A functional characterization of almost greedy and partially greedy bases

In 2003, S. J. Dilworth et al. ([8]) introduced the notion of almost-greedy (resp. partially-greedy) bases. These bases were characterized in terms of quasi-greediness and democracy (resp. conservativeness). In this paper we will show a new functional characterization of these type of bases in general Banach spaces following the spirit of the characterization of greediness proved in [5].


Introduction and background
Assume that (X, · ) is a Banach space over the field F = R or C. Throughout the paper, we assume that B = (e n ) ∞ n=1 is a semi-normalized Markushevich basis, that is, there exists a unique sequence (e * n ) ∞ n=1 ⊂ X * such that • span(e n : n ∈ N) = X, Hereinafter, by a basis for X, we mean a semi-normalized Markushevich basis. Under these conditions, for each f ∈ X, we have that f ∼ ∞ n=1 e * n (f )e n where (e * n (f )) n ∈ c 0 . The support of f ∈ X is denoted by supp(f ), where supp(f ) = {n ∈ N : |e * n (f )| = 0}. Finally, we will use the following notation: X f in is the subspace of X with the elements with finite support, if f, g ∈ X, f · g = 0 means that supp(f ) ∩ supp(g) = ∅,f = (e * n (f )) n∈N and f ∞ = sup n∈N |e * n (f )|. Moreover, if A and B are finite sets of natural numbers, A < B means that max n∈A n < min j∈B j, P A is the projection operator, that is, P A (f ) = n∈A e * n (f )e n , and S k is the partial sum of order k, that is, S k (f ) = P {1,··· ,k} (f ).
In 1999, S. V. Konyagin and V. N. Temlyakov ( [11]) introduced one of the most studied algorithms in the field of non-linear approximation, the so called Thresholding Greedy Algorithm: for f ∈ X and m ∈ N, we define a greedy sum of order m as The collection (G m ) m∈N is the Thresholding Greedy Algorithm. As for every algorithm, one of the first question that we can ask to the audience is when the algorithm converges. To solve that question, S. V. Konyagin and V. N. Temlyakov introduced in [11] the notion of quasi-greediness. Definition 1.1. We say that B is quasi-greedy if there is a positive constant C such that The least constant verifying (1) is denoted by C q = C q [X, B] and we say that B is C q -quasigreedy.
Although this definition only talks about the boundedness of the greedy sums, P. Wojtaszczyk showed in [12] that quasi-greediness is equivalent to the convergence of the algorithm.
Then, quasi-greediness is the minimal condition in the convergence of the algorithm, but we are interested in others types of convergence. For instance, when does the algorithm produce the best possible approximation? To study this question, in [11], the authors introduced the notion of greediness: a basis B is greedy if there is a positive constant C g such that n∈B a n e n : a n ∈ F, |B| ≤ m}, ∀m ∈ N, ∀f ∈ X.
There are several examples of greedy bases, for instance the canonical basis in the spaces ℓ p with 1 ≤ p < ∞, the Haar system in L p ((0, 1)) with 1 < p < ∞ or the trigonometric system in L 2 (T). To study greedy bases, S. V. Konyagin and V. N. Temlyakov gave a characterization in terms of unconditional and democratic bases, where a basis is unconditional if the projection operator is uniformly bounded, that is, there is K > 0 such that, for any finite set A, Consider A a finite set and define the set of the collection of signs in A, E A = {ε = (ε n ) n∈A : |ε n | = 1}, and take the indicator sum If ε ≡ 1, we will use the notation 1 A . Definition 1.3. [1,4,8,9] We say that B is symmetric for largest coefficients if there is a positive constant C such that for any pair of sets |A| ≤ |B| < ∞, A ∩ B = ∅, for any f ∈ X such that supp(f ) ∩ (A ∪ B) = ∅, |e * n (f )| ≤ 1 for all n ∈ N and for any choice of signs ε ∈ E A , ε ′ ∈ E B . The least constant verifying (2) is denoted by ∆ = ∆[X, B] and we say that B is ∆-symmetric for largest coefficients. If (2) is satisfied with the extra condition that A < supp(f ) ∪ B, then we say that B is partially symmetric for largest coefficients with constant ∆ pc . Definition 1.4. We say that B is super-democratic if there is a positive constant C such that for any pair of sets A, B ⊂ N, |A| ≤ |B| < ∞ and for any choice of signs ε ∈ E A , η ∈ E B . The least constant verifying (3) is denoted by ∆ s = ∆ s [X, B] and we say that B is ∆ s -superdemocratic.
