## Some inequalities for the tensor product of greedy bases and weight-greedy bases

**A 2005 Preprint by
G. Kerkyacharian,
D. Picard, and
V. Temlyakov
**

- 2005:15
In this paper we study properties of bases that are important in nonlinear m-term approximation with regard to these bases. It is known that the univariate Haar basis is a greedy basis for $L _ p([0,1))$ , $1<p<\infty$. This means that a greedy type algorithm realizes nearly best

*m-*term approximation for any individual function. It is also known that the multivariate Haar basis that is a tensor product of the univariate Haar bases is not a greedy basis. This means that in this case a greedy algorithm provides a*m-*term approximation that may be equal to the best*m-*term approximation multiplied by a growing (with*m*) to infinity factor. There are known results that describe the behavior of this extra factor for the Haar basis. In this paper we extend these results to the case of a basis that is a tensor product of the univariate greedy bases for $L _ p([0,1))$ , $1<p<\infty$. Also, we discuss weight-greedy bases and prove a criterion for weight-greedy bases similar to the one for greedy bases.