---
title: Quasi-Greedy Parameter in Approximation Theory
url: https://www.emergentmind.com/topics/quasi-greedy-parameter
type: topic
---

# Quasi-Greedy Parameter in Approximation Theory

In greedy approximation theory, the quasi-greedy parameter is the least uniform constant controlling thresholding greedy approximants selected by the Thresholding Greedy Algorithm (TGA). For a basis $(e_n)$ and $x=\sum_n a_n e_n$, the TGA chooses the indices of the largest coefficients in modulus and forms the $m$-term greedy approximant $G_m(x)$. A basis is quasi-greedy when these greedy approximants are uniformly bounded and converge to the original vector; the associated parameter is usually denoted $C_{\mathrm{qg}}$ or $K_{qg}$, depending on the source [1504.04368] [2510.13693]. In recent work, this parameter has become a central invariant linking nonlinear approximation, conditionality, democracy, truncation operators, duality, and, in quasi-Banach settings, the structure of the Banach envelope itself [1903.11651] [2510.06398].

## 1. Definition and equivalent formulations

For a semi-normalized basis $(e_n)$ in a Banach space, the $m$-th thresholding greedy approximant is
\[
G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,
\]
where $\Lambda_m(x)$ is any set of $m$ indices corresponding to the largest $m$ magnitudes of the coefficients of $x$ [1504.04368]. The basis is quasi-greedy if there exists $C_{\mathrm{qg}}<\infty$ such that
\[
\|G_m(x)\| \le C_{\mathrm{qg}}\,\|x\|
\quad \text{for all } x \text{ and } m,
\]
and the least such constant is the quasi-greedy parameter [1504.04368]. In operator notation, one often writes $S_A(x)=\sum_{n\in A} e_n^*(x)e_n$ for a coordinate projection onto a greedy set $A$, and then quasi-greediness is equivalently the uniform boundedness of $S_A$ over greedy sets [2510.13693].

Several equivalent formulations recur across the literature. In Banach spaces, quasi-greediness is equivalent to the convergence of $G_m(x)$ to $x$ for every $x$ [1207.0946]. In the general Markushevich-basis framework for quasi-Banach spaces, quasi-greediness is equivalent to convergence of the greedy series and to uniform boundedness of the family $(G_m)$ [1903.11651]. In the notation of [2510.13693], a basis is quasi-greedy if greedy projections along $\mathcal{G}(x)$ converge to $x$, equivalently if
\[
\sup\{\|x-S_A(x)\|:x\in B_X,\ A\in \mathcal{G}(x)\}<\infty.
\]

Different papers package the parameter in slightly different but equivalent ways. One common variant controls both $\|G_m(x)\|$ and $\|x-G_m(x)\|$; in the extremal case $C_{\mathrm{qg}}=1$, these formulations coincide [1504.04368]. In $p$-Banach spaces, one also uses residual versions such as
\[
\mathbf{g}_m^c:=\sup_{k\le m}\sup_{f\neq 0}\frac{\|f-\mathcal{G}_k(f)\|}{\|f\|},
\]
which are equivalent, up to $p$-dependent constants, to the boundedness of greedy projections themselves [2508.16893].

A foundational rigidity phenomenon occurs at constant $1$. A semi-normalized basis in a Banach space is quasi-greedy with quasi-greedy constant equal to $1$ if and only if it is unconditional with suppression-unconditional constant equal to $1$ [1504.04368]. Thus the “isometric” quasi-greedy regime collapses to the strongest suppression-unconditional regime in Banach spaces.

## 2. Relation to democracy, almost greediness, and Lebesgue-type estimates

The quasi-greedy parameter does not by itself measure optimal approximation quality; its full effect is revealed when combined with democracy. In the Banach-space theory, and in its quasi-Banach extension to $W$-bases and Markushevich systems, the basic characterization is
\[
\text{almost greedy} \iff \text{quasi-greedy}+\text{democratic}
\]
[2510.13693] [1903.11651]. Quantitatively, democracy is encoded by upper and lower democracy functions such as
\[
\Phi_u(m)=\sup\{\|\varepsilon_A\|:|A|\le m\},
\qquad
\Phi_l(m)=\inf\{\|\varepsilon_A\|:|A|\ge m\},
\]
and almost greediness corresponds to their comparability together with finite quasi-greedy parameter [2510.13693].

Lebesgue-type inequalities make this interaction explicit. For a quasi-greedy basis in a real Banach space, Hernández proved
\[
\|x-G_N(x)\|\le 8K^5\,v(N)\,\sigma_N(x),
\]
where $K$ is the quasi-greedy constant, $\sigma_N(x)$ is the best $N$-term error, and
\[
v(N)=\sum_{k=1}^N \frac{p(k)}{k},
\qquad
p(N)=\sup_{1\le k\le N}\frac{D(k)}{d(k)},
\]
with $D$ and $d$ the democracy functions [1111.0460]. In democratic situations, $p(N)\approx 1$, so $v(N)\approx \log N$, yielding logarithmic Lebesgue bounds [1111.0460]. A later Banach-space formulation sharpened the comparison to
\[
C_N \approx \max\{\mu(N),K_N\},
\]
where $\mu(N)$ is the democracy ratio and $K_N=\sup_{|A|\le N}\|S_A\|$ is the coordinate projection bound [1207.0946].

A more recent parameterization replaces coarse democracy by finer modulators. The squeeze symmetry parameter $A_m$ and its disjoint version $A_m^{\mathrm d}$ were introduced so that, for arbitrary bases in Banach or quasi-Banach spaces,
\[
L_m \approx \max\{A_m,k_m\},
\qquad
L_a(m)\approx \max\{A_m^{\mathrm d},g_m\},
\]
thereby answering Temlyakov’s question for a natural greedy-type parameter that combines linearly with unconditionality to determine Lebesgue constants [2104.10912]. This suggests that the quasi-greedy parameter is best viewed not as a standalone approximation constant, but as one term in a larger system of nonlinear structural parameters.

## 3. Conditionality growth and ambient geometry

The quasi-greedy parameter imposes strong restrictions on conditionality, but the sharp form of those restrictions depends on the geometry of the ambient space. For a semi-normalized Schauder basis $\mathcal{B}$, the conditionality constants are
\[
k_m[\mathcal{B}]=\sup_{|A|\le m}\|S_A\|.
\]
For every quasi-greedy basis in a Banach space one has the universal bound
\[
k_m[\mathcal{B}] \lesssim \log m
\quad (m\ge 2)
\]
[1702.06326]. This logarithmic bound is optimal in non-superreflexive settings: a Banach space $X$ is non-superreflexive if and only if there exists a Banach space finitely representable in $X$ with a quasi-greedy basis $\mathcal{B}$ satisfying
\[
k_m[\mathcal{B}] \approx \log m
\]
[1702.06326].

Superreflexivity improves this picture. If $X$ is superreflexive, then every quasi-greedy basis in $X$ satisfies
\[
k_m[\mathcal{B}] \lesssim (\log m)^a
\quad \text{for some } 0<a<1,
\]
and this rate is optimal [1702.06326]. In Hilbert spaces, a stronger statement is available: if a quasi-greedy basis has quasi-greedy constant $K$, then there exist $a=a(K)<1$ and $c>0$ such that
\[
k_N \le c(\log N)^a,
\]
while for every $\alpha<1$ there are quasi-greedy bases in Hilbert spaces with
\[
k_N \ge C_\alpha (\log N)^\alpha
\]
[1301.4844]. Thus the quasi-greedy parameter controls conditionality, but the ambient geometry determines whether the logarithmic barrier can be improved.

These results motivated systematic constructions of highly conditional quasi-greedy and almost greedy bases. In non-superreflexive classical spaces, one can realize the maximal asymptotic growth $k_m\approx \log m$, while in superreflexive spaces one can realize sublogarithmic growth of the form $(\log m)^{1-\epsilon}$ [1712.04004] [1803.08351]. A plausible implication is that the quasi-greedy parameter functions as a local stability bound, whereas the asymptotic profile of $k_m$ records the interaction between greedy selection and large-scale geometry.

