Greedy algorithms and bestm-term approximation with respect to biorthogonal systems

Simultaneous Approximation by Greedy Algorithms

“…The problem of m-term approximation with regard to a basis has been studied thoroughly and rather complete results have been established (see [2], [4]- [6], [9]- [11], [15], [19]- [23], [25]- [27], [31], [34]- [37], 1 Part of this work was done while the first author visited the University of South Carolina in January 2003. 2 This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003.…”

Greedy Type Bases in Banach Spaces

“…The first result was first proved in [33] but we present argument given in [22] and [40] which is a bit easier. (27) For each t 1 we apply the inductive hypothesis (note that the number of different B's is at most J) and we continue the estimates (28) Now we apply the estimate (24) for d = 1 and we continue as (29) Due to Proposition 4 we can complete the proof of (24). The inequality (25) follows by duality from (24) …”

“…The dual system is the system considered in the space VMO. It was proved in [29] that in the space VMO, so we have a natural example of a greedy system whose dual is not greedy. Actually one can show that the space VMO does not have any greedy system.…”

Lebesgue inequalities for the greedy algorithm in general bases

“…This is for instance the case for the trigonometric system, and the other examples in §5. In this respect, we can mention the results of Oswald [17], who obtains additive estimates of the form L N ≈ k N + B N , but with constants B N of a more complicated nature.…”