Sharp conditions for the convergence of greedy expansions with prescribed coefficients

Abstract: Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients { } = ∞ cn n 1 is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined set – a dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form ∑ = ∞ = ∞ c n n 1 and ∑ < ∞ = ∞ c n n 1 2 . In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the

Abstract: Greedy expansions with prescribed coefficients were introduced by V. N. Temlyakov in a general case of Banach spaces. In contrast to Fourier series expansions, in greedy expansions with prescribed coefficients, a sequence of coefficients { } = ∞ c n n 1 is fixed in advance and does not depend on an expanded element. During the expansion, only expanding elements are constructed (or, more precisely, selected from a predefined seta dictionary). For symmetric dictionaries, V. N. Temlyakov obtained conditions on a sequence of coefficients sufficient for a convergence of a greedy expansion with these coefficients to an expanded element. In case of a Hilbert space these conditions take the form ∑ = ∞ In this paper, we study a possibility of relaxing the latter condition. More specifically, we show that the convergence is guaranteed for

Introduction
We consider greedy expansions with prescribed coefficients in Hilbert spaces. This type of greedy expansion was initially introduced by V. N. Temlyakov [1,2] (see also [3]) in a more general case of Banach spaces. In case of Hilbert spaces, the definition of greedy expansions with prescribed coefficients (furtherjust greedy expansions) takes the following form. and hence the convergence of the expansion to an expanded element is equivalent to the convergence of remainders r n to zero.
As a selection of an expanding element e n is potentially not unique, there may exist different realizations of a greedy expansion for a given expanded element f and a given dictionary D. Furthermore, if t n = 1 for at least one ∈ n , greedy expansion may turn out to be nonrealizable due to the absence of an element e ∈ D which provides ( ) . However, the absence of convergence has been shown only for one of the possible realizations of this expansion.
In this paper, we present improvements for both the negative and the positive results.

Main results
An improvement of the negative result can be stated as follows.
Hence, there exists a number M, that for all n > M the inequality (2) holds. Thus, we can reenumerate (shift) a sequence: and as a result obtain a sequence for which the inequalities (1) and (2) hold for all ∈ n .
We construct the example inductively. First, as e 1 we take an (arbitrary) element of H which satisfies the following conditions: (2) the projection of e 1 on the plane ⟨e −1 , e 0 ⟩ is collinear to r 0 .
We then set r 1 to r 0 − c 1 e 1 .
Let vectors e −1 , e 0 , e 1 , …, e n and r 0 , r 1 , r 2 , …, r n have already been constructed. Then as e n+1 we take an (arbitrary) element of H which satisfies the following conditions: (2) the projection of e n+1 on the subspace ⟨e −1 , e 0 , …, e n ⟩ is collinear to r n ; and define r n+1 as r n − c n+1 e n+1 .
We note that in this construction the angle α n+1 is equal to the angle between e n+1 and the subspace ⟨e −1 , e 0 , …, e n ⟩.
e e n n n n 1 1 . It is sufficient to show that for each n = 0, 1, 2, … the element e n+1 is the only vector from D that can be selected as an expanding element at the step n, and that ∥r n ∥ ↛ 0 as n → ∞.
We split the proof of these assertions into the following steps. First, we show that ∥r n ∥ ↛ 0. Second, we show that at the step (n + 1) the vector −e n cannot be selected as an expanding element. Third, we show the same for vectors e k and −e k , k > n + 1. As (− ) ,  is the angle between vectors a and b), vectors −e n+1 and e n also cannot be selected as expanding elements at the step (n + 1). And finally, we show that it also holds for vectors e k and −e k , k < n.
1. Due to the law of cosines Using (2), (3) and (4) we can note that the following inequalities hold:  Combining (7) and (6), we see that it is sufficient to prove that    . The norm of each functional equals 1. As a unit sphere is a weak compact (according to the Banach-Alaoglu theorem), there exists a weakly converging subse- . For simplicity we denote F rn k i as F i . As noted above, there exists a weak limit Due to the fact that the dictionary D is symmetric, for all sufficiently large i the following inequality holds: Passage to the limit results in an estimate F( f ) ≥ r, which implies that F ≠ 0. On the other hand, for every g from the dictionary D ( ) ( ) Hence, F(g) equals 0 for all g ∈ D and, due to completeness of D, we get that F = 0. We have come to a contradiction, which proved the equality n n Thus, we completed the first part of the proof of Theorem 2.2.
2. Now it remains to prove the following two lemmas.

Generalization
One of the disadvantages of a greedy expansion with the prescribed coefficients in case of t n ≡ 1 is that there might be no greedy expansion for an expanded element. Another disadvantage is that (irrespective of t n ) in a general case selection of expanding element is not constructive.
We consider the following generalization of the greedy algorithm that eliminates these undesirable properties. Let symmetric sets { } = ∞ D n n 1 be an exhaustion of a dictionary D, i.e., In this paper, we presented improvements for both negative and positive results on the convergence of greedy expansions with prescribed coefficients. These improvements, in particular, allowed to remove completely a gap between positive and negative results. We expect that the technique we used in this research is applicable also to greedy expansions with errors in coefficient calculation [7], where a gap between negative and positive results still exists, and we are going to present results for greedy expansions with errors in coefficient calculation in our subsequent publications. We are also going to study a possibility of generalizing positive results to the case of t n ≡ t ∈ (0, 1).