Geodesic
Greedy bases in $L^p$ spaces
Informatics and Automation, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 188-197.

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We consider a weighted $L^p$ space $L^p(w)$ with a weight function $w$. It is known that the Haar system $\mathcal H_p$ normalized in $L^p$ is a greedy basis of $L^p$, $1$. We study a question of when the Haar system $\mathcal H_p^w$ normalized in $L^p(w)$ is a greedy basis of $L^p(w)$, $1$. We prove that if $w$ is such that $\mathcal H_p^w$ is a Schauder basis of $L^p(w)$, then $\mathcal H_p^w$ is also a greedy basis of $L^p(w)$, $1$. Moreover, we prove that a subsystem of the Haar system obtained by discarding finitely many elements from it is a Schauder basis in a weighted norm space $L^p(w)$; then it is a greedy basis. 
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     author = {K. Kazarian and V. N. Temlyakov},
     title = {Greedy bases in $L^p$ spaces},
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     year = {2013},
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     url = {http://geodesic.mathdoc.fr/item/TRSPY_2013_280_a11/}
}
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K. Kazarian; V. N. Temlyakov. Greedy bases in $L^p$ spaces. Informatics and Automation, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 188-197. http://geodesic.mathdoc.fr/item/TRSPY_2013_280_a11/

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