A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1

Pablo M. Berná; Andrea García; David González

doi:10.1515/dema-2024-0047

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A note on 1-semi-greedy bases in p-Banach spaces with 0 < p ≤ 1

Pablo M. Berná , Andrea García und David González

Veröffentlicht/Copyright: 4. Dezember 2024

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Abstract

The purpose of this article is to discuss about the so-called semi-greedy bases in p -Banach spaces. Specifically, we will review existing results that characterize these bases in terms of almost-greedy bases, and, also, we analyze quantitatively the behavior of certain constants. As new results, by avoiding the use of certain classical results in p -convexity, we aim to quantitatively improve specific bounds for bi-monotone 1-semi-greedy bases.

Keywords: non-linear approximation; greedy algorithm; semi-greedy bases

MSC 2010: 41A65; 46B15

1 Introduction and background

In [1], Konyagin and Temlyakov introduced the thresholding Greedy algorithm (TGA), where, given a basis in a particular space X (a p -Banach or a quasi-Banach space) and one element f ∈ X , the algorithm selects the largest coefficients in modulus of f with respect to the given basis. To formalize the algorithm, we introduce the notion of p -Banach spaces and bases (see, e.g., [1–3]).

1.1 p -Banach spaces and bases

Let X be a vector space over the field F = R or C with a quasi-norm ‖ ⋅ ‖ : X → [ 0 , + ∞ ) , where the quasi-norm is an application verifying the following three conditions:

‖ f ‖ ≥ 0 for all f ∈ X and ‖ f ‖ = 0 if and only if f = 0 .
‖ λ f ‖ = ∣ λ ∣ ‖ f ‖ for all f ∈ X and for all λ ∈ F .
There is κ ≥ 1 such that for every f , g ∈ X ,
‖ f + g ‖ ≤ κ ( ‖ f ‖ + ‖ g ‖ ) .
Given 0 < p ≤ 1 , a p-norm is a map ‖ ⋅ ‖ : X → [ 0 , + ∞ ) satisfying conditions (a), (b), and
for every f , g ∈ X ,
‖ f + g ‖ p ≤ ‖ f ‖ p + ‖ g ‖ p .

Of course, (d) implies (c) with κ = 2 1 ⁄ p − 1 . If ‖ ⋅ ‖ is a quasi-norm (resp. a p -norm) on X such that defines a complete metrizable topology, then X is called a quasi-Banach space (resp. a p-Banach space) and thanks to the Aoki-Rolewicz’s theory (see [4,5]), we know that a quasi-Banach space is p -convex for some p ∈ ( 0 , 1 ] , that is, there is a constant C > 0 such that

∑ i = 1 m f i p ≤ C ∑ i = 1 m ‖ f i ‖ p , ∀ m ∈ N , ∀ f i ∈ X .

Then, a quasi-Banach space becomes p -Banach under a suitable renorming.

Now, given a quasi-Banach (or a p -Banach) space, a Schauder basis ℬ = ( x n ) n ∈ N is any collection of vectors in the space such that for every f ∈ X , there is a unique sequence ( a j ( f ) ) j ∈ N such that

lim m → + ∞ f − ∑ j = 1 m a j ( f ) x j = 0 .

In other words, for every f ∈ X , we have

f = ∑ j = 1 ∞ a j ( f ) x j .

Moreover, given a Schauder basis, we have the collection of biorthogonal functionals (called dual basis) ℬ * = ( x n * ) n ∈ N ⊆ X * such that x n * ( x m ) = δ n , m . Using these functionals,

x n * ∑ j = 1 ∞ a j ( f ) x j = a n ( f ) ,

so every element could be represented by

f = ∑ j = 1 ∞ x j * ( f ) x j .

Equivalently, it is well known that ℬ = ( x n ) n ∈ N is a Schauder basis if and only if the partial sums are bounded, that is,

K b ≔ sup m ‖ S m ‖ < ∞ ,

where, given m ∈ N and f ∈ X , S m ( f ) is the mth partial sum of f:

S m [ ℬ , X ] ( f ) = S m ( f ) ≔ ∑ j = 1 m x j * ( f ) x j ,

and K b is called the basis constant. To know more about Schauder bases in quasi-Banach spaces, we cite the book [6].

Definition 1.1

We say that a Schauder basis ℬ in a p -Banach space (or quasi-Banach space) X is monotone if

(1.1) ‖ S m ( f ) ‖ ≤ ‖ f ‖ , ∀ m ∈ N , ∀ f ∈ X .

