Frames generated by double sequences in Hilbert spaces

. In this paper, we introduce frames generated by double sequences ( d -frame) in Hilbert spaces and describe some of their properties. Furthermore, we discuss frame operators, alternate dual frames and stability for d -frames.


Introduction
Throughout this paper, the notations H and F will intend for an infinite dimensional Hilbert space and scalar field of real and complex numbers, respectively. N, R and C have their usual meanings.
The positive constants λ 1 and λ 2 are called the lower and upper frame bounds respectively. If λ 1 = λ 2 , then {x n } n∈N is said to be a tight frame and if λ 1 = λ 2 = 1, then {x n } n∈N is called Parseval frame.
Here, we recall that every Bessel sequence in a Hilbert space is not necessarily a frame. But by including some additional elements or by sparsing the elements of such sequences one can convert these sequences into frames. Considering this fact, recently Sharma et al. [8] tried to construct frames for the Hilbert spaces using such Bessel sequences which are not frames for the given spaces. In fact, they gave the following definition. |⟨x, x n,i ⟩| 2 ≤ λ 2 ∥x∥ 2 , x ∈ H.
The positive constants λ 1 and λ 2 are called the lower and upper approximative frame bounds, respectively. If λ 1 = λ 2 , then {x n,i } i=1,2,...,mn n∈N is a tight approximative frame and if λ 1 = λ 2 = 1, then it is called a Parseval approximative frame. A sequence {x n,i } i=1,2,...,mn n∈N is said to be an approximative Bessel sequence if righthand side of inequality (2) is satisfied.
Again, Sharma et. al. published a corrigendum [9] on the same paper [8] with following statement: "The paper [8] requires some clarifications, throughout the paper [8], {x n,i } i=1,2,...,mn n∈N is a sequence of special index in H such that x n,i = x n+1,i , i = 1, 2, . . . , m n ; n ∈ N and {α n,i } i=1,2,...,mn n∈N is a sequence of special index in F such that α n,i = α n+1,i , i = 1, 2, . . . , m n ; n ∈ N." The idea of construction of approximative frame is quite interesting, but there is some ambiguity in the definition of approximative frame and examples discussed in [8] and corrigendum [9]. We have the following dissensions on the Definition 2 and examples given in [8]: (i) Considering {x n,i } i=1,2,...,mn n∈N as a sequence then it contains same terms repeated infinitely many times. Hence, lim n→∞ mn i=1 |⟨x, x n,i ⟩| 2 becomes unbounded.
(ii) As considered by the authors in examples [8], n a fixed number and m n , a function of n and the claim that the sequence satisfies the equation (2) is contradictory. (iii) Since, {m n } is an increasing sequence of natural numbers and from the Definition 2, it is clear that m n is dependent on n such that m n ≥ n, hence in equation (2), lim n→∞ implies m n → ∞. Which is also a contradictory statement in case of a sequence. (iv) Considering {x n,i } i=1,2,...,mn n∈N a sequence of finite sequences, m n an increasing sequence depending on n, then the statement x n,i = x n+1,i , i = 1, 2, . . . , m n ; n ∈ N given in corrigendum [9] is again contradictory because n th term, i.e., {x n,i } i=1,2,...,mn contains m n terms while (n + 1) th term i.e. {x n+1,i } i=1,2,...,m n+1 contains m n+1 terms. Now, we discuss above infusions with the help of Examples (3.6) and (3.2) of [8] and [9] in details.
Example 3.6 ([8]): "Let H be a Hilbert space and {e n } be an orthonormal basis for H. Define a sequence {x n } by x n = en √ n , n ∈ N. Then it is elementary to observe that {x n } is not a frame for H. Define a sequence . . .

