TY - JOUR

T1 - Simultaneous approximation by greedy algorithms

AU - Leviatan, D.

AU - Temlyakov, V. N.

N1 - Funding Information:
★Part of this work was done while the first author visited the University of South Carolina in January 2003. ★★This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003.

PY - 2006/7

Y1 - 2006/7

N2 - We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f H and any dictionary D an expansion into a series f = ∑j=1∞cj(f) ψj(f), ψj(f)∈ D, j = 1, 2,..., with the Parseval property: ∥f∥2 = ∑j|c j(f)|2. Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of functions f 1,.∈.∈.,f N with a requirement that the dictionary elements φj of these expansions are the same for all f i, i=1,.∈.∈.,N. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.

AB - We study nonlinear m-term approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each f H and any dictionary D an expansion into a series f = ∑j=1∞cj(f) ψj(f), ψj(f)∈ D, j = 1, 2,..., with the Parseval property: ∥f∥2 = ∑j|c j(f)|2. Following the paper of A. Lutoborski and the second author we study analogs of the above expansions for a given finite number of functions f 1,.∈.∈.,f N with a requirement that the dictionary elements φj of these expansions are the same for all f i, i=1,.∈.∈.,N. We study convergence and rate of convergence of such expansions which we call simultaneous expansions.

KW - Convergence

KW - Greedy algorithms

KW - Simultaneous approximation

UR - http://www.scopus.com/inward/record.url?scp=33745640090&partnerID=8YFLogxK

U2 - 10.1007/s10444-004-7613-4

DO - 10.1007/s10444-004-7613-4

M3 - מאמר

AN - SCOPUS:33745640090

VL - 25

SP - 73

EP - 90

JO - Advances in Computational Mathematics

JF - Advances in Computational Mathematics

SN - 1019-7168

IS - 1-3

ER -