Thresholded Weak Convergence in L1 Spaces

• Thresholded weak convergence is defined by dual parameters that control the greedy approximation process, ensuring both convergence and uniform boundedness in L1.
• The method employs a dyadic chain selection and a two-stage weak thresholding approach that filters Haar coefficients based on predefined ratios.
• This framework distinguishes itself from classical greedy methods by enabling effective approximation even when the multivariate Haar basis is not quasi-greedy.

Thresholded weak convergence is a concept arising from the study of greedy approximation algorithms for the multivariate Haar basis in $L_{1}([0,1]^d)$, specifically in the context where the underlying basis is not quasi-greedy. The thresholded weak greedy algorithm introduces two real parameters, $0 < t < s < 1$, governing the weakness (threshold) and a secondary chain-length threshold. The central result is that, for this algorithm, the sequence of greedy approximants converges uniformly and is bounded for all $f \in L_{1}([0,1]^d)$, in contrast to the failure of classical thresholding greedy algorithms in this setting (Dilworth et al., 2012).

1. Multivariate Haar Basis and Haar Coefficients

Let $d \ge 1$. The setting is the Banach space $X = L_{1}([0,1]^d)$, with the normalized multivariate Haar system $$\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$$, augmented by the constant $h_{[0,1]^d}^{(0)} = 1$. The set $$\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$$ dyadic of length $2^{-n}$}$indexes dyadic cubes of all scales. For$f \in L_{1}([0,1]^d)$0 < t < s < 1$0I$0 < t < s < 1$1i</sup>isdefinedby</p><p></sup> is defined by</p> <p>0 < t < s < 1$2

A fixed total order $0 < t < s < 1$3 is imposed on the collection of multi-indexed Haar functions.

2. Weak Thresholding Greedy Algorithm: Construction

The weak thresholding greedy algorithm (WTGA) is parameterized by two real numbers $0 < t < s < 1$4. The approximation to $0 < t < s < 1$5 is constructed inductively via

$0 < t < s < 1$6

with $0 < t < s < 1$7 and $0 < t < s < 1$8. The iteration at $0 < t < s < 1$9 proceeds via three substeps:

• (A) Branch Selection: Find the minimal $f \in L_{1}([0,1]^d)$0 for which $f \in L_{1}([0,1]^d)$1.
• (B) Weak Thresholding on Dyadic Chain: Define $f \in L_{1}([0,1]^d)$2 as the maximal dyadic ancestor of $f \in L_{1}([0,1]^d)$3 such that all cubes $f \in L_{1}([0,1]^d)$4 in the dyadic chain $f \in L_{1}([0,1]^d)$5 from $f \in L_{1}([0,1]^d)$6 to $f \in L_{1}([0,1]^d)$7 admit some Haar direction $f \in L_{1}([0,1]^d)$8 with $f \in L_{1}([0,1]^d)$9.
• (C) Threshold-$d \ge 1$0 Selection in $d \ge 1$1: Among directions $d \ge 1$2 in $d \ge 1$3, select the smallest index $d \ge 1$4 such that

$d \ge 1$5

Set $d \ge 1$6, $d \ge 1$7. This selection rule is branch-greedy, focused on $d \ge 1$8-admissible large coefficients, and is independent of minor variants of the selection order (Dilworth et al., 2012).

3. Main Results: Convergence and Uniform Boundedness

The primary theorem establishes:

For $d \ge 1$9, for every $X = L_{1}([0,1]^d)$0:

• Convergence: $X = L_{1}([0,1]^d)$1 in $X = L_{1}([0,1]^d)$2.
• Uniform Boundedness: There exists $X = L_{1}([0,1]^d)$3 such that for all $X = L_{1}([0,1]^d)$4,

$X = L_{1}([0,1]^d)$5

An explicit bound,

$X = L_{1}([0,1]^d)$6

is provided. If either $X = L_{1}([0,1]^d)$7 or $X = L_{1}([0,1]^d)$8, the algorithm fails to remain bounded and diverges on suitable examples. Thus, the multivariate Haar basis is not quasi-greedy in $X = L_{1}([0,1]^d)$9 (Dilworth et al., 2012).

