Semi-continuous g-frames in Hilbert spaces

SEMI-CONTINUOUS G-FRAMES IN HILBERT SPACES

arXiv:1808.08445v1 [math.FA] 25 Aug 2018

ANIRUDHA PORIA Abstract. In this paper, we introduce the concept of semi-continuous g-frames in Hilbert spaces. We first construct an example of semi-continuous g-frames using the Fourier transform of the Heisenberg group and study the structure of such frames. Then, as an application we provide some fundamental identities and inequalities for semi-continuous gframes. Finally, we present a classical perturbation result and prove that semi-continuous g-frames are stable under small perturbations.

1. Introduction Discrete and continuous frames arise in many applications in mathematics and, in particular, they play important roles in scientific computations and digital signal processing. The concept of a frame in Hilbert spaces has been introduced in 1952 by Duffin and Schaeffer [13], in the context of nonharmonic Fourier series (see [28]). After the work of Daubechies et al. [11] frame theory got considerable attention outside signal processing and began to be more broadly studied (see [8, 21]). A frame for a Hilbert space is a redundant set of vectors in Hilbert space which provides non-unique representations of vectors in terms of frame elements. The redundancy and flexibility offered by frames has spurred their application in several areas of mathematics, physics, and engineering such as wavelet theory, sampling theory, signal processing, image processing, coding theory and many other well known fields. Applications of frames, especially in the last decade, motivated the researcher to find some generalization of frames like continuous frames [1, 22], g-frames [26], Hilbert−Schmidt frames [24, 25], K-frames [19] and etc. Our main purpose in this paper is to study a generalization of frames, name as semi-continuous g-frames, which are natural generalizations of g-frames and continuous g-frames. We investigate the structure of semi-continuous g-frames and establish some identities and inequalities of these frames. