Greedy approximation with regard to non-greedy bases

Temlyakov, Vladimir; Yang, Mingrui; Ye, Peixin

doi:10.1007/s10444-010-9155-2

Cited by 21 publications

(26 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Then for any finite set of indices Λ we have for all f ∈ X , We now formulate a result concerning quasigreedy bases in L p spaces. The following theorem is from [19]. We note that for the case p = 2, Theorem 4.1 was proved in [23].…”

Section: This Impliesmentioning

confidence: 92%

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Temlyakov

2014

Forum math. Sigma

Self Cite

View full text Add to dashboard Cite

We study sparse approximation by greedy algorithms. We prove the Lebesgue-type inequalities for the weak Chebyshev greedy algorithm (WCGA), a generalization of the weak orthogonal matching pursuit to the case of a Banach space. The main novelty of these results is a Banach space setting instead of a Hilbert space setting. The results are proved for redundant dictionaries satisfying certain conditions. Then we apply these general results to the case of bases. In particular, we prove that the WCGA provides almost optimal sparse approximation for the trigonometric system in L p , 2 p < ∞.

show abstract

Section: This Impliesmentioning

confidence: 92%

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Temlyakov

2014

Forum math. Sigma

Self Cite

View full text Add to dashboard Cite

show abstract

“…When p = 2, this argument only gives C N log N, a result which goes back to [22]. [20,5]). Investigate whether, for quasi-greedy bases in a Hilbert space, the inequality C N log N can be replaced by a slower growing factor.…”

Section: Some Questionsmentioning

confidence: 83%

“…The purpose of this paper is to study such inequalities for quasi-greedy bases in general Banach spaces, thus complementing and in some cases improving the results in [20,21,5]. Towards this end we define the sequence…”

Section: Introductionmentioning

confidence: 93%

Lebesgue-Type Inequalities for Quasi-greedy Bases

2013

View full text Add to dashboard Cite

The final publication is available at Springer via http://dx.doi.org/10.1007/s00365-013-9209-zWe show that for quasi-greedy bases in real or complex Banach spaces the error of the thresholding greedy algorithm of order N is bounded by the best N-term error of approximation times a function of N which depends on the democracy functions and the quasi-greedy constant of the basis. If the basis is democratic this function is bounded by C log N. We show with two examples that this bound is attained for quasi-greedy democratic basesFirst and second authors supported by Grant MTM2010-16518 (Spain). Third author supported by a travel grant from Simons Foundation, and by a COR grant from University of California Syste

show abstract

“…Theorem 2.1 was proved in [26] under assumption that Ψ is a normalized basis. That proof works for a semi-normalized basis as well.…”

Section: Quasi-greedy Basesmentioning

confidence: 99%

“…The proof of Theorem 2.1 in [26] gives the following inequalities. Let Ψ = {ψ k } ∞ k=1 be a quasi-greedy basis of X.…”

Section: Quasi-greedy Basesmentioning

confidence: 99%

Quasi-greedy Bases and Lebesgue-type Inequalities

Temlyakov¹

2015

Advanced Courses in Mathematics - CRM Barcelona

View full text Add to dashboard Cite

We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on this study in the L p spaces. The novelty of the paper is in obtaining better Lebesgue-type inequalities under extra assumptions on a quasigreedy basis than known Lebesgue-type inequalities for quasi-greedy bases. We consider uniformly bounded quasi-greedy bases of L p , 1 < p < ∞, and prove that for such bases an extra multiplier in the Lebesgue-type inequality can be taken as C(p) ln(m + 1). The known magnitude of the corresponding multiplier for general (no assumption of uniform boundedness) quasi-greedy bases is of order m | 1 2 − 1 p | , p = 2. For uniformly bounded orthonormal quasi-greedy bases we get further improvements replacing ln(m + 1) by (ln(m + 1)) 1/2 .

show abstract

Greedy approximation with regard to non-greedy bases

Cited by 21 publications

References 15 publications

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Sparse Approximation and Recovery by Greedy Algorithms in Banach Spaces

Lebesgue-Type Inequalities for Quasi-greedy Bases

Quasi-greedy Bases and Lebesgue-type Inequalities

Contact Info

Product

Resources

About