Moreover, we say that ℬ is bi-monotone if (1.1) holds and

(1.2) ‖ f − S m ( f ) ‖ ≤ ‖ f ‖ , ∀ m ∈ N , ∀ f ∈ X .

A stronger condition than Schauder is unconditionality: Given a finite set of indices A ⊂ N and f ∈ X , the projection operator P A is defined as

P A [ ℬ , X ] ( f ) = P A ( f ) ≔ ∑ n ∈ A x n * ( f ) x n .

Using that operator, we say that ℬ is K -unconditional if

K ≔ sup ∣ A ∣ < ∞ ‖ P A ‖ < ∞ .

Remark 1.2

There are several examples of bi-monotone bases. For instance, any 1-unconditional basis in a Banach space is bi-monotone. A particular basis that is bi-monotone but not unconditional could be the basis that appears in [7, Proposition 6.10].

From now on, we will consider that X is a p -Banach space for 0 < p ≤ 1 and ℬ is a semi-normalized Schauder basis in X .

1.2 TGA

In 1999, in [1], the authors introduced the TGA ( G m ) m = 1 ∞ . To define the algorithm, we take an element f ∈ X and m ∈ N . A greedy sum of order m is a sum of the form

G m [ ℬ , X ] ( f ) = G m ( f ) ≔ ∑ n ∈ G x n * ( f ) x n ,

where G is a finite set called a greedy set of f of order m and verifies that ∣ G ∣ = m and

min n ∈ G ∣ x n * ( f ) ∣ ≥ max n ∉ G ∣ x n * ( f ) ∣ .

The collection ( G m ) m ∈ N is the TGA. Of course, any greedy sum is a particular projection since given a greedy sum of order m of f ∈ X , using the corresponding greedy set G ,

G m ( f ) = P G ( f ) .

To talk about the convergence of the algorithm, we need the following definition.

Definition 1.3

We say that a basis ℬ is quasi-greedy if for every f ∈ X , there is C > 0 such that

(1.3) ∥ f − P G ( f ) ∥ ≤ C ‖ f ‖ ,

whenever G is a finite greedy set of f .

Thanks to a famous result of Wojtaszczyk (see [2,8]), we know that a basis is quasi-greedy if and only if

lim m → + ∞ ∥ f − G m ( f ) ∥ = 0 , ∀ f ∈ X .

Remark 1.4

Since every greedy sum is a particular projection, if the basis is unconditional, automatically, the basis is quasi-greedy.

Although for quasi-greedy bases we have the convergence of the algorithm, we can ask when the algorithm produces the best possible approximation, that is, when ‖ f − G m ( f ) ‖ is comparable with the best mth possible error in the approximation of f

σ m [ ℬ , X ] ( f ) = σ m ( f ) ≔ inf y ∈ X : ∣ supp ( y ) ∣ ≤ m ‖ f − y ‖ ,

where, for a general g ∈ X , supp ( g ) = { n ∈ N : x n * ( g ) ≠ 0 } .

In [1], we find greedy bases, where a basis is greedy if there is C > 0 such that for every element f ∈ X ,

‖ f − P G ( f ) ‖ ≤ C σ ∣ G ∣ ( f ) ,

whenever G is a finite greedy set of f . Of course, every greedy basis is quasi-greedy since we can take y = 0 in the definition of the error σ m ( f ) .

Here, we study an intermediate notion between quasi-greediness and greediness, the so-called almost-greedy bases introduced in [9].

Definition 1.5

We say that ℬ is almost-greedy if for every f ∈ X , there is C > 0 such that

(1.4) ∥ f − P G ( f ) ∥ ≤ C inf B ⊂ N : ∣ B ∣ ≤ ∣ G ∣ ‖ f − P B ( f ) ‖ ,

whenever G is a finite greedy set of f . The smallest constant verifying (1.4) is denoted by C a l = C a l [ ℬ , X ] and we say that ℬ is C a l -almost-greedy.

The existence of almost-greedy and non-greedy bases is well known (see [10,11]). In fact, if ℬ is almost-greedy in a Banach space, for every f and every finite greedy set G ,

‖ f − P G ( f ) ‖ ≲ ln ( ∣ G ∣ ) σ ∣ G ∣ ( f ) ,

and the logarithm is an optimal bound [12].

