The «test space and pairing» idea for frames and some generalized characterizations and topological properties of Euclidean continuous frames

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Introduction
Frames, introduced by Duffin and Shaeffer in [7], have recently received great attention due to their wide range of applications in both mathematics and engineering science.The classical definition of a frame is that of a sequence of vectors in a Hilbert space H that satisfies a double inequality involving the scalar product and the norm maps.Specifically, and generalizing slightly from sequences to families indexed by general sets (possibly uncountably infinite), we have Definition 1.We say that a family of vectors u = (u i ) i∈I with u i ∈ H for all i ∈ I is a classical (or discrete) frame if We denote by F H I the set of I-indexed frames with values in H.
Remark 1. Classical (or discrete) Bessel families are the families for which only the second inequality holds.
Plan of the article.In section 2, we introduce the «test space and pairing» idea for frames and apply it to the ℓ 2 and L 2 spaces.First, we show that for every non-empty set J, the notions of a classical I-frame with values in H and a J-extended classical I-frame with values in H are the same.The definition of a J-extended classical I-frame with values in H, u = (u i ) i∈I , utilizes the «test space and pairing» idea by replacing the usual «test space» H with ℓ 2 (J; H) and the usual «pairing» P : Secondly, we prove a similar result when the space ℓ 2 (J; H) is replaced with the space L 2 (Y, ν; H) and the frame u = (u x ) x∈X is (X, µ)-continuous.Besides, we define the J-extended and (Y, ν)-extended analysis, synthesis, and frame operators of the frame u and note that they are just natural block-diagonal operators.After that, we generalize in section 3 in a quite straightforward manner the well-known characterizations of Euclidean finite frames and Parseval frames to the corresponding characterizations of Euclidean continuous frames and Parseval frames.In section 3.1, we show some useful rewritings of the quotients In section 3.2, we generalize the classical characterizations of Euclidean finitely indexed frames and normalized tight frames (Parseval frames) to the Euclidean continuous case.In section 3.3, we give a simple sufficient condition for having a frame with values in F 2 and use it to provide an example of a classical frame indexed by N * with values in C 2 .Finally, in section 4, we generalize some topological properties of the set of frames and normalized tight frames (Parseval frames) from the Euclidean finitely indexed case to the more general Euclidean continuous one.
2 The «test space and pairing» idea: application to the ℓ 2 and L 2 spaces Let I be some set and H = (H i ) i∈I be a family of Hilbert spaces over the same field F := R or C. We define ℓ 2 (I; Hi < ∞}.We endow ℓ 2 (I; H) with the pointwise scalar product: u, v = i∈I u i , v i Hi .It is well-known that ℓ 2 (I; H) is a Hilbert space.It is also a (vector) subspace of the product × i∈I H i .We now fix the family {H i } i∈I such that each H i is equal to the same Hilbert space H.We define ℓ 2 (I; H) := ℓ 2 (I; H).Similarly, for every measure space (X, Σ, µ), we have the Hilbert space L 2 (X, µ; H) of Bochner square-integrable functions from X to H.
In the frame inequality given in the introduction, the variable x runs through the Hilbert space H. Therefore, H may be seen as a «test space» over which we are testing the frame inequality.Now, we could raise the following question: what happens of the notion of a frame if we replace H with some other test space and change the map by which we pair an element of this test space to the frame?In the classical case, this «pairing» is which corresponds closely to the expression of the analysis operator T u of u: In this section, we first consider the case of the test space ℓ 2 (J; H) with the pairing which now agrees with the expression of an analysis operator T J u of u in ℓ 2 (J; H): and even more generally, if u is a frame (u x ) x∈X in H associated to the measure space (X, Σ, µ), we consider the test space L 2 (Y, ν; H) with the pairing which matches the expression of an analysis operator The two following propositions assert that, in the ℓ 2 and L 2 cases, the resulting notions of extended frames are the same as the usual ones.We start with the ℓ 2 case ; however, this case can obviously be seen as a particular case of L 2 case.Proposition 1. Suppose that u = (u i ) i∈I with u i ∈ H for all i ∈ I. Then the following are equivalent.

