Greedy Expansions with Prescribed Coefficients in Hilbert Spaces

Greedy expansions with prescribed coefficients, which have been studied by V. N. Temlyakov in Banach spaces, are considered here in a narrower case of Hilbert spaces.We show that in this case the positive result on the convergence does not require monotonicity of coefficient sequenceC. Furthermore,we show that the condition sufficient for the convergence, namely, the inclusionC ∈ l2 \l1, can not be relaxed at least in the power scale. At the same time, in finite-dimensional spaces, the conditionC ∈ l2 can be replaced by convergence of C to zero.


Introduction
When dealing with Hilbert spaces H, the most standard way to expand a generic element of H with respect to a fixed set of vectors D ⊂ H is to fix D as an (orthonormal) basis and consider the usual (unique) expansion with respect to such a basis. Greedy algorithms are different ways of expanding elements in Hilbert spaces, with respect to different fixed sets of vectors, called dictionaries. Using such methods it is possible to obtain good properties of the corresponding expansions, related to different aspects; one of the main goals of greedy algorithms is related to the convergence rate of the expansions themselves, but there are aspects related to the 'philosophy' in the way an element is expanded, that may be different with respect to the classical basis expansion. In fact, in the classical basis expansion the dictionary (i.e., the basis) is fixed and ordered independently from the expanded element; the coefficients are then chosen depending on the element we want to represent, and each element of the basis appears only once in the expansion. In the greedy algorithms we still have a fixed set of expanding elements, but without a prescribed order, and in the expansion it could happen that an element from the dictionary appears more than once (still with coefficients that in general depend on the element we want to represent). In the greedy algorithms with prescribed coefficients, that constitute the object of the present work, both the dictionary and the coefficients are fixed, independently from the vector we want to expand, and the only thing that depends on the element we want to represent is how the coefficients are associated to the corresponding elements from the dictionary; when the algorithm converges, the convergence of the corresponding series expansion is in general not unconditional.
There is a large literature on greedy expansion, both in Hilbert and in Banach spaces. We refer in particular to the works of Temlyakov [4], [1], [2], where a large amount of material on different kind of greedy algorithms can be found. The greedy expansions with prescribed coefficients in Banach spaces have been introduced in [3]. Here we study a version of such algorithm for real Hilbert spaces. Let then H be a real Hilbert space; we indicate with ·, · its inner product. We say that D ⊂ H is a dictionary if (i) for every g ∈ D we have g = 1 (ii) D is complete, in the sense that span{D} = H.
We say that a dictionary D is symmetric if for every g ∈ D we have −g ∈ D. For a generic dictionary D we indicate with D ± its symmetrized, i.e., D ± := g∈D {g, −g}. We shall suppose throughout the paper that the dictionary D is symmetric.
Greedy Expansion with prescribed coefficients. Now we describe how the algorithm works. We fix a coefficient sequence C = {c n } n with c n > 0 for every n, and a weakening sequence τ , that is, a sequence τ = {t k } ∞ k=1 with 0 < t k 1 for every k. For f ∈ H we define: then, for every m 1 we procede as follows. If f m−1 = 0: • We choose an element ϕ m ∈ D such that • We define In this way, G m represents the approximation of f , and f m is the remainder of such an approximation at the step m. Then the greedy expansion of f is the series ∞ n=1 c n ϕ n .
If f m−1 = 0 then the algorithm stops, and the corresponding approximation at this step coincides with f . Since the choice of ϕ m satisfying (1) is not unique, we may have different expansions of the same element.
We observe that if the weakening sequence τ satisfies t k = 1 for some k, the greedy expansion could not exist, since at some step it could be impossible to find ϕ m satisfying (1). On the other hand, the case t k < 1 is much more difficult to treat, while for t k = 1 there are more complete results in the literature. Moreover, even when the greedy expansion exists, it does not need to converge to f . Of course the interesting case is when this happens, so we say that the greedy algorithm is convergent if for every f ∈ H we have lim for every approximating sequence {G m } ∞ m=1 obtained by the greedy algorithm. When the weakening sequence satisfies t k = t for every k we say that the greedy algorithm has weakening parameter t. In this paper we deal with greedy algorithms with prescribed coefficients and weakening parameter 1, assuming that at least one greedy expansion always exists. In this case the convergence of the algorithm is characterized by the following two theorems, proved in [6]. Theorem 1.1 ([6]). Let D be a symmetric dictionary in H, and C be a sequence of positive coefficients satisfying Then, for every f ∈ H, all the possible greedy expansions of f with prescribed coefficients C and weakening parameter t = 1 with respect to the dictionary D converge to f . The previous results leave open the question if the greedy algorithm can converge with weaker conditions on the coefficients for particular Hilbert spaces or for particular dictionaries. In fact, in [5] it is proved that in the case of finite dimensional spaces the greedy algorithm is convergent for a larger set of prescribed coefficients with respect to Theorem 1.1, as stated in the following result. Then for every f ∈ H, all the possible greedy expansions of f with prescribed coefficients C and weakening parameter t = 1 converge to f .
In this paper we extend the previous results in several directions.
-Concerning the finite dimensional case, we prove that the result of Theorem 1. 3 holds without requiring that C is non increasing, and moreover it can be extended to an arbitrary weakening sequence τ = {t k } ∞ k=1 with t k ∈ (0, 1] under the following conditions on the coefficients (cf.
Concerning the infinite dimensional case we prove the following results.
-When the dictionary is the symmetrized of an orthonormal basis and we consider a weakening parameter t = 1 (i.e., t k = 1 for every k), the conditions (2) are sufficient to guarantee the convergence of the algorithm (cf. Theorem 3.2).
-The fact that in the previous point we consider t = 1 is essential; indeed when the dictionary is the symmetrized of an orthonormal basis and we consider a weakening parameter t < 1 (i.e., t k = t < 1 for every k) and coefficients satisfying (2), the convergence of the algorithm fails (cf. Theorem 3.4).
-When the dictionary is the symmetrized of the union of an orthonormal basis E and an arbitrary number of elements that are linear combinations of a finite subset of E, the convergence of the algorithm (with weakening parameter t = 1) is guaranteed under the hypotheses (2) (cf. Corollary 4.3).
The last result is a consequence of a general result on greedy expansions on direct sums of Hilbert spaces, given in Theorem 4.2.

