for quasi-greedy bases in real or complex Banach spaces, an optimal bound for the ratio between greedy N-term approximation ∥x-G N x∥ and the best N-term approximation σ N(x) is controlled by max{μ(N),k N}, where μ(N) and k N are well-known constants that quantify the democracy and conditionality of the basis. In particular, for democratic bases this bound is O(logN). We show with various examples that these bounds are actually attained.
KW - Bounded variation
KW - Democracy functions
KW - Lebesgue-type inequalities
KW - Non-linear approximation
KW - Quasi-greedy bases
KW - Thresholding greedy algorithm
UR - http://www.scopus.com/inward/record.url?scp=84886773975&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84886773975&partnerID=8YFLogxK
U2 - 10.1007/s00365-013-9209-z
DO - 10.1007/s00365-013-9209-z
M3 - Article
AN - SCOPUS:84886773975
SN - 0176-4276
VL - 38
SP - 447
EP - 470
JO - Constructive Approximation
JF - Constructive Approximation
IS - 3
ER -