New inequalities for weaving frames in Hilbert spaces

In this paper, we establish Parseval identities and surprising new inequalities for weaving frames in Hilbert space, which involve scalar $\lambda\in\rs$. By suitable choices of $\lambda$, one obtains the previous results as special cases. Our results generalize and improve the remarkable results which have been obtained by Balan et al. and G\u{a}vru\c{t}a.


Introduction
Frames in Hilbert spaces were first introduced in 1952 by Duffin and Schaeffer [8] to study some deep problems in nonharmonic Fourier series, reintroduced in 1986 by Daubechies, Grossmann and Meyer [6], and today frames play important roles in many applications in several areas of mathematics, physics, and engineering, such as coding theory [14,17], sampling theory [23,20], quantum measurements [9],filter bank theory [13] and image processing [7].
Let H be a separable space and I a countable index set. A sequence {φ i } i∈I of elements of H is a frame for H if there exist constants A, B > 0 such that The number A, B are called lower and upper frame bounds, respectively. If A = B, then this frame is called an A-tight frame, and if A = B = 1, then it is called a Parseval frame.
Suppose {φ i } i∈I is a frame for H , then the frame operator is a self-adjoint positive invertible operators, which is given by The following reconstruction formula holds: where the family { φ i } i∈I = {S −1 φ i } i∈I is also a frame for H , which is called the canonical dual frame of {φ i } i∈I . The frame {ϕ i } i∈I for H is called an alternate dual frame of {φ i } i∈I if the following formula holds: Suppose that {φ i } i∈I and {ψ i } i∈I are woven, the frame operator of {φ i } i∈σ ∪ {ψ i } i∈σ c is defined by then S W is a bounded, invertible, self-adjoint and positive operator. A frame {ϕ i } i∈I is called an alternate dual frame of {φ i } i∈σ ∪ {ψ i } i∈σ c if for all f ∈ H the following identity holds: For every σ ⊂ I, define the bounded linear operators S σ W , S σ c W : H → H by It is easy to check that S σ W and S σ c W are self-adjoint. In [1], the authors solved a long-standing conjecture of the signal processing community. They showed that for suitable frames {φ i } i∈I , a signal f can (up to a global phase) be recovered from the phaseless measurements {| f, φ i |} i∈I . Note, that this only shows that reconstruction of f is in principle possible, but there is not an effective constructive algorithm. While searching for such an algorithm, the authors of [2] discovered a new identity for Parseval frames [3]. The authors in [10,24] generalized these identities to alternate dual frames and got some general results. The study of inequalities has interested many mathematicians. Some authors have extended the equalities and inequalities for frames in Hilbert spaces to generalized frames [16,18,22,19]. The following form was given in [3] (See [2] for a discussion of the origins of this fundamental identity). Theorem 1. Let {φ i } i∈I be a Parseval frame for H . For every J ⊂ I and every f ∈ H , we have Later on, the author in [10] generalized Theorem 1 to general frames.
Theorem 2. Let {φ i } i∈I be a frame for H with canonical dual frame { φ i } i∈I . Then for every J ⊂ I and every f ∈ H , we have Theorem 3. Let {φ i } i∈I be a frame for H and {ϕ i } i∈I be an alternate dual frame of {φ i } i∈I . Then for every J ⊂ I and every f ∈ H , we have (1.4) Motivated by these interesting results, the authors in [24] generalized the Theorem 3 to a more general form which does not involve the real parts of the complex numbers.
Theorem 4. Let {φ i } i∈I be a frame for H and {ϕ i } i∈I be an alternate dual frame of {φ i } i∈I . Then for every J ⊂ I and every f ∈ H , we have In this paper, we generalize the above mentioned results for weaving frames in Hilbert spaces. We generalize the above inequalities to a more general form which involve a scalar λ ∈ R which is different from the scalar λ ∈ [0, 1] in [19]. Since a frame is woven with itself, the previous equality and inequalities in frames can be obtained as a special case of the results we establish on weaving frames.

