# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Greedy approximation and the multivariate Haar system

### Tom 161 / 2004

Studia Mathematica 161 (2004), 199-223 MSC: 41A65, 41A46. DOI: 10.4064/sm161-3-1

#### Streszczenie

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 ${\mathcal H}$ in $L_p([0,1])$, $1< p< \infty$) 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 ${\mathcal H}^d={\mathcal H}\times \mathinner {\ldotp \ldotp \ldotp }\times {\mathcal H}$ in $L_p([0,1]^d)$, $1< p< \infty$, $p\not =2$). 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.

#### Autorzy

• A. KamontInstitute of Mathematics
Abrahama 18
81-825 Sopot, Poland
e-mail
• V. N. TemlyakovDepartment of Mathematics
University of South Carolina
Columbia, SC 29208, U.S.A.
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek