Quasi-Greedy Parameter in Approximation Theory

• The quasi-greedy parameter is defined as the least uniform constant that bounds thresholding greedy approximants, ensuring their uniform boundedness and convergence.
• It links nonlinear approximation with democracy, conditionality, and duality, highlighting its role in the stability of bases across Banach and quasi-Banach spaces.
• Its behavior reflects ambient geometric properties, showing optimal logarithmic bounds in non-superreflexive settings and enhanced stability in superreflexive spaces, but may fail in the Banach envelope.

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 (Albiac et al., 2015, Albiac et al., 15 Oct 2025). 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 (Albiac et al., 2019, Berasategui, 7 Oct 2025).

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 $x=\sum_n a_n e_n$0 indices corresponding to the largest $x=\sum_n a_n e_n$1 magnitudes of the coefficients of $x=\sum_n a_n e_n$2 (Albiac et al., 2015). The basis is quasi-greedy if there exists $x=\sum_n a_n e_n$3 such that

$x=\sum_n a_n e_n$4

and the least such constant is the quasi-greedy parameter (Albiac et al., 2015). In operator notation, one often writes $x=\sum_n a_n e_n$5 for a coordinate projection onto a greedy set $x=\sum_n a_n e_n$6, and then quasi-greediness is equivalently the uniform boundedness of $x=\sum_n a_n e_n$7 over greedy sets (Albiac et al., 15 Oct 2025).

Several equivalent formulations recur across the literature. In Banach spaces, quasi-greediness is equivalent to the convergence of $x=\sum_n a_n e_n$8 to $x=\sum_n a_n e_n$9 for every $m$0 (Garrigós et al., 2012). 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 $m$1 (Albiac et al., 2019). In the notation of (Albiac et al., 15 Oct 2025), a basis is quasi-greedy if greedy projections along $m$2 converge to $m$3, equivalently if

$m$4

Different papers package the parameter in slightly different but equivalent ways. One common variant controls both $m$5 and $m$6; in the extremal case $m$7, these formulations coincide (Albiac et al., 2015). In $m$8-Banach spaces, one also uses residual versions such as

$m$9

which are equivalent, up to $G_m(x)$0-dependent constants, to the boundedness of greedy projections themselves (Berasategui et al., 23 Aug 2025).

A foundational rigidity phenomenon occurs at constant $G_m(x)$1. A semi-normalized basis in a Banach space is quasi-greedy with quasi-greedy constant equal to $G_m(x)$2 if and only if it is unconditional with suppression-unconditional constant equal to $G_m(x)$3 (Albiac et al., 2015). 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 $G_m(x)$4-bases and Markushevich systems, the basic characterization is

$G_m(x)$5

(Albiac et al., 15 Oct 2025, Albiac et al., 2019). Quantitatively, democracy is encoded by upper and lower democracy functions such as

$G_m(x)$6

and almost greediness corresponds to their comparability together with finite quasi-greedy parameter (Albiac et al., 15 Oct 2025).

Lebesgue-type inequalities make this interaction explicit. For a quasi-greedy basis in a real Banach space, Hernández proved

$G_m(x)$7

where $G_m(x)$8 is the quasi-greedy constant, $G_m(x)$9 is the best $C_{\mathrm{qg}}$0-term error, and

$C_{\mathrm{qg}}$1

with $C_{\mathrm{qg}}$2 and $C_{\mathrm{qg}}$3 the democracy functions (Hernández, 2011). In democratic situations, $C_{\mathrm{qg}}$4, so $C_{\mathrm{qg}}$5, yielding logarithmic Lebesgue bounds (Hernández, 2011). A later Banach-space formulation sharpened the comparison to

$C_{\mathrm{qg}}$6

where $C_{\mathrm{qg}}$7 is the democracy ratio and $C_{\mathrm{qg}}$8 is the coordinate projection bound (Garrigós et al., 2012).

A more recent parameterization replaces coarse democracy by finer modulators. The squeeze symmetry parameter $C_{\mathrm{qg}}$9 and its disjoint version $K_{qg}$0 were introduced so that, for arbitrary bases in Banach or quasi-Banach spaces,

$K_{qg}$1

thereby answering Temlyakov’s question for a natural greedy-type parameter that combines linearly with unconditionality to determine Lebesgue constants (Albiac et al., 2021). 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 $K_{qg}$2, the conditionality constants are

$K_{qg}$3

For every quasi-greedy basis in a Banach space one has the universal bound

$K_{qg}$4

(Albiac et al., 2017). This logarithmic bound is optimal in non-superreflexive settings: a Banach space $K_{qg}$5 is non-superreflexive if and only if there exists a Banach space finitely representable in $K_{qg}$6 with a quasi-greedy basis $K_{qg}$7 satisfying

$K_{qg}$8

(Albiac et al., 2017).

Superreflexivity improves this picture. If $K_{qg}$9 is superreflexive, then every quasi-greedy basis in $(e_n)$0 satisfies

$(e_n)$1

and this rate is optimal (Albiac et al., 2017). In Hilbert spaces, a stronger statement is available: if a quasi-greedy basis has quasi-greedy constant $(e_n)$2, then there exist $(e_n)$3 and $(e_n)$4 such that

$(e_n)$5

while for every $(e_n)$6 there are quasi-greedy bases in Hilbert spaces with

$(e_n)$7

(Garrigos et al., 2013). 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 $(e_n)$8, while in superreflexive spaces one can realize sublogarithmic growth of the form $(e_n)$9 (Albiac et al., 2017, Albiac et al., 2018). A plausible implication is that the quasi-greedy parameter functions as a local stability bound, whereas the asymptotic profile of $m$0 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 $m$1-norm, obtained after Aoki–Rolewicz renorming, satisfying

$m$2

(Albiac et al., 15 Oct 2025). 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 (Albiac et al., 2019).

