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• # Artykuł - szczegóły

## Studia Mathematica

2004 | 161 | 3 | 199-223

## Greedy approximation and the multivariate Haar system

EN

### Abstrakty

EN
We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis 𝓗 in $L_{p}([0,1])$, 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis $𝓗^{d} = 𝓗 × ... × 𝓗$ in $L_{p}([0,1]^{d})$, 1 < p < ∞, p ≠ 2). We prove some convergence results and also some results on convergence rate of weak type greedy algorithms. Our results are expressed in terms of properties of the basis with respect to a given weakness sequence.

199-223

wydano
2004

### Twórcy

autor
• Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, 81-825 Sopot, Poland
autor
• Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.