Finite-dimensional case
In this section we prove a general result on convergence of greedy algorithm for finite dimensional spaces.
Then, for every f ∈ H, all the possible greedy expansions of f with prescribed coefficients C and weakening sequence τ with respect to the dictionary D converge to f.
Proof. Due to the fact that in the finite-dimensional Hilbert space a unit sphere is a compact and the completeness of D, there exist a constant c > 0 such that for every Then from the definition of greedy expansions with prescribed coefficients we immediately have that Let us fix an arbitrary ε > 0. Due to the second condition of the theorem, there exist N > 0, such that for every n > N the following inequalities hold Let us assume that f k > ε c for some k > N . In this case, we will show that there exists l > k such that Assume the contrary. Then from (3) and (4) for every n > k we get It implies which contradicts with the assumption. Then for some l > k we have (5). Now, if the inequality holds for all n l, the proof of the theorem is complete. So let us suppose it exists j such that and j is the first step for which the above inequality holds after l. We observe that from (6) we get From now on, (7) holds for the sequential steps n > j till a possible step d (which exists) such that f d < ε c , so during all these steps the sequence is not increasing. Moreover, since the above pattern is replicable after step d, we can finally deduce that for all n l, and this completes the proof of the theorem.

Symmetrized orthonormal bases
In this section we analyze the case when the dictionary is the symmetrized of an orthonormal basis (in infinite dimensional spaces), proving that when the weakening parameter is t = 1, the hypotheses on the coefficients C given in Theorem 2.1 are enough to guarantee the convergence of the algorithm, while for t < 1 the convergence fails, in the sense that there exist an example where greedy expansion does not converge.
be an orthonormal basis for a (separable) Hilbert space H. We consider the following dictionary for H Let moreover C be a sequence of positive coefficients such that Then for every f ∈ H, all the possible greedy expansions of f with prescribed coefficients C and weakening parameter t 1 satisfy lim inf n→∞ ||f n || = 0.
Proof. In the following we write for simplicity the dictionary as is an orthonormal basis, we can write f as x i e i for a unique choice of the coefficients x i . Without loss of generality we can assume that the sequence {|x i |} ∞ i=1 is non increasing. Similarly, for every remainder f n we can write for a unique choice of the coefficients x i,n . For convenience we put x i,0 = x i . We notice that the algorithm, at each step m, modifies one single coefficient of the orthonormal expansion of the remainder f m−1 .
Let e km be an element from the dictionary that is touched by the algorithm at step m. We note that As {x i } ∞ i=1 ∈ l 2 , we have that lim inf n→∞ √ n|x n | = 0. Let now ε be an arbitrary positive number, and let n be such that |x n | ε √ n . Also, let k be the first step such that such a number exists due to (8). Then x n has not been modified in the first k − 1 steps; since {|x i |} ∞ i=1 is non increasing, also x m , m n, has not been modified in the first k − 1 steps, and so x m,k−1 = x m for every m n. Now, for every m we have the following estimation Therefore, using the monotonicity of the sequence Since ε > 0 is arbitrary and n → ∞, we get that lim inf n→∞ ||f n || = 0, which completes the proof of Theorem 3.1.

