Semi-Greedy Bases in Banach Spaces

• The topic defines semi-greedy bases as Schauder bases where the greedy support, paired with optimal recomputation of coefficients, approximates any vector within a controlled error bound.
• It establishes that semi-greedy and almost-greedy bases are equivalent in Banach spaces through the interplay of quasi-greediness and (super-)democratic properties.
• Extensions to weighted and Markushevich bases, and refinements using Schreier families, open up new avenues in greedy approximation theory and algorithmic analysis.

Searching arXiv for recent and foundational papers on semi-greedy bases, almost-greedy bases, and Schreier-refined greedy notions. Semi-greedy bases are bases for which the greedy choice of support, followed by optimal recomputation of coefficients on that support, yields approximants comparable with best $m$-term approximation. In the standard Banach-space setting of a semi-normalized Schauder basis $\mathcal B=(e_n)$, if $A_m(x)$ is a greedy set of size $m$ for $x$, then a Chebyshev-greedy sum $\mathcal C\mathcal G_m(x)$ is any best approximant in $\operatorname{span}\{e_n:n\in A_m(x)\}$, and $\mathcal B$ is semi-greedy when there exists $C_{sg}>0$ such that

$$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$

where $\mathcal B=(e_n)$0 is the best unrestricted $\mathcal B=(e_n)$1-term approximation error (Berná, 2019). A central conclusion of the modern theory is that, for Schauder bases in arbitrary Banach spaces, semi-greediness is equivalent to almost-greediness; weighted and Markushevich-basis extensions preserve this equivalence under the hypotheses stated in the corresponding papers (Berná, 2018, Berná, 2019, Berasategui et al., 2021).

1. Definition and algorithmic framework

Let $\mathcal B=(e_n)$2 be a Banach space and $\mathcal B=(e_n)$3 a semi-normalized Schauder basis with biorthogonal functionals $\mathcal B=(e_n)$4. For $\mathcal B=(e_n)$5, a set $\mathcal B=(e_n)$6 is a greedy set of order $\mathcal B=(e_n)$7 if

$\mathcal B=(e_n)$8

The associated thresholding greedy approximant is

$\mathcal B=(e_n)$9

Two standard error functionals are

$A_m(x)$0

the best unrestricted $A_m(x)$1-term error, and

$A_m(x)$2

the best coordinate-projection error, where $A_m(x)$3 (Berná, 2019).

The Chebyshev version of greedy approximation fixes the greedy support $A_m(x)$4 but reoptimizes the coefficients. A Chebyshev-greedy sum $A_m(x)$5 satisfies

$A_m(x)$6

Semi-greediness therefore measures the quality of the greedy support itself, rather than the raw projection $A_m(x)$7. This places the notion between pure support selection and best $A_m(x)$8-term approximation in a precise algorithmic sense (Berná, 2019).

The surrounding classes are defined by progressively weaker comparison principles. A basis is quasi-greedy if greedy sums are uniformly bounded, for instance by

$A_m(x)$9

It is almost greedy if

$m$0

It is greedy if the same comparison is made with $m$1 rather than $m$2 (Berná, 2018, Berná, 2019).

2. Characterizations and equivalence with almost-greedy bases

The decisive structural theorem is that semi-greedy and almost-greedy bases coincide for Schauder bases in arbitrary Banach spaces. In the formulation with basis constant $m$3, if $m$4 is $m$5-quasi-greedy and $m$6-super-democratic, then $m$7 is $m$8-semi-greedy with

$m$9

Conversely, if $x$0 is $x$1-semi-greedy, then it is $x$2-super-democratic and $x$3-quasi-greedy, with

$x$4

Together with the characterization of almost-greedy bases as quasi-greedy plus super-democratic, this yields

$x$5

for Schauder bases in Banach spaces (Berná, 2018).

This result removes an earlier finite-cotype hypothesis. The classical theorem stated that almost-greedy and semi-greedy bases were equivalent for Schauder bases in Banach spaces with finite cotype; the later theorem established the equivalence in general Banach spaces (Berná, 2019, Berná, 2018). In particular, semi-greediness is not a genuinely distinct intermediate class in the Banach-space Schauder setting once the full characterization is taken into account.

The almost-greedy side of the equivalence is itself structural. A basis is almost-greedy if and only if it is quasi-greedy and democratic; equivalently, quasi-greedy and super-democratic, or quasi-greedy and disjoint-super-democratic (Berasategui et al., 2022). Combining this with the semi-greedy equivalence produces the standard synthesis: $x$6 This identification is one of the central organizing principles of greedy approximation theory (Berná, 2018, Berasategui et al., 2022).