∃A, B >
2. There exists A, B > 0 such that for one or every non-empty set J, (2 ⇒ 1) Suppose there exists constants A, B > 0 such that : Take v ∈ H and choose (v j ) j∈J such that v j• = v for only one fixed index j • ∈ J and v j = 0 for other indices j ∈ J \ {j • }.We obtain Proposition 2. Let (X, Σ, µ) be a σ-finite measure space with positive measure µ and (u x ) x∈X a family of elements in H such that ∀h ∈ H, v → v, u x is a measurable function on X.The following are equivalent : x∈X is a frame associated with the measure space (X, Σ, µ) (or a (X, µ)-continuous frame)) 2. There exist constants A, B > 0 such that for one or every measure space (Y, S, ν) such that ν is a non-zero positive σ-finite measure, we have This part follows immediately by integration and Tonelli's theorem.
Applying the hypothesis to (v y ) y∈Y := (f (y)v) y∈Y yields the result.
Besides, based on the idea of changing the test spaces to ℓ 2 (J; H) and L 2 (Y, ν; H), we can define extended versions of the analysis, synthesis and frame operators.
For a classical I-indexed frame (u i ) i∈I with values in H and bounds A, B > 0, we have called respectively the J-extended analysis, synthesis and frame operators.We have : which shows that S J u is an invertible, positive definite operator on ℓ 2 (J; H), that T J u is a continuous and bounded below operator and thus a continuous and injective operator with closed range, and that (T J u ) * is a continuous and surjective operator as the adjoint of a continuous and injective operator with closed range.
For a (Y, S, ν)-continuous frame (u y ) y∈Y with values in H and bounds A, B > 0, we have called respectively the (Y, ν)-extended analysis, synthesis and frame operators.We have : is a continuous and bounded below operator and thus a continuous and injective operator with closed range, and that (T ) * is a continuous and surjective operator as the adjoint of a continuous and injective operator with closed range.In fact all of these extended analysis, synthesis, and frame operators are natural block-diagonal operators, so their importance compared to the classical analysis, synthesis, and frame operators is not yet clear.
3 Some characterizations of Euclidean continuous frames The objective of this subsection is to propose some useful rewritings of the quotients Definition 2. In a Hilbert space H over F, let (w k ) k∈[ [1,n]] be a sequence of n vectors in H.The Gramian matrix of (w k ) k∈[ [1,n]] is defined as the matrix W ∈ M n (F) whose k, l-th component is W k,l = w k , w l .This concept will be used in the following proposition where different Hilbert spaces are in play.
Proof.First, we have On the one hand, we have On the other hand, we have obviously n k,l=1 v k v l u l , u k = Tr(V U ).Moreover, In the same way, we find that