1.3 Thresholding Chebyshev Greedy algorithm (TCGA)

In 2003, Dilworth et al. introduced in [13] the notion of semi-greedy bases as an enhancement of the TGA to improve the rate of convergence. To define these bases, we need to introduce the TCGA: let f ∈ X and consider now G a finite greedy set of f . Then, a Chebyshev greedy sum of order m ≔ ∣ G ∣ of f is any element CG m [ ℬ , X ] ( f ) = CG m ( f ) of the form ∑ n ∈ G a n x n ∈ span { x n : n ∈ G } such that

‖ f − CG m ( f ) ‖ = min f − ∑ n ∈ G a n x n : a n ∈ F ∀ n ∈ G .

Definition 1.6

We say that ℬ is semi-greedy if for every f ∈ X , there is C > 0 such that

(1.5) min f − ∑ n ∈ G a n x n : a n ∈ F ∀ n ∈ G ≤ C σ ∣ G ∣ ( f ) ,

whenever G is a finite greedy set of f . The smallest constant verifying (1.5) is denoted by C s g = C s g [ ℬ , X ] and we say that ℬ is C s g -semi-greedy.

In the last few years, some authors have studied that these bases are equivalent to almost-greedy bases. Concretely, we can find the following results:

In [13], the authors proved that if ℬ is a Schauder basis in a Banach space with finite cotype, then ℬ is almost-greedy if and only if the basis is semi-greedy.
In [14], the author improved the last result by removing the condition of finite cotype.
In [15], the authors improved the result from [14] by extending the condition of Schauder bases to semi-normalized Markushevich bases in Banach spaces.
In [3], the authors showed the equivalence between almost-greedy and semi-greedy bases in the context of Schauder bases in p -Banach spaces using intermediate properties as quasi-greediness and democracy.

To introduce the notion of democracy, we need the indicator sums: let A ⊂ N be a finite set of indices and ε ∈ ℰ A a sign, where ℰ A is defined as follows:

ℰ A = { ε = ( ε n ) n ∈ A : ∣ ε n ∣ = 1 ∀ n ∈ A } .

An indicator sum is any finite combination of the form

1 ε , A [ ℬ , X ] = 1 ε , A ≔ ∑ n ∈ A ε n x n ,

where ε ∈ ℰ A and A is a finite set of indices. When ε ≡ 1 , we use the notation 1 A .

Definition 1.7

We say that ℬ is super-democratic if for every pair of finite sets A , B with ∣ A ∣ ≤ ∣ B ∣ and every choice of signs ε ∈ ℰ A and η ∈ ℰ B , there is C > 0 such that

(1.6) ‖ 1 ε , A ‖ ≤ C ‖ 1 η , B ‖ .

The smallest constant verifying (1.6) is denoted by Δ s = Δ s [ ℬ , X ] and we say that ℬ is Δ s -super-democratic. When (1.6) is satisfied for ε ≡ η ≡ 1 , we say that ℬ is democratic.

Thanks to the combination of the results proved in [2,3], we have the following result in the context of p -Banach spaces for Schauder bases.

Theorem 1.8

Let ℬ be a Schauder basis with basis constant K b in a p-Banach space with 0 < p ≤ 1 . If ℬ is C s g -semi-greedy, then the basis is C a l -almost-greedy with

(1.7) C a l ≤ C 1 ( p ) ( 1 + [ A p C 2 ( p ) C 1 ( p ) η p ( C 1 ( p ) ) ] p ) 1 ⁄ p ,

where C 1 ( p ) = K b C s g ( 1 + ( 1 + K b ) p C s g p ) 1 ⁄ p , C 2 ( p ) = K b ( 1 + K b ) C s g 2 ,

A p = ( 2 p − 1 ) − 1 ⁄ p , 0 < p ≤ 1

and

η p ( u ) = min 0 < t < 1 ( 1 − t p ) − 1 ⁄ p ( 1 − ( 1 + A p − 1 u − 1 t ) − p ) − 1 ⁄ p .

We want to focus our attention on the case C s g = 1 . The study of 1-greedy-like bases started in 2006 [16] where the authors characterized 1-greedy bases introducing the so-called Property (A) (or symmetry for largest coefficients). After this year, we can find some articles discussing the 1-almost-greedy and 1-quasi-greedy bases, but in no case is there mention of the 1-semi-greedy bases. In this article, we aim to combine known techniques about the characterization of semi-greediness to produce a new quantitative version for bounding certain constants in the general case of p -Banach spaces using bi-monotone and 1-semi-greedy bases. This new quantitative version implies a more refined bound for the constant of almost-greedy bases using semi-greediness in the general context of p -Banach spaces. Furthermore, the previous bound (1.7) is established by leveraging the concept of p -convexity, which introduces the constant A p in p -Banach spaces, as we can see in [2]. In this work, we will avoid the use of this property, and we give an improvement of the constant C a l for bi-monotone 1-semi-greedy bases. Concretely, the main result that we show is the following one.

