Greedy approximation and the multivariate Haar system

• Autorzy: A. Kamont , V. N. Temlyakov
• Języki publikacji: 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.

Kategorie tematyczne

• 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
• 41A46: Approximation by arbitrary nonlinear expressions; widths and entropy

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2004
• Tom: 161
• Numer: 3
• Strony: 199-223

Daty

wydano

2004

Twórcy

autor

A. Kamont

• Institute of Mathematics, Polish Academy of Sciences, Abrahama 18, 81-825 Sopot, Poland

autor

V. N. Temlyakov

• Department of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

Identyfikatory

DOI

10.4064/sm161-3-1