2
, hence for n = 1, i = 1; for n = 2, i = 1, 2, 3; for n = 3, i = 1, 2, 3, 4, 5, 6 and so on. So the construction of the sequence is as: then the (n + 1) th term of the above sequence will have (n + 1) more terms than the total terms of n th term. Hence the statement given in [9] is erroneous. Further, it cannot be claimed that such sequence is an approximative frame while only the infine'th term   So, as discussed in previous example, the sequence {y n,i } i=1,2,...,2n n∈N is neither a Bessel sequence nor an approximative Bessel sequence. Case 2. If we consider {y n,i } i=1,2,...,2n n∈N as a sequence of sequences, then the (n+1) th term will have 2 more terms than the total terms of n th term. Which is again a contradiction to the statement given in corrigendum [9]. Similarly, as in previous example, only the infine'th term of the sequence {y n,i } i=1,2,...,2n n∈N , i.e., {e 1 , e 1 , e 2 , e 2 , e 3 , e 3 , . . . , e n , e n , e n+1 e n+1 , . . . } is a frame for infinite dimensional Hilbert space H.
The same is the case for Example 3.2, Example 3.3, Example 3.5 and Example 3.6 in [8].
Considering the above discussions, it is concluded that the authors have not been able to construct the synthesis operator, analysis operator, frame operator for the entire terms of the sequence i.e., the results given in [8] are erroneous. From the above discussion, if we consider limit for both the suffices, then it becomes a case of double sequences. Now, we define a new generalization of frame with the help of double sequences and named it as d-frame.
The theory of double sequence and double series is an extension of single or ordinary sequences and series. In 1900, Pringsheim [6] introduced the concept of real double sequences and their convergence. A double sequence is a function or mapping x : x ij is known as double series. We use following definition and concepts to define d-frame and to prove the results on the properties of d-frame and frame operators.
The sequence of partial sums of double sequence {x ij } i,j∈N is defined by x ij , for all m, n ∈ N.
If no such limit exists then the double series is divergent.
If every x ij is non-negative then The constants λ 1 and λ 2 are called lower and upper d-frame bounds respectively. If Remark 2. Let {y i } i∈N is a frame for Hilbert space H with lower and upper frame bounds λ 1 and λ 2 , respectively. Then, we define a double sequence which is a d-frame for H with the same bounds λ 1 and λ 2 .
Let {e i } i∈N be an orthonormal basis for H. Following examples vindicate the Definition 4. Then We know that every Bessel sequence is not a frame always. One can construct a double sequence from a given Bessel sequence, which becomes a d-frame.
Example 3. Given a sequence {x n } such that x n = en √ n , for all n ∈ N. Then, {x n } is a Bessel sequence but not a frame for H because it does not satisfy the lower condition of frame. Define a sequence {x ij } i,j∈N in H by . . .
Example 4. The sequence {x n } n∈N such that x n = e n +e n+1 , for all n ∈ N is a Bessel sequence for H, but not a frame for H. Define a sequence {x ij } i,j∈N in H by which is a d-frame for H with lower and upper d-frame bounds λ 1 = 4 and λ 2 = 8 respectively.
For the rest part of this paper, we define the space as Then l 2 (N × N) is a Hilbert space with the norm induced by the inner product which is given by α ij x ij , for all {α ij } i,j∈N ∈ l 2 (N × N).
If {x ij } i,j∈N is a d-frame then operator T is called pre d-frame (synthesis) operator and the adjoint operator T * of T is called analysis operator for d-frame. Proof. From the definition of T , it is obvious that T is linear. Let {α ij } i,j∈N ∈ l 2 (N × N). For any m, n, m ′ , n ′ ∈ N with m > m ′ , n > n ′ , we have m,n i,j=1 α ij x ij exists. Hence, T is well defined. Further, Conversely, for any x ∈ H, we have Hence, Thus, ⟨x, x ij ⟩x ij , for all x ∈ H.
Since T and T * both are linear, so S is also linear. |⟨x, x ij ⟩| 2 .
Using definition of d-frame, we have Hence, Thus, S is a positive operator. Further, Generalizing Theorem 1, if we take an extra condition i.e., operator T is surjective then following theorem gives a necessary and sufficient condition for a double sequence to be a frame Proof. It is clear from Theorem 1 that, the operator T is well defined, bounded and linear. Since {x ij } i,j∈N is a d-frame hence the d-frame operator S = T T * is invertible(bijective) which implies T is also surjective.
Conversely, let T is well defined, bounded, linear and surjective. From Theorem 1 it is already clear that {x ij } i,j∈N is a Bessel sequence. Now we prove the lower d-frame inequality.
Since T is surjective and T * is one-one operator, then the operator S = T T * is invertible and positive. For any a, b ∈ H, |⟨a, b⟩| ≤ ∥a∥ ∥b∥ (Cauchy Schwarz inequality).
Hence, since S is bounded, Finally, Hence {x ij } i,j∈N is a d-frame for H. □ Now, we establish following result to characterize d-frames in terms of bounded linear operators. Proof. Since T is a linear bounded operator hence T * is also linear bounded. Taking T * (x) ∈ H and using the definition of d-frame {x ij } i,j∈N , By the given condition, we get In the following theorem, we prove that one can also construct a d-frame with the help of d-frame operator. Proof. From equation (5) of Theorem 2, we have Frames generated by double sequences in Hilbert spaces = lim m,n→∞ m,n i,j=1 x, Hence, by equation (6), {S −1 (x ij )} i,j∈N is a d-frame for H with lower and upper bound λ −1 2 and λ −1 1 respectively i.e., Remark 4. In above theorem, equations (6) and (7) show that S −1 is a d-frame operator for the d-frame {S −1 (x ij )} i,j∈N . And for any x ∈ H,

Alternate dual d-frames
In this section, we study alternate/canonical dual d-frame and its properties. ⟨x,x ij ⟩x ij , Proof. For x ∈ H and N × M ⊆ N × N, define the operator T N ×M as It is obvious that the operator T N ×M (x) is well defined, linear and bounded. And by the definition of dual d-frame, we have Therefore, Hence, □ Remark 6. Every Parseval d-frame is dual d-frame of itself. Hence identity (8) becomes Which is called Parseval d-frame identity.

Stability of d-frames
In this section, we study the stability of d-frames, which is similar version of Paley-Wiener Theorem for frames [2]. Also, we prove similar results for the stability of canonical dual d-frame.
Then, {y ij } i,j∈N is also a d-frame for H with lower and upper d-frame Now, define an another operator U : So, for m > m ′ and n > n ′ , where m, n, m ′ , n ′ ∈ N, we have m,n i,j=1 α ij y ij .
Using (10), we get m,n i,j=1 α ij y ij exists. Which implies Therefore, operator U is linear, well defined and bounded. Thus, by Theo- Now, using T and U in equation (9) (14) By Theorem 5, we know that S = T T * is a d-frame operator for {x ij } i,j∈N with upper bound λ −1 1 .

Conclusion
The paper introduces frames generated by double sequences in Hilbert spaces and named as d-frames. Some of the properties of d-frames, frame operators, alternate dual d-frames and stability for d-frames are also discussed in details. Applications of d-frames in other areas of study, specially in signal processing, can be considered as future scope of the work.