Theorem 1.9

Let ℬ be a bi-monotone basis in a p-Banach space with 0 < p ≤ 1 . If ℬ is C s g -semi-greedy, then the basis is 3 2 ⁄ p C s g 5 -almost-greedy. In particular, for 1-semi-greedy bases, C a l ≤ 3 2 ⁄ p .

The structure of this article is as follows: In Section 2, we prove some technical and necessary results to prove the main theorem. In Section 3, we show the proof of Theorem 1.9, and in Section 4, we discuss the bound proved in Theorem 1.9 in connection with some known results in the literature.

2 Some technical results about the TCGA

Proposition 2.1

Let ℬ be a C s g -semi-greedy Schauder basis in a p-Banach space X with 0 < p ≤ 1 . Let f , g ∈ X with A = supp ( f ) , B = supp ( g ) , ∣ supp ( f ) ∣ ≤ ∣ supp ( g ) ∣ < ∞ , and min n ∈ B ∣ x n * ( g ) ∣ ≥ max j ∈ A ∣ x j * ( f ) ∣ .

If the basis is monotone and supp ( f ) < supp ( g ) , then
‖ f ‖ ≤ C s g ‖ g ‖ .
If the basis is bi-monotone and A ∩ B = ∅ , then
‖ f ‖ ≤ C s g 2 ‖ g ‖ .

Proof

We start with (a). We construct the element

h ≔ f + g .

Using the hypothesis

min n ∈ B ∣ x n * ( g ) ∣ ≥ max j ∈ A ∣ x j * ( f ) ∣ ,

B is a greedy set of h . Hence, applying the TCGA, we obtain some coefficients ( a n ) n ∈ B such that

h − CG m ( h ) = f + ∑ n ∈ B a n x n ,

where m ≔ ∣ B ∣ . Then,

‖ f ‖ ≤ A < B and ( 1.1 ) f + ∑ n ∈ B a n x n ≤ semi-greediness C s g σ m ( h ) ≤ C s g ‖ h − f ‖ = C s g ∥ g ∥ .

Hence, (a) is proved. Now, we show (b). For that, since we are playing with elements with finite support, we can take a set C such that C > A ∪ B , ∣ C ∣ = ∣ B ∣ , and hence, taking α ≔ min n ∈ B ∣ x n * ( g ) ∣ ≥ max j ∈ A ∣ x j * ( f ) ∣ and applying (a),

(2.1) ‖ f ‖ ≤ C s g ∥ α 1 C ∥ .

Now, we define the element

δ ≔ g + α 1 C .

It is clear now that B is a greedy set for δ , and then, applying the TCGA, there is a sequence ( b n ) n ∈ B such that

δ − CG k ( δ ) = ∑ n ∈ B b n x n + α 1 C ,

where k ≔ ∣ B ∣ . Thus,

(2.2) ∥ α 1 C ∥ ≤ ( 1.2 ) ∥ δ − CG k ( δ ) ∥ ≤ semi-greediness C s g σ k ( δ ) ≤ C s g δ − ∑ n ∈ C α x n = C s g ‖ g ‖ .

By (2.1) and (2.2), we conclude the result.□

Corollary 2.2

Any 1-semi-greedy and bi-monotone basis in a p-Banach space with 0 < p ≤ 1 is 1-super-democratic.

Proof

Consider A , B two finite sets and ε ∈ ℰ A and η ∈ ℰ B . Consider now a set C > A ∪ B with ∣ C ∣ = ∣ B ∣ . Applying (a) of Proposition 2.1,

‖ 1 ε , A ‖ ≤ ‖ 1 C ‖ .

Using now item (b) of Proposition 2.1,

‖ 1 C ‖ ≤ ‖ 1 η , B ‖ .

Thus, by both inequalities,

‖ 1 ε , A ‖ ≤ ‖ 1 η , B ‖ ,

so the basis is 1-super-democratic.□

Proposition 2.3

Let X be a p-Banach space with 0 < p ≤ 1 and ℬ a bi-monotone C s g -semi-greedy basis in X . Then, for every greedy set G of f with ∣ supp ( f ) ∣ < ∞ and ε ∈ ℰ G ,

‖ P G c ( f ) + α 1 ε , G ‖ ≤ 3 1 ⁄ p C s g 3 ‖ f ‖ ,

where α ≔ min n ∈ G ∣ x n * ( f ) ∣ .

