Existence of Almost Greedy Bases in Mixed-Norm Sequence and Matrix Spaces, Including Besov Spaces

We prove that the sequence spaces ℓ p ⊕ ℓ q and the spaces of infinite matrices ℓ p ( ℓ q ) , ℓ q ( ℓ p ) and ( ⨁ n = 1 ∞ ℓ p n ) ℓ q , which are isomorphic to certain Besov spaces, have an almost greedy basis whenever 0 < p < 1 < q < ∞ . More precisely, we custom-build almost greedy bas...

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Bibliographic Details
Published in:Constructive approximation Vol. 60; no. 2; pp. 253 - 283
Main Authors: Albiac, Fernando, Ansorena, José L., Bello, Glenier, Wojtaszczyk, Przemysław
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2024
Springer Nature B.V
Subjects:
Analysis
Approximation
Banach spaces
Function space
Mathematics
Mathematics and Statistics
Numerical Analysis
Terminology
Subsymmetric basis
Almost greedy basis
46B15
Spaces
41A65
Quasi-greedy basis
Thresholding greedy algorithm
Conditional basis
ISSN:0176-4276, 1432-0940
Online Access:Get full text
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Summary:We prove that the sequence spaces ℓ p ⊕ ℓ q and the spaces of infinite matrices ℓ p ( ℓ q ) , ℓ q ( ℓ p ) and ( ⨁ n = 1 ∞ ℓ p n ) ℓ q , which are isomorphic to certain Besov spaces, have an almost greedy basis whenever 0 < p < 1 < q < ∞ . More precisely, we custom-build almost greedy bases in such a way that the Lebesgue parameters grow in a prescribed manner. Our arguments critically depend on the extension of the Dilworth–Kalton–Kutzarova method from Dilworth et al. (Stud Math 159(1):67–101, 2003), which was originally designed for constructing almost greedy bases in Banach spaces, to make it valid for direct sums of mixed-normed spaces with nonlocally convex components. Additionally, we prove that the fundamental functions of all almost greedy bases of these spaces grow as ( m 1 / q ) m = 1 ∞ .
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ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-023-09662-0