If (3) is satisfied for A < B, we say that B is ∆ sc -super-conservative.
If (3) is satisfied for ε ≡ η ≡ 1, we say that B is ∆ d -democratic and if in addition, A < B, we say that B is ∆ c -conservative.
With these definitions, we can find the following characterizations of greedy bases. Theorem 1.5. Assume that B is a basis in a Banach space X.
• B is greedy if and only if B is democratic and unconditional (see [11]). Moreover, • B is greedy if and only if B is super-democratic and unconditional (see [6]). Moreover, • B is greedy if and only if B is symmetric for largest coefficients and unconditional (see [9]). Moreover, The last two characterizations were studied with the objective to improve the boundedness constant of greedy bases. Moreover, all the characterizations were given under the assumption of unconditionality and one of the democracy-like properties but, in [5], we find a new and interesting property that is so useful to give a new characterization of greediness (see [5,Corollary 1.8]). This property is the so called Property (Q): there is a C > 0 such that In that paper, we focus our attention in a closed inequality to characterize the so called almost-greedy and partially-greedy bases. Definition 1.6 ([8]). We say that B is almost-greedy if there is a positive constant C such that The least constant verifying (4) is denoted by C al = C al [X, B] and we say that B is C al -almostgreedy. 4,8]). We say that B is partially-greedy if there is positive constant C such that The least constant verifying (5) is denoted by C p = C p [X, B] and we say that B is C p -partiallygreedy.
Remark 1.8. In [8], the condition of partially-greediness was introduced as follows: Under the condition of Schauder bases, (6) and (5) are equivalent notions and in [4], the authors proved that if (6) is satisfied with C = 1, then the basis is partially-greedy.
Of course, every greedy basis is almost-greedy and every almost-greedy basis is partiallygreedy. One example of an almost-greedy basis that is not greedy is the Lindestrauss basis in ℓ 1 ( [10]). Recently, one basis that is partially-greedy and not almost-greedy is presented in [7,Proposition 6.10].
It is well known that a basis is almost-greedy if and only if the basis is quasi-greedy and democratic and a basis is partially-greedy if the basis is quasi-greedy and conservative ( [8,7]). Moreover, as for greedy bases, we have the following characterizations. Theorem 1.9. Assume that B is a basis in a Banach space.
• B is almost-greedy if and only if B is democratic and quasi-greedy ( [8]). Moreover, is almost-greedy if and only if B is super-democratic and quasi-greedy ( [6]). Moreover, • B is almost-greedy if and only if B is symmetric for largest coefficients and quasi-greedy Theorem 1.10. [4,8] Assume that B is a basis in a Banach space.
• B is partially-greedy if and only if B is conservative and quasi-greedy. Moreover, is partially-greedy if and only if B is super-conservative and quasi-greedy. Moreover, • B is partially-greedy if and only if B is partially-symmetric for largest coefficients and quasi-greedy. Moreover, The purpose of this paper is to get a new characterization of almost-greedy and partiallygreedy bases following the ideas of [5, Corollary 1.8] for greedy bases. Definition 1.11. We say that B has the Property (F) if there is a positive constant C such that for any A, B, f, g satisfying the following conditions: The least constant verifying (7) is denoted by F = F [X, B] and we say that B has the Property (F) with constant F . Also, if (7) is satisfied with the extra condition that A < supp(g) ∪ B, we say that B has the Property (F p ) with constant F p . Definition 1.12. We say that B has the Property (F * ) if there is a positive constant C such that for any f, z, y ∈ X f in satisfying the following conditions: and we say that B has the Property (F * ) with constant F * .