3. Weighted and Markushevich extensions

Weighted versions replace cardinality by a weight $x$7, with $x$8. The weighted best errors are

$x$9

A basis is $\mathcal C\mathcal G_m(x)$0-almost-greedy if

$\mathcal C\mathcal G_m(x)$1

and $\mathcal C\mathcal G_m(x)$2-semi-greedy if

$\mathcal C\mathcal G_m(x)$3

For Schauder bases in arbitrary Banach spaces, the weighted theory parallels the unweighted one: $\mathcal C\mathcal G_m(x)$4-semi-greediness is equivalent to $\mathcal C\mathcal G_m(x)$5-almost-greediness, and both are equivalent to quasi-greediness plus $\mathcal C\mathcal G_m(x)$6-super-democracy, or quasi-greediness plus $\mathcal C\mathcal G_m(x)$7-disjoint-super-democracy (Berná, 2019).

More precisely, if $\mathcal C\mathcal G_m(x)$8 is $\mathcal C\mathcal G_m(x)$9-$\operatorname{span}\{e_n:n\in A_m(x)\}$0-semi-greedy, then it is $\operatorname{span}\{e_n:n\in A_m(x)\}$1-quasi-greedy and $\operatorname{span}\{e_n:n\in A_m(x)\}$2-$\operatorname{span}\{e_n:n\in A_m(x)\}$3-super-democratic, with

$\operatorname{span}\{e_n:n\in A_m(x)\}$4

where $\operatorname{span}\{e_n:n\in A_m(x)\}$5 and $\operatorname{span}\{e_n:n\in A_m(x)\}$6. Conversely, quasi-greediness plus $\operatorname{span}\{e_n:n\in A_m(x)\}$7-super-democracy yields $\operatorname{span}\{e_n:n\in A_m(x)\}$8-semi-greediness with

$\operatorname{span}\{e_n:n\in A_m(x)\}$9

and quasi-greediness plus $\mathcal B$0-disjoint-super-democracy yields $\mathcal B$1-semi-greediness with

$\mathcal B$2

Thus the weighted equivalence is quantitative as well as qualitative (Berná, 2019).

The theory also extends beyond Schauder bases to Markushevich bases. In that setting an M-basis is a fundamental minimal system whose biorthogonal functionals are total. Weak $\mathcal B$3-greedy sets are defined by

$\mathcal B$4

and this leads to weak weight-semi-greedy and weak weight-almost-greedy notions. The weak weight-almost-greedy property is equivalent to weight almost-greediness, and Theorems 3.22 and 3.23 give conditions under which weak weight semi-greedy Markushevich bases are weight almost greedy. The same work shows that weight semi-greedy Markushevich bases are truncation quasi-greedy and $\mathcal B$5-superdemocratic, hence have the $\mathcal B$6-Property (A) (Berasategui et al., 2021).

4. Position within the greedy-type hierarchy

Semi-greedy bases sit inside a broader hierarchy of greedy-type conditions. Greedy bases compare the raw greedy projection $\mathcal B$7 with $\mathcal B$8. Almost-greedy bases compare $\mathcal B$9 with $C_{sg}>0$0. Semi-greedy bases instead compare the Chebyshev improvement on the greedy support with $C_{sg}>0$1. Quasi-greedy bases ask only for boundedness or convergence of the thresholding greedy approximants (Berná, 2018, Berasategui et al., 2022).

A useful clarification comes from the theory of partially greedy bases. A basis is partially greedy if

$C_{sg}>0$2

and strongly partially greedy if

$C_{sg}>0$3

The paper on strengthened partial greediness introduces consecutive almost greedy (CAG) bases by replacing initial segments with arbitrary intervals, and proves that CAG is equivalent to almost-greediness. It also defines super-strong partially greedy bases; for Schauder bases, partially greedy, strong partially greedy, and super-strong partially greedy coincide, but for general bases super-strong partially greedy is strictly stronger than strong partially greedy (Berasategui et al., 2022).

These results sharply separate the various intermediate notions. In particular, the modern equivalence theorems show that semi-greedy is not merely adjacent to almost-greedy; in the Banach-space Schauder setting, and in the Markushevich settings cited above, it is the same class. A common misconception is therefore to treat semi-greedy as permanently distinct from almost-greedy. The earlier finite-cotype theorem and the later general equivalence theorem show that this distinction disappears under the standard hypotheses of the subject (Berná, 2018, Berná, 2019).

5. Restricted admissibility, Schreier families, and lower-order refinements

A different direction restricts the competitor supports rather than the greedy support. If $C_{sg}>0$4 is hereditary, a basis is $C_{sg}>0$5-greedy when

$C_{sg}>0$6

and $C_{sg}>0$7-almost-greedy when

$C_{sg}>0$8

These notions are characterized by $C_{sg}>0$9-unconditionality together with $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$0-disjoint democracy in the greedy case, and by quasi-greediness together with $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$1-disjoint democracy in the almost-greedy case (Beanland et al., 2022).