Proof

Let f ∈ X with finite support and take G a greedy set of f, α ≔ min n ∈ G ∣ x n * ( f ) ∣ and ε ∈ ℰ G . Define now the element

h ≔ P G c ( f ) + α 1 ε , G + α 1 C ,

where C > supp ( f ) and ∣ C ∣ = ∣ G ∣ . Hence, it is clear that C is a greedy set with cardinality m ≔ ∣ C ∣ of h . Applying the TCGA, there exists a sequence of coefficients ( a n ) n ∈ C where

h − CG m ( h ) = P G c ( f ) + α 1 ε , G + ∑ n ∈ C a n x n .

Then,

(2.3) ‖ P G c ( f ) + α 1 ε , G ‖ p ≤ ( 1.1 ) P G c ( f ) + α 1 ε , G + ∑ n ∈ C a n x n p = ‖ h − CG m ( h ) ‖ p ≤ semi-greediness C s g p ( σ m ( h ) ) p ≤ C s g p ‖ h + P G ( f ) ‖ p = C s g p ‖ f + α 1 ε , G + α 1 C ‖ p ≤ C s g p ‖ f ‖ p + C s g p ( ‖ α 1 ε , G ‖ p + ‖ α 1 C ‖ p ) ≤ Proposition 2.1 C s g p ‖ f ‖ p + 2 C s g 2 p ‖ α 1 C ‖ p .

Now, take the element g ≔ f + α 1 C . Hence, G is a greedy set of g and then, applying the TCGA over G ,

g − CG m ( g ) = ∑ n ∈ G d n x n + P G c ( f ) + α 1 C ,

where the sequence ( d n ) n ∈ G is given by the TCGA. Thus,

(2.4) ‖ α 1 C ‖ p ≤ ( 1.2 ) ∑ n ∈ G d n x n + P G c ( f ) + α 1 C p = ‖ g − CG m ( g ) ‖ p ≤ semi-grediness C s g p ( σ m ( g ) ) p ≤ C s g p ‖ g − α 1 C ‖ p = C s g p ‖ f ‖ p .

Combining (2.3) and (2.4), we obtain that

□ ‖ P G c ( f ) + α 1 ε , G ‖ ≤ 3 1 ⁄ p C s g 3 ‖ f ‖ .

Proposition 2.4

Let X be a p-Banach space with 0 < p ≤ 1 and ℬ a bi-monotone 1-semi-greedy basis in X . For every f ∈ X with finite support,

‖ f ‖ ≤ 3 1 ⁄ p C s g 2 ‖ f − P A ( f ) + t 1 ε , B ‖ ,

whenever ∣ A ∣ ≤ ∣ B ∣ < ∞ , B ∩ supp ( f ) = ∅ , A ⊂ supp ( f ) , t ≥ max n ∈ supp ( f ) ∣ x n * ( f ) ∣ , and ε ∈ ℰ B .

Proof

Consider f ∈ X with finite support and t , A , B , ε as in the statement.

(2.5) ‖ f ‖ p ≤ ‖ f − P A ( f ) + t 1 ε , B ‖ p + ‖ P A ( f ) ‖ p + ‖ t 1 ε , B ‖ p .

Now, take an element g ≔ t 1 C ∈ X such that supp ( g ) > supp ( f ) ∪ B and ∣ B ∣ = ∣ C ∣ . Applying Proposition 2.1,

(2.6) ‖ P A ( f ) ‖ ≤ C s g ‖ g ‖ .

Also, applying the same result,

(2.7) ‖ t 1 ε , B ‖ ≤ C s g ‖ g ‖ .

Now, since supp ( g ) > supp ( f ) ∪ B , we can define the element

y ≔ f − P A ( f ) + t 1 ε , B + t 1 C ,

where one of the greedy sets of cardinality n ≔ ∣ B ∣ is B so, applying the TGCA,

y − CG n ( y ) = f − P A ( f ) + ∑ n ∈ B c n x n + t 1 C .

Hence,

(2.8) ‖ t 1 C ‖ ≤ ( 1.2 ) f − P A ( f ) + ∑ n ∈ B c n x n + t 1 C ≤ semi-greediness C s g σ n ( y ) ≤ ‖ y − t 1 C ‖ = C s g ‖ f − P A ( f ) + t 1 ε , B ‖ .