Also, if (8) is satisfied with the extra condition that supp(z) < supp(f + y), we say that B has the Property (F * p ) with constant F * p . The main theorems that we will prove are the following.
then the basis is partially-greedy with constant C p ≤ (F * p ) 2 . The structure of the paper is the following: in Section 2, we will show some basics about the Properties (F) and (F * ). In Section 3, we prove Theorem 1.13. In Section 4 we give a brief summary about the Properties (F p ) and (F * p ), in Section 5 we prove Theorem 1.14 and, finally, in Section 6, we give some density results that we use in the paper.

Properties (F) and (F * )
This section is focused in the study of the Properties (F) and (F*). In fact, we will show that these properties are equivalent. To show that we will need some auxiliary lemmas about convexity.
Lemma 2.1. [1, Corollary 2.3] Let X be a Banach space, let B be a basis for X and J a finite set.
(i) For any scalars (a j ) j∈J with 0 ≤ a j ≤ 1 and any g ∈ X, g + j∈J a j e j ≤ sup{ g + 1 A : A ⊆ J}.
(ii) For any scalars (a j ) j∈J with |a j | ≤ 1 and any g ∈ X, g + j∈J a j e j ≤ sup Proof. If F = R, following the result [5, Lemma 2.3], we know that We prove now the result for the complex case. In that case, where A i are the corresponding subsets of A. Then, Theorem 2.3. Let B be a basis in a Banach space X. The basis is democratic (or symmetric for largest coefficients) and quasi-greedy if and only if the basis has the Property (F). Concretely: (1) If B has the Property (F) with constant F , then the basis is C q -quasi-greedy and ∆ ddemocratic with max{C q , ∆ d } ≤ F .
(2) If B has the Property (F) with constant F , then the basis is C q -quasi-greedy and ∆symmetric for largest coefficients with (3) If B is ∆ d -democratic and C q -quasi-greedy, then the basis has the Property (F) with constant F ≤ C q (1 + (1 + C q )∆ d ).
(4) If B is ∆-symmetric for largest coefficients and C q -quasi-greedy, then the basis has the Property (F) with constant F ≤ 3∆C q .
Proof. First of all, we show (1). Assume that the basis has the Property (F) with constant F . To show that B is quasi-greedy, we take f ∈ X f in with t = f ∞ and m ∈ N. Then, if we take in the definition of the Property so the basis is quasi-greedy with C q ≤ F for elements with finite support. To obtain that B is quasi-greedy for any f ∈ X, we use Corollary 6.2. Prove now that the basis is democratic. For that, we take C and D two finite sets such that |C| ≤ |D|. Now, we do the following decomposition: Thus, B is democratic with ∆ d ≤ F . Prove now (2). We only have to show that B is symmetric for largest coefficients. For that, Also, respect to the set A ′ , we can have the following: Finally, applying convexity, So, the basis is symmetry for largest coefficients for elements with finite support with constant Applying Lemma 6.5, the result follows for any f ∈ X.
(3) Assume now that B is C q -quasi-greedy and ∆ d -democratic and take f, g ∈ X f in with If we take h := f + g + 1 B , it is clear that supp(g + 1 B ) is a greedy set of h. Then, if |supp(g + 1 B )| = n, Since inf n inf n∈supp(g) |e * n (g)| ≥ f ∞ and f ∞ ≤ 1, we can decompose g as g = g 1 + g 2 , where supp(g 1 ) = {n ∈ supp(g) : |e * n (g)| ≥ 1} and supp(g 2 ) = {n ∈ supp(g) : |e * n (g)| < 1}. Then, if we take u := f + g 2 + 1 B , B is a greedy set for u with of order k := |B|, and taking v = u + g 1 , supp(g 1 ) is a greedy set of v of order p := |supp(g 1 )|. Thus, Adding up (13) and (14) in (12), we obtain the result, that is, the basis has the Property (F) with F ≤ C q (1 + (1 + C q )∆ d ).
(4) Finally, assume that B is ∆-symmetric for largest coefficients and C q -quasi-greedy. Take f, g, A and B as in the Property (F). Then, Thus, the basis has the Property (F) with constant F ≤ 3C q ∆. Proof. Assume that we have the Property (F * ) with constant F * and take f, g, A and B as in the Property (F), that is, f · g = 0, A ∩ B = ∅, |A| ≤ |B|, supp(f + g) ∩ (A ∪ B) = ∅, f ∞ ≤ 1 and f ∞ ≤ inf n∈supp(g) |e * n (g)|. Taking z = 1 A and y = g + 1 B in the Property (F * ), f, z and y verify the conditions established in the Property (F * ). Then, so the basis has the Property (F) with F ≤ F * .
