Strong Partially Greedy Bases: Theory & Extensions

• Strong partially greedy bases are defined by controlling the thresholding greedy algorithm error via the best residual from initial segments, linking quasi-greediness with conservativeness.
• Lebesgue-type parameters and the constant-1 case offer quantitative insights into the approximation performance, highlighting conditions for optimal greedy behavior.
• Sequence-dependent and weighted extensions generalize the classical theory, revealing precise geometric conditions through prescribed gap patterns and broader greedy sums.

Strong partially greedy bases are bases for which the Thresholding Greedy Algorithm (TGA) is controlled, up to a uniform constant, by the best residual obtained from initial coordinate segments. In the now standard formulation, a basis $B=(e_n)$ is strong partially greedy if there exists $C>0$ such that for every vector $x$, every $m\in\mathbb N$, and every greedy set $A_m(x)$ of size $m$,

$\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$

where $G_m(x)$ is the $m$-term greedy approximant and $S_k$ is the $C>0$0-th partial sum operator. This places the notion between the fully symmetric almost-greedy theory and the one-sided partially greedy theory, and it admits a structural characterization in terms of quasi-greediness and conservativeness (Berasategui et al., 2020, Berasategui et al., 2023).

1. Classical formulation and structural characterization

The TGA begins with a basis $C>0$1, biorthogonal functionals $C>0$2, and the coordinate projections

$C>0$3

A set $C>0$4 with $C>0$5 is greedy for $C>0$6 if

$C>0$7

The corresponding greedy approximant is

$C>0$8

In the classical strong partially greedy inequality, the greedy error is compared not with arbitrary $C>0$9-term approximants, but with the best residual among partial sums $x$0. This is the “strong” form because the comparison is taken over all $x$1, rather than a single prescribed index. In the Banach-space Markushevich setting and in the $x$2-Banach setting, the central structural theorem is that strong partial greediness is equivalent to quasi-greediness plus conservativeness (Berasategui et al., 2020, Berasategui et al., 2023).

The conservativeness condition is one-sided. It requires a constant $x$3 such that

$x$4

where $x$5 means that the smaller set lies entirely to the left of the larger one. In the Markushevich framework, the same class is also characterized by quasi-greediness together with superconservativeness, and by quasi-greediness together with partial greediness (Berasategui et al., 2020).

This characterization is the basic organizing principle of the subject. It shows that strong partial greediness is not defined merely by a comparison estimate for greedy errors: it is exactly the conjunction of TGA stability, encoded by quasi-greediness, and a left-to-right size comparison property on basis blocks, encoded by conservativeness.

2. Lebesgue-type parameters and the constant-$x$6 case

A quantitative version of the theory is expressed through strong residual errors and Lebesgue-type parameters. For a semi-normalized Markushevich basis, the strong residual error is

$x$7

and the strong residual Lebesgue-type parameter $x$8 is the smallest constant such that

$x$9

for every $m\in\mathbb N$0 and every greedy operator $m\in\mathbb N$1. A basis is strong partially greedy precisely when $m\in\mathbb N$2 (Berasategui et al., 2020).

The main quantitative estimates relate $m\in\mathbb N$3 to quasi-greedy and conservative parameters. Among the basic inequalities are

$m\in\mathbb N$4

$m\in\mathbb N$5

and, in particular, if the basis is $m\in\mathbb N$6-quasi-greedy, then

$m\in\mathbb N$7

The parameters $m\in\mathbb N$8 are the usual quasi-greedy Lebesgue constants, while $m\in\mathbb N$9 are the superconservative parameters. The same paper also proves

$A_m(x)$0

linking $A_m(x)$1 to the partially symmetric for largest coefficients constants $A_m(x)$2 (Berasategui et al., 2020).

The extremal case $A_m(x)$3 for all $A_m(x)$4 is especially rigid. A basis is $A_m(x)$5-strong partially greedy if and only if it is $A_m(x)$6-PSLC, where PSLC denotes partial symmetry for largest coefficients. This is further equivalent to concrete pointwise monotonicity conditions: one condition compares $A_m(x)$7 with $A_m(x)$8 whenever $A_m(x)$9 dominates the coefficients of $m$0, and another compares $m$1 with $m$2 for $m$3 and $m$4 (Berasategui et al., 2020).

