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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.

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