Theorem 3.2. Let
be an orthonormal basis for a (separable) Hilbert space H. We consider the following dictionary for H Let moreover C be a sequence of positive coefficients such that where the components are subdivided in groups, and the j-th group contains k + j components, all equal to t k+j .
In order to prove the theorem, we build the sequence C = {c n } ∞ n=1 so that the following conditions hold: We construct c n consequently for each group, starting from the first one, in the following way: -Increase the subnorm of the group to one. First, we take any component of the group and choose c i in the coefficient sequence C in order that the component changes its sign and is multiplied by 1 t (so, t h becomes −t h−1 ); then we do the same for the remaining elements of the group (it is possible since we are applying the algorithm with weakening parameter t). We repeat this procedure on the group until each component of the group in the remainder has modulus between t √ h and At this point we choose the next coefficients to consequently change all the components of the group to 1 √ h (or − 1 √ h ), that means that at this step the subnorm of the group equals one, and so condition (i) holds.
Finally we choose the next coefficients equal to 1 √ h in order that all the components of the group become 0. Then we pass to the next group of components and repeat the same procedure.
We observe that all the coefficients c i , in these steps, are less than 2 √ h , and since h → ∞ we have that condition (ii) holds. Moreover, since for every h we choose at least one coefficient equal to 1 √ h we have that condition (iii) is satisfied, and this completes the proof.

Direct sum of Hilbert spaces
In this section we prove that the greedy algorithm with prescribed coefficients in infinite dimensional Hilbert spaces is still convergent (with the same hypotheses on the coefficients as in Theorem 3.2) if we consider as a dictionary the symmetrized of the union of an orthonormal basis E and an arbitrary number of elements that are linear combinations of a finite subset of E. In order to do this we prove a result on greedy algorithm in direct sums of Hilbert spaces.
Let H be a Hilbert space and M ⊂ H a closed subspace of H. It is well-known that where M ⊥ ⊂ H is the orthogonal of M in H. Indeed, from the Projection Theorem we have that each x ∈ H can be uniquely decomposed as x = P x + Qx, with P x ∈ M and Qx ∈ M ⊥ . Then the map is an isometric isomorphism. Of course we can consider the direct sum of a finite number of Hilbert spaces. Now we want to prove that the convergence of the greedy algorithm is maintained by isometric isomorphisms, in the following sense.

Lemma 4.1.
Let H be a Hilbert space, D a dictionary for H, and C a prescribed sequence of coefficients. We suppose that, for every f ∈ H, all the possible greedy expansions of f with respect to the dictionary D with prescribed coefficients C and weakening parameter t = 1 converge to f . Let moreover K be another Hilbert space and α : H → K be an isometric isomorphism. Define E := {α(ϕ) : ϕ ∈ D}. Then E is a dictionary for K (symmetric if D is symmetric), and for every g ∈ K all the possible greedy expansions of g with respect to the dictionary E with prescribed coefficients C and weakening parameter t = 1 converge.
Proof. The fact that E is a dictionary (symmetric if D is symmetric) is straightforward since α is an isometric isomorphism. Now let g ∈ K and consider a greedy expansion of g, that is an expression of the kind ∞ p=1 c p ψ p with ψ p := α(ϕ p ) ∈ E. We observe that at the m-th step the greedy approximation is obtained by choosing ψ m ∈ D in such a way that Applying α −1 to both sides we then obtain with f = α −1 (g). This holds for every positive integer m, so by definition it follows that Then the considered greedy expansion of g converges to g, and the proof is completed.
Now we analyze greedy expansions in direct sums of Hilbert spaces.
and consider in H the set Then D is a symmetric dictionary for H; moreover for every f ∈ H and for every sequence of coefficients C as above, all the greedy expansions of f with respect to the dictionary D with prescribed coefficients C and weakening parameter t = 1 converge to f .
Proof Then for every l ∈ {1, . . . , N } the sequence {c np(l) } p is finite, and this means that at a certain step m the greedy algorithm stops, and this happens only if f m−1 = 0. In this case the greedy expansion is a finite sum that equals f , and so the proof is complete. We have to analyze the case Now, {n p (l ′ )} p is an infinite sequence, the right-hand side of (19) is maximum among all l ′′ such that {n p (l ′′ )} p is finite and we have already proved that if {n p (l ′′ )} p is infinite we have convergence; then at some step the greedy algorithm will select an element of the dictionary D whose non vanishing component is the l ′′ -th one. This would add a new term in the sum (18), and so {n p (l ′′ )} p would contain q + 1 elements. Also in this case we have a contradiction, due to the assumption that there exists l ′′ such that p c np(l ′′ ) ϕ l ′′ p is different from f l ′′ . Then the greedy expansion of f converges to f and the proof is complete.
where E ± and Y ± are the symmetrized of E and Y , respectively. Let moreover C be a sequence of prescribed coefficients such that with prescribed coefficients C and weakening parameter t = 1 converge. We then conclude from Lemma 4.1, since the isomorphism α : H → M ⊕M ⊥ given by (10) transforms D in the dictionary (20).

Conclusions
In this paper we have extended the convergence result of the greedy algorithm with prescribed coefficients known for finite dimensional Hilbert spaces, and we have proved that such a result, in the case of a weakening parameter t = 1, holds also in the case of infinite dimensional spaces when the dictionary is the symmetrized of an orthonormal basis, to which linear combinations of elements from a finite subset of the basis itself can eventually be joined. These are sufficient conditions on the dictionary to have convergence of the algorithm; the question to characterize the dictionaries for which the greedy algorithm converges under the considered hypotheses on the coefficients remains open.