The constant-$m$5 theory also clarifies what strong partial greediness does not imply. There exists a normalized $m$6-unconditional Schauder basis that is $m$7-PSLC but not democratic. Consequently, $m$8-strong partially greedy does not imply democratic, and therefore does not imply $m$9-almost greedy (Berasategui et al., 2020).

3. Position within greedy-type approximation theory

Strong partially greedy bases belong to a hierarchy of greedy-type notions defined by the comparison class used on the right-hand side of the TGA error estimate. Greedy bases compare $\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$0 with best unrestricted $\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$1-term approximation; almost-greedy bases compare with best $\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$2-term coordinate projections; partially greedy and strong partially greedy bases compare with coordinate information coming from the initial segment structure (Berná et al., 2021, Berasategui et al., 2022).

Within this hierarchy, strong partial greediness is weaker than almost greediness in general, because the comparison is only with partial sums $\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$3, not with arbitrary $\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$4-term projections (Berasategui et al., 2023). It always implies partial greediness in the sense

$\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$5

but the two notions coincide for Schauder bases. In the Markushevich setting, this coincidence is replaced by the statement that strong partial greediness is the correct analogue of partial greediness, and it is still characterized by quasi-greediness plus conservativeness (Berasategui et al., 2020, Berasategui et al., 2022).

A further strengthening is the super-strong partially greedy property, defined by

$\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$6

where $\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$7 is the best approximation by arbitrary linear combinations supported on $\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$8. For Schauder bases, super-strong partial greediness is equivalent to strong partial greediness and to ordinary partial greediness, because one has

$\|x-G_m(x)\|\le C\inf_{0\le k\le m}\|x-S_k(x)\|,$9

For general bases, however, the equivalence fails: there exists a rearranged conditional almost greedy basis that is strong partially greedy but not super-strong partially greedy (Berasategui et al., 2022).

The terminology has also evolved. A 2018 characterization of partially greedy bases introduced a stronger-looking constrained inequality

$G_m(x)$0

where the competitor is supported strictly before the greedy block, and proved that this property is equivalent to ordinary partial greediness. Later work reserved the term “strong partially greedy” for the residual inequality involving $G_m(x)$1 (Dilworth et al., 2018, Berasategui et al., 2020).

4. Sequence-dependent strong partial greediness

A major extension replaces the standard initial segment $G_m(x)$2 by an arbitrary increasing sequence $G_m(x)$3. For such a sequence, one defines

$G_m(x)$4

and says that $G_m(x)$5 is $G_m(x)$6 strong partially greedy$G_m(x)$7 if

$G_m(x)$8

When $G_m(x)$9, this is exactly the classical strong partially greedy property (Chu, 2022).

The corresponding structure theorem is a direct analogue of the classical one. For a basis $m$0, the following are equivalent: $m$1 is $m$2 strong partially greedy$m$3; $m$4 is quasi-greedy and $m$5 PSLC$m$6; $m$7 is quasi-greedy and $m$8 superconservative$m$9; and $S_k$0 is quasi-greedy and $S_k$1 conservative$S_k$2 (Chu, 2022).

This sequence-dependent theory is tail-invariant. If two increasing sequences $S_k$3 and $S_k$4 have finite symmetric difference, then

$S_k$5

Conversely, if the difference set is infinite, the properties can differ. Thus the property depends only on the tail-equivalence class of the sequence (Chu, 2022).

The Lebesgue-type constants in this setting retain a sharp form. Writing $S_k$6 for the best constant in

$S_k$7

the theory gives upper bounds such as

$S_k$8

and culminates in the exact identification

$S_k$9

where $C>0$00 is the $C>0$01 PSLC$C>0$02 constant. The constant-$C>0$03 case again collapses to symmetry: a basis is $C>0$04-$C>0$05 strong partially greedy$C>0$06 if and only if it is $C>0$07-$C>0$08 PSLC$C>0$09 (Chu, 2022).

5. Sequential families, prescribed gaps, and the bounded–unbounded dichotomy

A different extension fixes a positive integer sequence $C>0$10 and builds a family of finite sets

$C>0$11

When $C>0$12, this family is exactly the family of finite intervals; when $C>0$13, it consists of finite arithmetic progressions of step $C>0$14 (Berasategui et al., 2023).