To finish this section, we give the following nice characterization of the Property (F * ) that will be useful to show our main theorem. Proposition 2.5. Let B be a basis in a Banach space X. The following are equivalent: i) There is a positive constant C such that for any f, g ∈ X f in such that f ·g = 0, f ∞ ≤ 1 and inf n∈supp(g) |e * n (g)| ≥ f ∞ , for any pair of finite sets A and B such that A ∩ B = ∅, |A| ≤ |B|, supp(f + g) ∩ (A ∪ B) = ∅, and for any ε ∈ E A , η ∈ E B .
ii) The basis has the Property (F*) with constant F * . iii) There is a positive constant C such that for any f, y ∈ X f in with f · y = 0 and . Moreover, if we denote by C 1 and C 2 the least constants verifying (19) and (20) respectively, we have Proof. First, we prove i) ⇒ ii). Take f, z, y ∈ X f in as in the definition of the Property (F * ): If z = 0, just take A = B = ∅ and the prove is over. Consider now that z = 0 and take supp(z) = A. If we divide y = 1 ηD + P D c (y) with η ≡ {sign(e * n (y))}, we have for all ε ∈ E A , f + 1 εA ≤ C 1 f + P D c (y) + 1 ηD = C 1 f + y (21) Applying now Lemma 2.1, we obtain the result with F * ≤ C 1 . Now, we show that ii) ⇒ iii). Of course, if A = ∅, the result is trivial. Take f, y and A as in iii) with A = ∅ and A ⊆ supp(f ). If in the Property (F*) we take f ′ = f − P A (f ), z ′ = 1 εA with ε ∈ E A and y ′ = y, Finally, we make the proof to show that iii) ⇒ i). Take f, g ∈ X f in such that f · g = 0, Taking f ′ = f + 1 εA and y = g + 1 ηB , so the proof is over and C 1 ≤ C 2 .

Proof of Theorem 1.13
To prove Theorem 1.13, we will use one of the most important tools in the world of quasigreedy bases: the truncation operator. To define this operator, we take α > 0 and define, first of all, the α-truncation of z ∈ C: Now, it is possible to extend T α to an operator in the space X by where the set ∆ α = {n ∈ N : |e * n (f )| > α}. Of course, since ∆ α is a finite set, T α is well-defined for all f ∈ X.
Lemma 3.1. [6, Lemma 2.5] Let B be a C q -quasi-greedy basis in a Banach space. Then, the truncation operator is uniformly bounded that is, Proof of Theorem 1.13. Assume that B is almost-greedy with constant C al and take f, z and y as in the Property (F * ) and decompose y = P B 1 (y) + P B 2 (y) + 1 ηB , where η ≡ {sign(e * n (y))}, B 1 ∪ B 2 = B c and Taking now h := f + 1 εA + P B 2 (y) + 1 ηB with A = supp(z), ε ∈ E A and n = |B 2 | + |B|, we obtain Thus, applying Lemma 2.1, the basis has the Property (F * ) with constant F * ≤ C al (1 + 2C al ). Assume now that the basis has the Property (F * ). Take f ∈ X f in , m ∈ N, G m (f ) = P G (f ) and |A| ≤ m.
Consider now the elements f ′ = 1 t (f − G m (f )) with t = min n∈G\A |e * n (f )|, B = A \ G, y = 1 η(G\A) and η ≡ {sign(e * n (f ))}. Of course, f ′ · y = 0, f ′ ∞ ≤ 1 since |e * n (f − G m (f ))| ≤ t for n ∈ G c and |G \ A| ≥ |B|. Then, applying these elements in the item iii) of Proposition 2.5, we obtain the following: Since the Property (F * ) implies that the basis is quasi-greedy with C q ≤ F * (Theorems 2.3 and 2.4), applying Lemma 3.1, Thus, by (23) and (22), the basis is almost-greedy with constant C al ≤ (F * ) 2 for elements f ∈ X f in . Now, applying Corollary 6.3, the results follows.