For Schreier families $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$2, this yields a strict hierarchy: $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$3 and none of these implications can be reversed. Moreover, for each countable ordinal $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$4, there exists a basis that is $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$5-greedy but not $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$6-greedy (Beanland et al., 2022). This restricted-family formalism is directly relevant to semi-greedy ideas, because it isolates how much admissible-support structure is needed for greedy approximation to remain effective.

The 2025 paper on lower-order refinements pushes this further by separating unconditionality and democracy levels. A basis is $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$7-quasi-greedy if it is quasi-greedy, $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$8-unconditional but not $$\|x-\mathcal C\mathcal G_m(x)\|\le C_{sg}\,\sigma_m(x),\qquad \forall x,\ \forall m,$$9-unconditional, and $\mathcal B=(e_n)$00-democratic but not $\mathcal B=(e_n)$01-democratic. It constructs $\mathcal B=(e_n)$02-quasi-greedy bases for all $\mathcal B=(e_n)$03, except the previously solved $\mathcal B=(e_n)$04 case (Beanland et al., 7 Apr 2025). That paper explicitly states that it does not define or mention semi-greedy bases, but it places semi-greedy behaviour in a natural broader framework. A plausible implication is that semi-greedy phenomena can be refined by replacing global democracy or unconditionality with Schreier-level analogues, thereby producing ordinally graded versions of the approximation properties usually associated with semi-greedy bases (Beanland et al., 7 Apr 2025).

6. Quantitative behavior, examples, and open directions

Because almost-greedy and semi-greedy coincide in the Banach-space Schauder setting, quantitative constructions of almost-greedy bases automatically supply semi-greedy examples. In particular, the paper on highly conditional almost-greedy and quasi-greedy bases constructs almost-greedy, hence semi-greedy, bases with extremal conditionality behaviour: in non-superreflexive classical spaces it produces examples with

$\mathcal B=(e_n)$05

while in superreflexive spaces it produces, for every $\mathcal B=(e_n)$06, almost-greedy bases with

$\mathcal B=(e_n)$07

These examples show that semi-greedy bases can be highly conditional, up to the sharp logarithmic constraints imposed by quasi-greediness (Albiac et al., 2018).

Weighted semi-greedy bases also interact strongly with the geometry of the weight. When $\mathcal B=(e_n)$08, the weighted theory can force $\mathcal B=(e_n)$09-type behaviour. The weighted almost-greedy paper proves, for example, that if $\mathcal B=(e_n)$10 and a basis is $\mathcal B=(e_n)$11-almost greedy, then every tail subsequence is uniformly equivalent to the canonical basis of $\mathcal B=(e_n)$12, and every subsequence has a further subsequence equivalent to $\mathcal B=(e_n)$13. Since $\mathcal B=(e_n)$14-almost-greedy and $\mathcal B=(e_n)$15-semi-greedy coincide under finite cotype and, in the Schauder setting, $\mathcal B=(e_n)$16-semi-greedy and $\mathcal B=(e_n)$17-almost-greedy are equivalent in arbitrary Banach spaces, these phenomena are part of the weighted semi-greedy landscape as well (Dilworth et al., 2018, Berná, 2019).

Not every nearby structural condition implies semi-greediness. Bidemocratic bases, for example, retain a strong unconditionality flavor, but they need not be quasi-greedy; for each $\mathcal B=(e_n)$18, $\mathcal B=(e_n)$19 has a bidemocratic basis which is not quasi-greedy. Since semi-greedy implies quasi-greedy in the settings discussed above, bidemocracy alone does not force semi-greedy behaviour (Albiac et al., 2021).

Several open directions remain explicit in the cited literature. In the weighted Markushevich setting, open questions ask whether every $\mathcal B=(e_n)$20-semi-greedy Markushevich basis with $\mathcal B=(e_n)$21 must be quasi-greedy, whether one can bound $\mathcal B=(e_n)$22-superdemocracy and truncation quasi-greedy constants directly from the semi-greedy constant in that regime, and whether $\mathcal B=(e_n)$23-$\mathcal B=(e_n)$24-semi-greediness is stable under equivalent weights (Berasategui et al., 2021). In the Schreier-refined setting, the open problem

$\mathcal B=(e_n)$25

suggests a corresponding unresolved region for any future ordinal hierarchy of semi-greedy bases (Beanland et al., 7 Apr 2025). Taken together, these problems indicate that the classical Banach-space theory of semi-greedy bases is structurally complete, while weighted, weak, and admissibility-restricted versions continue to generate new phenomena and unresolved classification questions.