For a general family $C>0$15, the paper defines $C>0$16-strong partial greediness by the estimate

$C>0$17

valid for all $C>0$18, all $C>0$19, and every greedy set $C>0$20. This compares the greedy error with projections onto a restricted family of admissible coordinate sets rather than with ordinary initial segments (Berasategui et al., 2023).

The exact analogue of the classical structure theorem survives in this family-based setting. A basis is $C>0$21-strong partially greedy if and only if it is quasi-greedy and $C>0$22-strong disjoint superconservative, and this is also equivalent to quasi-greediness plus $C>0$23-strong disjoint conservative. The disjoint conservative condition requires

$C>0$24

whenever $C>0$25, $C>0$26, $C>0$27, and $C>0$28 for some $C>0$29; the superconservative version allows arbitrary signs (Berasategui et al., 2023).

For the concrete family $C>0$30, the decisive issue is whether the gap sequence $C>0$31 is bounded. Theorem 2.6 states that the following are equivalent:

1. $C>0$32 is bounded.
2. A basis is $C>0$33-strong partially greedy if and only if it is strong partially greedy.

If $C>0$34 is unbounded, the new property can be strictly weaker: there exists a basis that is $C>0$35-strong partially greedy but not strong partially greedy. The bounded–unbounded dichotomy is therefore exact (Berasategui et al., 2023).

This suggests a precise geometric principle. When the prescribed family $C>0$36 has uniformly bounded gaps, its sequential restriction does not alter the classical class; when the gaps are unbounded, the restriction changes the admissible comparison geometry enough to produce genuinely different strong greedy-type behavior.

6. Larger greedy sums, gaps, and weighted extensions

Another direction enlarges the greedy approximant itself. Writing

$C>0$37

and

$C>0$38

a basis is $C>0$39-partially greedy, for $C>0$40, if

$C>0$41

The correct structural replacement for conservativeness is $C>0$42-max conservativeness, defined by

$C>0$43

whenever

$C>0$44

The main theorem states that a basis is $C>0$45-partially greedy if and only if it is quasi-greedy and $C>0$46-max conservative. At $C>0$47, this reduces to the classical characterization of strong partially greedy bases by quasi-greediness and conservativeness (Chu, 2022).

For $C>0$48, the class expands strictly. For each $C>0$49, there exists an unconditional basis that is $C>0$50-partially greedy but not strong partially greedy. There is also an unconditional basis that is not strong partially greedy with constant $C>0$51, but is $C>0$52-partially greedy with constant $C>0$53 for some $C>0$54. The enlarged-greedy-sum formulation therefore preserves the classical theory only at $C>0$55 (Chu, 2022).

The arbitrary-sequence theory admits two further extensions. First, one can restrict the inequality to a subsequence $C>0$56 of greedy orders. Under bounded quotient gaps, an $C>0$57-$C>0$58 strong partially greedy$C>0$59 basis is characterized by quasi-greediness plus $C>0$60-order-$C>0$61 superconservativeness$C>0$62 in the Schauder and $C>0$63-Schauder setting; under bounded additive gaps, the same remains valid for $C>0$64-Schauder Markushevich bases. If the additive gaps are arbitrarily large, there are examples of $C>0$65-$C>0$66 strong partially greedy$C>0$67 bases that are not $C>0$68 conservative$C>0$69, and hence not $C>0$70 strong partially greedy$C>0$71 (Chu, 2022).

Second, the theory can be weighted. For an arbitrary weight $C>0$72, one has the equivalence

$C>0$73

and this is also equivalent to quasi-greediness plus weighted superconservativeness or weighted conservativeness. A notable corollary is that a basis is quasi-greedy if and only if it is $C>0$74-$C>0$75 strong partially greedy$C>0$76 for some weight $C>0$77. For sequence weights $C>0$78, if

$C>0$79

then the weighted and unweighted $C>0$80 strong partially greedy$C>0$81 properties coincide; if $C>0$82 or $C>0$83, the theory yields canonical $C>0$84-type subsequences (Chu, 2022).

Strong partially greedy bases thus form a stable core notion with a precise classical characterization, but they also support a large family of controlled deformations. Sequence restrictions, prescribed gap patterns, larger greedy sums, and weights can either reproduce the classical class or produce strictly different ones, depending on the exact geometry imposed on admissible comparison sets and residuals.