## 4. Quasi-Banach extensions

In quasi-Banach spaces, quasi-greediness requires additional structure because convexity is unavailable. The standard replacement is a $p$-norm, obtained after Aoki–Rolewicz renorming, satisfying
\[
\|x+y\|^p \le \|x\|^p+\|y\|^p
\qquad (0<p\le 1)
\]
[2510.13693]. Within this setting, the quasi-greedy parameter is again the least constant controlling greedy projections, but its interaction with truncation operators, suppression phenomena, and democracy becomes more delicate [1903.11651].

A systematic theory for biorthogonal systems in quasi-Banach spaces was developed in [1903.11651]. There, a semi-normalized $M$-bounded basis is quasi-greedy if there exists $C_{\mathrm{qg}}<\infty$ such that
\[
\|S_A(f)\|\le C_{\mathrm{qg}}\|f\|
\]
for every $f$ and every finite greedy set $A$ of $f$; equivalently, $C_{\mathrm{qg}}=\sup_m\|G_m\|=\sup_m\|H_m\|$ when the quasi-norm is continuous [1903.11651]. The same paper proves quasi-greedy $\iff$ convergence of the greedy series, extends the almost greedy characterization to quasi-Banach spaces, and obtains $p$-dependent estimates for truncation operators and related constants [1903.11651].

The quasi-greedy parameter also appears in characterizations of other greedy-like notions. For partially-greedy bases in quasi-Banach spaces, one has
\[
C_{qg}\le 2^{1/p} C_{pg},
\]
while the partially-greedy constant can be bounded in terms of quasi-greediness, partial symmetry, and truncation bounds [2004.01128]. For strong partially greedy bases, a new conservative squeeze symmetry parameter $\lambda_m^c$ yields the exact growth law
\[
\mathbf{L}_m^s \asymp \max\{\lambda_m^c,\mathbf{g}_m\},
\]
and for semi-greedy bases one has
\[
\mathbf{L}_m^{ch}\lesssim
\begin{cases}
\max\{\mathbf{g}_m,\lambda_m\}, & p=1,\\[4pt]
\max\{\mathbf{g}_m^{\,1+1/p},\lambda_m\}, & 0<p<1.
\end{cases}
\]
[2508.16893].

A major structural theorem in the nonlocally convex regime states that every quasi-greedy basis in $\ell_p$ for $0<p<1$ is democratic, with fundamental function of the same order as $m^{1/p}$ [2004.05206]. The same paper extends this to separable $\mathcal{L}_p$-spaces, $0<p<1$, with the bounded approximation property [2004.05206]. This shows that, in some quasi-Banach spaces, the quasi-greedy parameter forces democracy rather than merely coexisting with it.

## 5. Banach envelopes and the fragility of the parameter

A decisive development is the discovery that quasi-greediness is not stable under passage to the Banach envelope. The paper “When Greedy Approximation Breaks: Counterexamples in Quasi-Banach Spaces” constructs a quasi-Banach space $\mathbb{X}$ with an almost greedy basis such that the transported basis in the Banach envelope $\widehat{X}$ is no longer quasi-greedy [2510.13693]. In that construction, the space
\[
X=E\cap \mathcal{B}_0
\]
is built from a symmetric unconditional sequence lattice $E$ with $\Phi_E(m)\approx m$ and a TGA-sensitive gauge $\|\cdot\|_{\mathcal B}$, and the canonical basis satisfies
\[
K_{qg}(X)\le 1
\]
[2510.13693].

The key estimate is that greedy truncation is contractive in the $\mathcal{B}$-component:
\[
\|x-S_A(x)\|_{\mathcal B}\le \|x\|_{\mathcal B},
\]
while the $E$-component is also contractive because $E$ is $1$-unconditional and symmetric [2510.13693]. However, after passing to the Banach envelope, the image of the canonical basis remains democratic with fundamental function $\approx m$ but fails to be unconditional for constant coefficients [2510.13693]. Since quasi-greedy implies unconditional for constant coefficients, the basis in the envelope is not quasi-greedy, and the quasi-greedy parameter becomes
\[
K_{qg}(\widehat{X})=\infty
\]
[2510.13693].