Hence, combining (2.6), (2.7), and (2.8) in (2.5), we obtain the result.□

3 Proof of Theorem 1.9

Proof of Theorem 1.9

Consider f ∈ X , G a finite greedy set of f and B ⊂ N such that ∣ B ∣ ≤ ∣ G ∣ . Applying Proposition 2.4 over the element f − P G ( f ) , we can obtain

(3.1) ‖ f − P G ( f ) ‖ ≤ 3 1 ⁄ p C s g 2 ‖ f − P G ( f ) − P B \ G ( f ) + t 1 ε , G \ B ‖ = 3 1 ⁄ p C s g 2 ‖ P ( G ∪ B ) c ( f − P B ( f ) ) + t 1 ε , G \ B ‖ ,

where ε ≡ sign { ( x n * ( f ) ) } and t = min n ∈ ( G \ B ) ∣ x n * ( f ) ∣ . Now, taking into account that if G is a greedy set for f , then G \ B is a greedy set for f − P B ( f ) , applying Proposition 2.3, we have

(3.2) ‖ P ( G ∪ B ) c ( f − P B ( f ) ) + t 1 ε , G \ B ‖ ≤ 3 1 ⁄ p C s g 3 ‖ f − P B ( f ) ‖ .

Combining (3.1) and (3.2), we conclude the proof for elements with finite support. To conclude the result for general elements f ∈ X , we can apply the density argument proved in [17, Lemma 3.7].□

4 A comparison with previously known results

As we have commented in the introduction, in [3], the authors proved that for Schauder bases with basis constant K b in general p -Banach spaces, if ℬ is C s g -semi-greedy, then

C a l ≤ C 1 ( p ) ( 1 + [ A p C 2 ( p ) C 1 ( p ) η p ( C 1 ( p ) ) ] p ) 1 ⁄ p ,

where C 1 ( p ) = K b C s g ( 1 + ( 1 + K b ) p C s g p ) 1 ⁄ p , C 2 ( p ) = K b ( 1 + K b ) C s g 2 ,

A p = ( 2 p − 1 ) − 1 ⁄ p , 0 < p ≤ 1

and

η p ( u ) = min 0 < t < 1 ( 1 − t p ) − 1 ⁄ p ( 1 − ( 1 + A p − 1 u − 1 t ) − p ) − 1 ⁄ p .

Since η p ( u ) ≈ u 1 ⁄ p (see [2, Remark 4.9]), in the case when the basis is bi-monotone and 1-semi-greedy, it is possible to derivate from the last bound that

(4.1) C a l ≲ 2 1 ⁄ p ( 1 + ( A p ) p 2 1 + 1 ⁄ p ) 1 ⁄ p ≕ C .

Using our Theorem 1.9, we have that C a l ≤ 3 2 ⁄ p , that it is a better constant for a general 0 < p ≤ 1 than (4.1) as Figure 1 shows.

$Figure 1 Comparation between 3 2 ⁄ p {3}^{2/p} and C {\bf{C}} of (4.1).$

Figure 1

Comparation between 3 2 ⁄ p and C of (4.1).

In the case of Banach spaces (i.e., a p -Banach space with p = 1 ), we can found in [14] some different estimates. Concretely, it is possible to deduce from this article that if ℬ is bi-monotone and C s g -semi-greedy, then the basis is almost-greedy with

C a l ≤ 3 C s g 2 ( 1 + 4 C s g 2 ) .

In particular, for C s g = 1 , we have that C a l ≤ 15 , where, in our case, using the bound of Theorem 1.9 for p = 1 , C a l ≤ 9 .

Hence, to conclude, in any possible case, our estimates for the case of bi-monotone and 1-semi-greedy bases are the best so far.

Acknowledgements

The researcher would like to thank the reviewers for the valuable comments and suggestions.

Funding information: P. M. Berná and D. González are partially supported by Grant PID2022-142202NB-I00 (Agencia Estatal de Investigación, Spain) and D. González was partially supported by ESI International Chair@ CEU-UCH.
Author contributions: All authors contributed in the same proportion to the elaboration of this article.
Conflict of interest: Dr. Pablo M. Berná is a member of the Editorial Board of the Demonstratio Mathematica but was not involved in the review process of this article.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to this article as no data sets were generated or analyzed during this study.

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Received: 2023-10-06

Revised: 2024-03-20

Accepted: 2024-07-09

Published Online: 2024-12-04

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https://doi.org/10.1515/dema-2024-0047

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non-linear approximation; greedy algorithm; semi-greedy bases

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