Greedy approximation and the multivariate Haar system

Kamont, Anna; Temlyakov, Vladimir

doi:10.4064/sm161-3-1

Cited by 16 publications

(21 citation statements)

References 8 publications

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“…This completes the proof of Theorem 2.3. In the paper [11] the following modification of the above weak type greedy algorithm in a way of further weakening the restriction (2.1) has been studied. We call this modification the Weak Thresholding Greedy Algorithm (WTGA) with a weakness sequence τ = {t k }.…”

Section: It Is Clear Thatmentioning

confidence: 99%

“…In [11] the following three theorems on convergence of the WTGA have been proved. The first one deals with an arbitrary Banach space X and any basis Ψ.…”

Section: It Is Clear Thatmentioning

confidence: 99%

“…In the case X = L p ([0, 1] d ) one can derive from Theorem 2.4 a more specific condition in terms of τ (see [11]). (i) The sequence τ = {t k } does not converge to 0.…”

Section: It Is Clear Thatmentioning

confidence: 99%

“…Along with convergence of the WTGA efficiency of approximation by G τ m (·, Ψ) has been studied in [11]. The accuracy of the WTGA was compared with best m-term approximation.…”

Section: It Is Clear Thatmentioning

confidence: 99%

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Greedy Approximation with Regard to Bases and General Minimal Systems

Konyagin¹,

Temlyakov²

2002

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Abstract. This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasi-Banach space. The last decade was very successful in studying nonlinear approximation. This was motivated by numerous applications. Nonlinear approximation is important in applications because of its increased efficiency. Two types of nonlinear approximation are employed frequently in applications. Adaptive methods are used in PDE solvers. The m-term approximation considered here is used in image and signal processing as well as the design of neural networks. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being approximated. The fundamental question of nonlinear approximation is how to construct good methods (algorithms) of nonlinear approximation. In this paper we discuss greedy type and thresholding type algorithms.

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Section: It Is Clear Thatmentioning

confidence: 99%

“…In [11] the following three theorems on convergence of the WTGA have been proved. The first one deals with an arbitrary Banach space X and any basis Ψ.…”

Section: It Is Clear Thatmentioning

confidence: 99%

“…In the case X = L p ([0, 1] d ) one can derive from Theorem 2.4 a more specific condition in terms of τ (see [11]). (i) The sequence τ = {t k } does not converge to 0.…”

Section: It Is Clear Thatmentioning

confidence: 99%

“…Along with convergence of the WTGA efficiency of approximation by G τ m (·, Ψ) has been studied in [11]. The accuracy of the WTGA was compared with best m-term approximation.…”

Section: It Is Clear Thatmentioning

confidence: 99%

See 2 more Smart Citations

Greedy Approximation with Regard to Bases and General Minimal Systems

Konyagin¹,

Temlyakov²

2002

Self Cite

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show abstract

“…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.…”

Section: Introductionmentioning

confidence: 99%