The same paper gives a second counterexample: an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space $\mathbb{Y}$ which is never a Schauder basis under any reordering [2510.13693]. Here too, the basis has finite quasi-greedy constant in the original space, but the envelope destroys key greedy features. The paper’s summary states that these examples resolve negatively the questions of whether quasi-greediness must survive the Banach envelope and whether an almost greedy basis must become Schauder after some reordering in the quasi-Banach class [2510.13693].

This shows that the quasi-greedy parameter is not merely sensitive to linear or lattice structure. The envelope preserves linear and lattice features, yet can turn a finite quasi-greedy parameter into an infinite one [2510.13693]. This suggests that the parameter detects nonlinear features of the quasi-norm that are invisible to the Banach envelope construction.

## 6. Duality, extremal regimes, and newer refinements

The quasi-greedy parameter also has direct dual consequences. For a quasi-greedy Markushevich basis $\mathcal{X}$ of a Banach space $\mathbb{X}$, the closed span of the dual basis $\mathcal{X}^*$ is a norming subspace of $\mathbb{X}^*$ [2510.06398]. More precisely, if $\mathcal{X}$ is $K$-quasi-greedy, then $\langle \mathcal{X}^*\rangle$ is $K^{-1}$-norming, and for each $f\in \langle \mathcal{X}\rangle$ there exists $f^*\in \langle \mathcal{X}^*\rangle$ with
\[
f^*(f)=\|f\|
\quad \text{and} \quad
\|f^*\|\le K
\]
[2510.06398]. Thus the quasi-greedy parameter governs a concrete norming constant in dual space.

Stronger lattice-based refinements have also been introduced. In Banach lattices, uniformly quasi-greedy bases are characterized by uniform order boundedness of maximal greedy partial sums; the least constant in
\[
\Big\|\sup_{n=1}^m \mathcal{G}_n(x)\Big\|\le C\|x\|
\]
is the uniformly quasi-greedy constant $K_{uqg}$ [2606.10795]. In the “isometric” case $K_{uqg}=1$, strict monotonicity of the norm forces the basis vectors to be disjoint [2606.10795]. The same paper introduces an absolutely quasi-greedy constant $K_{aqg}$, requiring uniform order boundedness simultaneously over all greedy orderings [2606.10795]. These refinements are stronger than the classical quasi-greedy parameter and are tailored to order-sensitive settings.

Other variants place the quasi-greedy parameter inside restricted greedy frameworks. For hereditary families $\mathcal F$, an $\mathcal F$-greedy basis is always quasi-greedy, and the corresponding theory identifies quasi-greediness as the base level in the chain
\[
\text{greedy} \Rightarrow \mathcal S_\beta\text{-greedy} \Rightarrow \mathcal S_\alpha\text{-greedy} \Rightarrow \text{quasi-greedy}
\]
for Schreier families $\mathcal S_\alpha \subset \mathcal S_\beta$ [2211.01030]. For greedy algorithms with gaps, an $\mathbf n$-$t$-quasi-greedy parameter controls greedy truncations only at selected cardinalities; when the gap sequence has bounded quotient gaps, this restricted parameter still implies the ordinary quasi-greedy property [2009.02257].

At the level of interpretation, these developments separate three regimes. The first is the Banach-space extremal regime $C_{\mathrm{qg}}=1$, which is exactly suppression-unconditionality with constant $1$ [1504.04368]. The second is the quasi-Banach regime, where $C_{\mathrm{qg}}$ interacts with truncation, democracy, and $p$-geometry in ways that have no Banach analogue [1903.11651] [2004.05206]. The third is the envelope-sensitive regime, where finite quasi-greedy behavior in the original quasi-Banach space may become infinite in the Banach envelope [2510.13693]. Together, these results define the quasi-greedy parameter as a structural constant at the intersection of nonlinear approximation, basis geometry, and ambient space regularity.

Source: https://www.emergentmind.com/topics/quasi-greedy-parameter