A systematic theory for biorthogonal systems in quasi-Banach spaces was developed in (Albiac et al., 2019). There, a semi-normalized $m$3-bounded basis is quasi-greedy if there exists $m$4 such that

$m$5

for every $m$6 and every finite greedy set $m$7 of $m$8; equivalently, $m$9 when the quasi-norm is continuous (Albiac et al., 2019). The same paper proves quasi-greedy $G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$0 convergence of the greedy series, extends the almost greedy characterization to quasi-Banach spaces, and obtains $G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$1-dependent estimates for truncation operators and related constants (Albiac et al., 2019).

The quasi-greedy parameter also appears in characterizations of other greedy-like notions. For partially-greedy bases in quasi-Banach spaces, one has

$G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$2

while the partially-greedy constant can be bounded in terms of quasi-greediness, partial symmetry, and truncation bounds (Berná, 2020). For strong partially greedy bases, a new conservative squeeze symmetry parameter $G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$3 yields the exact growth law

$G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$4

and for semi-greedy bases one has

$G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$5

(Berasategui et al., 23 Aug 2025).

A major structural theorem in the nonlocally convex regime states that every quasi-greedy basis in $G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$6 for $G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$7 is democratic, with fundamental function of the same order as $G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$8 (Albiac et al., 2020). The same paper extends this to separable $G_m(x)=\sum_{n\in \Lambda_m(x)} a_n e_n,$9-spaces, $\Lambda_m(x)$0, with the bounded approximation property (Albiac et al., 2020). 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 $\Lambda_m(x)$1 with an almost greedy basis such that the transported basis in the Banach envelope $\Lambda_m(x)$2 is no longer quasi-greedy (Albiac et al., 15 Oct 2025). In that construction, the space

$\Lambda_m(x)$3

is built from a symmetric unconditional sequence lattice $\Lambda_m(x)$4 with $\Lambda_m(x)$5 and a TGA-sensitive gauge $\Lambda_m(x)$6, and the canonical basis satisfies

$\Lambda_m(x)$7

(Albiac et al., 15 Oct 2025).

The key estimate is that greedy truncation is contractive in the $\Lambda_m(x)$8-component: $\Lambda_m(x)$9 while the $x=\sum_n a_n e_n$00-component is also contractive because $x=\sum_n a_n e_n$01 is $x=\sum_n a_n e_n$02-unconditional and symmetric (Albiac et al., 15 Oct 2025). However, after passing to the Banach envelope, the image of the canonical basis remains democratic with fundamental function $x=\sum_n a_n e_n$03 but fails to be unconditional for constant coefficients (Albiac et al., 15 Oct 2025). Since quasi-greedy implies unconditional for constant coefficients, the basis in the envelope is not quasi-greedy, and the quasi-greedy parameter becomes

$x=\sum_n a_n e_n$04

(Albiac et al., 15 Oct 2025).

The same paper gives a second counterexample: an almost greedy Markushevich basis in a nonlocally convex quasi-Banach space $x=\sum_n a_n e_n$05 which is never a Schauder basis under any reordering (Albiac et al., 15 Oct 2025). 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 (Albiac et al., 15 Oct 2025).

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 (Albiac et al., 15 Oct 2025). 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 $x=\sum_n a_n e_n$06 of a Banach space $x=\sum_n a_n e_n$07, the closed span of the dual basis $x=\sum_n a_n e_n$08 is a norming subspace of $x=\sum_n a_n e_n$09 (Berasategui, 7 Oct 2025). More precisely, if $x=\sum_n a_n e_n$10 is $x=\sum_n a_n e_n$11-quasi-greedy, then $x=\sum_n a_n e_n$12 is $x=\sum_n a_n e_n$13-norming, and for each $x=\sum_n a_n e_n$14 there exists $x=\sum_n a_n e_n$15 with

$x=\sum_n a_n e_n$16

(Berasategui, 7 Oct 2025). 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

$x=\sum_n a_n e_n$17

is the uniformly quasi-greedy constant $x=\sum_n a_n e_n$18 (Yu, 9 Jun 2026). In the “isometric” case $x=\sum_n a_n e_n$19, strict monotonicity of the norm forces the basis vectors to be disjoint (Yu, 9 Jun 2026). The same paper introduces an absolutely quasi-greedy constant $x=\sum_n a_n e_n$20, requiring uniform order boundedness simultaneously over all greedy orderings (Yu, 9 Jun 2026). 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 $x=\sum_n a_n e_n$21, an $x=\sum_n a_n e_n$22-greedy basis is always quasi-greedy, and the corresponding theory identifies quasi-greediness as the base level in the chain

$x=\sum_n a_n e_n$23

for Schreier families $x=\sum_n a_n e_n$24 (Beanland et al., 2022). For greedy algorithms with gaps, an $x=\sum_n a_n e_n$25-$x=\sum_n a_n e_n$26-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 (Berasategui et al., 2020).

At the level of interpretation, these developments separate three regimes. The first is the Banach-space extremal regime $x=\sum_n a_n e_n$27, which is exactly suppression-unconditionality with constant $x=\sum_n a_n e_n$28 (Albiac et al., 2015). The second is the quasi-Banach regime, where $x=\sum_n a_n e_n$29 interacts with truncation, democracy, and $x=\sum_n a_n e_n$30-geometry in ways that have no Banach analogue (Albiac et al., 2019, Albiac et al., 2020). 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 (Albiac et al., 15 Oct 2025). Together, these results define the quasi-greedy parameter as a structural constant at the intersection of nonlinear approximation, basis geometry, and ambient space regularity.