Results and their proofs
We first state a simple result on operators, which is a distortion of [24, Lemma 2.1].
Proof. A simple computation shows that Now we state and prove a Parseval weaving frame identity.
Theorem 5. Suppose {φ i } i∈I and {ψ i } i∈I for a Hilbert space H are 1-woven. Then for all σ ⊂ I and Next, we prove the inequality of (2.1). A simple computation shows that and so Notice that operator S σ W is also self-adjoint and therefore (S σ W ) * = S σ W . Applying Lemma 1 to the operators P = S σ W and Q = S σ c W , we obtain Then equation (2.2) means that Therefore for all f ∈ H , we have This completes the proof.
Remark 6. If we take φ i = ψ i for all i ∈ I in Theorem 5, we can obtain the Theorem 1.
Lemma 2. Let P, Q ∈ L(H ) be two self-adjoint operators such that P + Q = I H . Then for any λ ∈ R , and all f ∈ H , we have Proof. For all f ∈ H , we have A simple computation of (2.3), we have This proves the desired result.
Theorem 7. Suppose two frames {φ i } i∈I and {ψ i } i∈I for a Hilbert space H are woven. Then for any λ ∈ R , for all σ ⊂ I and all f ∈ H , we have Proof. Since {φ i } i∈I and {ψ i } i∈I are woven, for all σ ⊂ I, Considering We have (2.9) Using equations (2.5)-(2.9) in the inequality (2.3), we obtain Remark 8. If we take φ i = ψ i for all i ∈ I and λ = 1 in Theorem 7, we can obtain Theorem 2 with scalar 3/4.
Theorem 9. Suppose two frames {φ i } i∈I and {ψ i } i∈I for a Hilbert space H are woven and {ϕ i } i∈I is an alternate dual frame of the weaving frame {φ i } i∈σ ∪ {ψ i } i∈σ c . Then for any λ ∈ R , for all σ ⊂ I and all f ∈ H , we have Proof. For all f ∈ H and all σ ⊂ I, define the operators Then the series converge unconditionally and E σ , E σ c ∈ L(H ). By (1.1), we have E σ + E σ c = I H .
Applying Lemma 3 to the operators P = E σ and Q = E σ c , for all f ∈ H , we obtain A simple computation of (2.11) and (2.12), we have Then, Hence, Since, (2.14) Re Using equations (2.13)-(2.17), we have We now prove the inequality of (2.10). From Lemma 3, we have Then Therefore, Using equations (2.14)-(2.17) and (2.19), we have The proof is completed.
Remark 10. Theorem 3 can be obtained from Theorem 9 by taking φ i = ψ i for all i ∈ I and λ = 1 2 .
Theorem 11. Suppose Φ = {φ i } i∈I and Ψ = {ψ i } i∈I for a Hilbert space H are woven and {ϕ i } i∈I is an alternate dual frame of the weaving frame {φ i } i∈σ ∪ {φ i } i∈σ c . Then for any λ ∈ R , for all σ ⊂ I and all f ∈ H , we have (2.20) Proof. For σ ⊂ I and f ∈ H , we define the operator E σ and E σ c as in Theorem 9. Therefore, we have Hence (2.20) holds. The proof is completed.
Proof. For all σ ⊂ I and f ∈ H , we define the operators and Note that these series converge unconditionally. Also we have Applying Lemma 1 to the operators P = E σ + E σ c and Q = F σ + F σ c and for every Hence the relation holds.
Observe that if we consider σ ⊂ I and then Theorem 11 follows from Theorem 12.
Remark 13. If we take φ i = ψ i for all i ∈ I in Theorem 11 and Theorem 12, we can obtain Theorem 4 and Theorem 2.3 of [24].
Theorem 14. Suppose two frames {φ i } i∈I and {ψ i } i∈I for a Hilbert space H are woven. Then for any λ ∈ R , σ ⊂ I and f ∈ H , we have Proof. Considering positive operators P = S and P Q = P (I H − P ) = P − P 2 = (I H − P )P = QP, Theorem 15. Suppose two frames {φ i } i∈I and {ψ i } i∈I for a Hilbert space H are woven. Then for any λ ∈ R , σ ⊂ I and f ∈ H , we have Proof. By (2.5), we have Next, we prove the last part. Let P = S Proof. Since {φ i } i∈I and {ψ i } i∈I are A-woven, we have S −1 W = 1 A I H , and then the results hold by Theorem 14 and Theorem 15.
Remark 16. If we take λ = 1 and φ i = ψ i for all i ∈ I in Theorem 14 and Theorem 15, we obtain the similar inequalities in Theorem 5 and Theorem 6 of [15].