4. Properties (F p ) and (F * p ) In all the results presented in Section 2 we can change democracy by conservativeness or super-conservativeness and Property (F) and (F * ) by Property (F p ) and (F * p ) and obtain the same results. Here, we only present the fundamental theorem that is the version of Theorem 2.3 to study how the constants change.
Proof. Assume that B has the Property (F p ) with constant F p . Taking A = ∅, we have that for any f, g and B as in the definition of the Property (F p ). Now, taking B = ∅ and considering so the basis is quasi-greedy for elements with finite support. Applying now Corollary 6.2, the basis is quasi-greedy with C q ≤ F p . Now, on the other hand, taking f = g = 0, we obtain conservativeness with constant ∆ c ≤ F p . Now, take f, g, A and B as in the definition of Property (F p ). If we have g = g 1 + g 2 where supp(g 1 ) = {n ∈ supp(g) : |e * n (g)| < 1},
Since partially-greediness implies quasi-greediness with constant C p (see Theorem 1.10), we have For the first element of the sum, consider w := f + y and we have f = w − G n (w) , with n = |supp(y)|. Then, applying quasi-greediness, we obtain f ≤ C p f + y . For the second one, we write w = f + y 1 + y 2 + 1 ηD and using quasi-greediness, we have where m = |supp(y 2 ) ∪ D|.
Prove now b). Without loss of generality we can assume that f ∈ X f in using Corollary 6.4 and that f ∞ ≤ 1. Start considering A = supp(G m (f )), k ≤ m and B = {1, . . . , k}. If A = B, then the result is trivial. If A = B, we can decompose t P (A∪B) c (f − S k (f )) and z = 1 t P B\A (f ) with t = min n∈A |e * n (f )| and y = 1 ε(A\B) with ε ≡ {sign(e * n (f )}. Of course, f ′ · z = 0, f ′ · y = 0 and y · z = 0, f ′ ∞ ≤ 1 since |e * n (P (A∪B) c (f − S k (f )))| ≤ t for n ∈ (A ∪ B) c and |A \ B| ≥ |B \ A|. Then, f ′ , z and y verify the items of the Property (F * p ), so

It turns out that
, where T t is the t-truncation operator. Now, since the Property (F * p ) implies the Property (F p ) with the same constant, because of Theorem 4.1 and Corollary 6.2, the basis is quasi-greedy with constant C q ≤ F * p . Then, applying Lemma 3.1, we have that for all k ≤ m and hence, B is partially-greedy.

Annex
In this annex, we write the main lemmas about density that we use in the paper.
Lemma 6.1. [3, Lemma 7.2] Let B be a basis for a Banach space X. If A is a greedy set for f ∈ X, for every ε > 0, there is y ∈ X f in such that f − y ≤ ε and A is a greedy set for y. Corollary 6.2. Assume that B is a C q -quasi-greedy basis of a Banach space X for elements with finite support. Then, B is quasi-greedy for every f ∈ X.
Proof. Take f ∈ X and A a greedy set of f with order m. By Lemma 6.1, there is y ∈ X f in such that f − y ≤ ε for every ε > 0 with A a greedy set for y. Then Taking ε → 0, we obtain the result. Corollary 6.3. Let B be a basis for a Banach space X. If B is an almost-greedy basis for all f ∈ X f in with constant C al , then the basis is almost-greedy for every f ∈ X with the same constant.
Proof. Assume that B is almost-greedy for elements with finite support. Take f ∈ X with A a greedy set of order m. Applying Lemma 6.1, for any ε > 0, there is g ∈ X f in such that f − g ≤ ε and A a greedy set for g. Consider the set B 1 the set such that Case 1: ≤ ε(1 + P A ) + C al g ≤ ε(1 + P A ) + C al f − g + C al f ≤ ε(1 + C al + P A ) + C al f .
With the same arguments, it is straightforward to show the next result.
Corollary 6.4. Let B be a basis for a Banach space X. If B is a partially-greedy basis for all f ∈ X f in with constant C p , then the basis is partially-greedy for every f ∈ X with the same constant.