[Paper Review] A Conditional Quasi-greedy Basis of l_1

S. J. Dilworth, David Mitra|ArXiv.org|Aug 15, 2000

Mathematical Analysis and Transform Methods4 citations

TL;DR

This paper constructs a conditional quasi-greedy basis for $\ell_1$ using Lindenstrauss's monotone basic sequence, demonstrating that $\ell_1$ admits a non-equivalent normalized basis that is quasi-greedy yet conditional. The key contribution is the existence of such a basis, resolving a question of Wojtaszczyk, while showing that the associated coefficient functionals fail to be quasi-greedy.

ABSTRACT

We show that the Lindenstrauss basic sequence in l_1 may be used to construct a conditional quasi-greedy basis of l_1, thus answering a question of Wojtaszczyk. We further show that the sequence of coefficient functionals for this basis is not quasi-greedy.

Motivation & Objective

• To resolve Wojtaszczyk's open question on the existence of a conditional quasi-greedy basis in $\ell_1$.
• To demonstrate that $\ell_1$ does not possess a unique normalized basis up to equivalence that is unconditional for constant coefficients.
• To analyze the quasi-greedy properties of the coefficient functionals associated with the constructed basis.
• To establish that the coefficient functionals of the new basis are not quasi-greedy, despite the basis being quasi-greedy.

Proposed method

• Utilizes Lindenstrauss's monotone, conditional basic sequence $\{x_i\}_{i=1}^\infty$ in $\ell_1$, defined as $x_i = e_i - \frac{1}{2}(e_{2i+1} + e_{2i+2})$.
• Employs coefficient functionals $x_i^*$ derived from $y_i^* \in \ell_\infty$, constructed via recursive sequences $\alpha_i$ with $y_i^* = \sum_{j=1}^{|α_i|} (1/2)^{j-1} e_{\alpha_i(j)}$.
• Applies a recursive induction process over sets $B_0, B_1, \dots, B_k$ to decompose the norm of a vector $x$ and analyze the action of greedy operators.
• Uses triangle inequality and disjoint support arguments to derive the key inequality $\|x+y\| \geq \|P_{B_0}x\| + \|P_{C_0}x\| + \frac{1}{2}\|P_{A_0}x\|$.
• Establishes quasi-greedy bounds by analyzing partial sums $\sum_{i=1}^{2^{n+1}-2} \epsilon_i x_i^*$ and comparing $\|\sum x_i^*\|$ to $\|\sum (-1)^i x_i^*\|$.
• Demonstrates failure of quasi-greediness for $\{x_i^* angle$ by showing $\|\sum x_i^*\| \geq n/2$ while $\|\sum (-1)^i x_i^*\| \leq 1$, contradicting uniform boundedness.

Experimental results

Research questions

• RQ1 Does $\ell_1$ admit a conditional quasi-greedy basis?
• RQ2 Can a quasi-greedy basis in $\ell_1$ be conditional, despite the space's strong uniqueness properties for unconditional bases?
• RQ3 Are the coefficient functionals of a quasi-greedy basis in $\ell_1$ necessarily quasi-greedy?
• RQ4 What is the relationship between the quasi-greedy property of a basis and that of its coefficient functionals?

Key findings

• A conditional quasi-greedy basis for $\ell_1$ exists, constructed from Lindenstrauss's basic sequence.
• The basis is quasi-greedy because the greedy operators $\mathcal{G}_m$ satisfy $\|\mathcal{G}_m(x)\| \leq C\|x\|$ uniformly.
• The coefficient functionals $\{x_i^*\}_{i=1}^\infty$ are not quasi-greedy, as shown by the norm ratio $\|\sum x_i^*\| / \|\sum (-1)^i x_i^*\| \to \infty$.
• Specifically, $\|\sum_{i=1}^{2^{n+1}-2} x_i^*\| \geq n/2$ while $\|\sum_{i=1}^{2^{n+1}-2} (-1)^i x_i^*\| \leq 1$, violating the quasi-greedy condition.
• The construction shows that $\ell_1$ has a normalized basis that is quasi-greedy but not equivalent to the standard basis, implying non-uniqueness up to equivalence.
• The method of recursive decomposition over sets $B_0, B_1, \dots, B_k$ enables control over norm estimates and proves the key inequality $\|x+y\| \geq \|P_{B_0}x\| + \|P_{C_0}x\| + \frac{1}{2}\|P_{A_0}x\|$.