Abstract
We consider a weighted L p space L p(w) with a weight function w. It is known that the Haar system H p normalized in L p is a greedy basis of L p, 1 < p < ∞. We study a question of when the Haar system H w p normalized in L p(w) is a greedy basis of L p(w), 1 < p < ∞. We prove that if w is such that H w p is a Schauder basis of L p(w), then H w p is also a greedy basis of L p(w), 1 < p < ∞. 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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Kazarian, K., Temlyakov, V.N. Greedy bases in L p spaces. Proc. Steklov Inst. Math. 280, 181–190 (2013). https://doi.org/10.1134/S0081543813010124
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DOI: https://doi.org/10.1134/S0081543813010124