Generalization of the well-known characterizations of Euclidean finitely indexed frames and normalized tight frames (Parseval frames) to the Euclidean continuous case
Let (u x ) x∈X be a (X, Σ, µ)-Bessel family with values in F n .Note again that this implies that each component We have the following characterization Theorem 1.Let u = (u x ) x∈X be a continuous family of vectors in F n .Then Proof.(⇒) First let's assume that u is a continuous frame.For the sake of contradiction, suppose that (u k ) k∈[ [1,n]] is dependant.So there exist λ 1 • • • , λ n not all zero such that n k=1 λ k u k = 0. We then have N (λ; (u x ) x∈X ) = 0 using formula 1, which is a contradiction.Hence, (u k ) k∈[ [1,n]] is free.(⇐) Suppose that (u k ) k∈[ [1,n]] is free.In this case, the nonnegative continuous function v ∈ F n \ {0} → N (v; (u x ) x∈X ) ∈ [0, +∞[ restricted to the unit sphere of F n has a global minimum, the unit sphere being compact.This minimum is nonnegative and different from 0 because (u k ) k∈[ [1,n]] is free (see formula 1).Let's denote it by A > 0. We then have for v ∈ F n \ {0} : Remark 3.This theorem has been proved in [6] in the case of a finite sequence of vectors u = (u i ) i∈[ [1,m]] with values in F n (see proposition 1.4.3 p. 19).
Another short proof of the direction (⇐) of the theorem is now given.Suppose that (u k ) k∈[ [1,n]] is free.So the Gramian matrix U of (u k ) k∈[ [1,n]] is invertible, i.e. det(U ) = 0.The matrix U is a positive semidefinite matrix, so the condition det(U ) = 0 is equivalent to the fact that all the eingenvalues of U are (strictly) positive.Now, using formula 2, we have where V Y denotes the Gramian matrix of (v Y,k ) k∈[ [1,n]] and λ k (U ) is the smallest eigenvalue of U (we have used a trace inequality that can be easily derived from the trace inequalities in [12]).
Definition 3. We say that a continuous family u = (u x ) x∈X of vectors in F n is a normalised tight frame (or a Parseval frame) if We have the following characterization which means that u is a normalised tight frame Remark 4. This theorem has been proved in [6] in the case of a finite sequence of vectors u = (u i ) i∈[ [1,m]] with values in F n (see proposition 1.4.7 p. 21).
If n = 2, a necessary and sufficient condition for u ∈ L 2 (X, µ; F 2 ) to be a continuous frame is that the determinant of the Gramian matrix is (strictly) positive (see theorem 1), that is In the next section, we will provide a sufficient but not necessary condition for u ∈ L 2 (X, µ; F 2 ) to be a frame.

A sufficient condition for a continuous family in F 2 to be a continuous frame
Proposition 4. Let u ∈ L 2 (X, µ; F 2 ).A sufficient condition for u to be a continuous frame is , we have : By symmetry, the same lower bound is obtained if valid for all a ∈ F, we then obtain The quantity min(||u 1 || 2 , ||u 2 || 2 ) − | u 1 , u 2 | being strictly positive, remark 2 shows that u is a continuous frame.
Example 1. Let's give an example of a classical frame in C 2 using the previous criterion.We'll define two sequences of complex numbers u 1 and u 2 indexed by N * that satisfy the previous criterion and that are square summable.We take ∀n ∈ N * : u 1 n = 1 n e 2πian and ∀n ∈ N * : u 2 n = 1 n e 2πibn with a and b two real numbers such that a − b is not an integer.u 1 and u 2 are square summable with sum π 2 6 .Moreover, by the strict triangle inequality, the equality case being excluded because {e 2πi(a−b)n } n∈N * are not positively proportional due to the conditions imposed on a and b.From this, we deduce that u = ( 1 n e 2πian , 1 n e 2πibn ) n∈N * is a classical frame in C 2 .

Remark 2 .
then it is a continuous Bessel family by the Cauchy-Schwarz inequality.Hence: Let u = (u x ) x∈X be a continuous family of vectors in F n .Then

Theorem 2 . 1 √ 2 and v l = 1 √ 2 1 √ 2
Let u = (u x ) x∈X be a continuous family of vectors in F n .Then u is a continuous Parseval frame ⇔ u ∈ L 2 (X, µ; F n ) and U = I n .Proof.(⇒) Suppose u is a normalised tight frame.Then : ∀v ∈ F n \ {0} : N (v; (u x ) x∈X ) = 1.By choosing v k = 1 for one index k and 0 otherwise, we have : ∀k ∈ [[1, n]] : ||u k || = 1.Next, by picking two indices k and l and choosing v k = and 0 otherwise, we have ℜ( u k , u l ) = 0. Choosing this time v k = and v l = i √ 2 and 0 otherwise, we have ℑ( u k , u l ) = 0 and so u k , u l = 0.This means that U = I n .(⇐) Suppose U = I n .Then by Pythagoras' theorem, ) y∈Y ; (u x ) x∈X ) in the Euclidean case Let (u x ) x∈X be a (X, Σ, µ)-continuous frame with values in F n .Note that this implies that each component u