4. Proof Structure and Technical Lemmas

The proof of boundedness and convergence follows three stages:

I. Norm-vs-Coefficient Estimates:

• Lemma 3.1: For $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$0, $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$1.
• Lemma 3.2: If $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$2 for all $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$3, then $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$4, with $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$5 as the Haar-projection.
• Lemma 3.3: For $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$6, $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$7,

$\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$8

II. Combinatorial Decomposition of Active Cubes:

• The minimal generalized chain representation (MGCR) provides a partition into dyadic chains whose tips correspond to cubes where ancestors drop below threshold.
• Lemma 4.6: For $\{h_{I}^{(i)} : I \in \mathcal D^d,\, 1 \le i < 2^d\}$9 with disjoint active coefficient sets and opposite $h_{[0,1]^d}^{(0)} = 1$0-weakness/$h_{[0,1]^d}^{(0)} = 1$1-smallness, $h_{[0,1]^d}^{(0)} = 1$2.

III. Symmetrization and Final Patching:

• Symmetrization via $h_{[0,1]^d}^{(0)} = 1$3 ensures all active cubes of a function lie in fixed dyadic siblings, facilitating lower bounds on norms.
• Iteration leads to $h_{[0,1]^d}^{(0)} = 1$4, with $h_{[0,1]^d}^{(0)} = 1$5.
• Once uniform boundedness is established, convergence is achieved by the standard “basis-projection” (gliding hump) argument (Wojtaszczyk [11]).

5. Algorithmic Quantities and Threshold Parameter Effects

No nontrivial asymptotic rate for $h_{[0,1]^d}^{(0)} = 1$6 as $h_{[0,1]^d}^{(0)} = 1$7 is provided, but explicit uniform bounds hold at each step: $h_{[0,1]^d}^{(0)} = 1$8 and consequently, $h_{[0,1]^d}^{(0)} = 1$9. The algorithm thus avoids blow-up. If parameter $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$0 is chosen too close to $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$1 or to $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$2, divergence results, demonstrating the criticality of a gap between the two thresholds for the algorithm's stability in $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$3 (Dilworth et al., 2012).

6. Comparison with Classical Greedy Approximations

The thresholded weak greedy algorithm contrasts with two classical paradigms in Banach space approximation:

• Thresholding Greedy Algorithm (TGA): Selects at each step the largest coefficient. The TGA fails to converge in $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$4 for the Haar basis due to lack of quasi-greediness.
• Weak Greedy Algorithm: At each step, selects coefficients within a factor $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$5 of the current maximum; this converges in $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$6 for $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$7 when the basis is quasi-greedy but fails in $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$8 without additional control.
• Thresholded Weak Algorithm (as in this context): Introduces two parameters $\mathcal D^d = \bigcup_{n \ge 0} \{I_1 \times \cdots \times I_d: I_j \subset [0,1)$9. The branch-greedy step, governed by a secondary threshold $2^{-n}$0, limits the climb in the dyadic tree and ensures selection of coefficients of sufficient size. Convergence in $2^{-n}$1 is attained without the Haar basis being quasi-greedy (Dilworth et al., 2012).

In Banach space terminology, this demarcates a new regime: even for bases lacking quasi-greediness, convergence of a greedy algorithm can be restored by suitably “throttling” the selection mechanism via dual thresholds.

7. Significance and Extensions

Thresholded weak convergence demonstrates that controlled weakening of the greedy step, combined with dyadic chain-length moderation via auxiliary threshold $2^{-n}$2, suffices for uniform approximation in $2^{-n}$3 by members of the Haar basis. This is structurally distinct from classical greedy and weakly greedy algorithms, underpinning further studies in approximation theory for bases failing standard (quasi-)greedy conditions. The framework suggests avenues for developing analogous strategies for other non-quasi-greedy systems in $2^{-n}$4 spaces and more general Banach spaces (Dilworth et al., 2012).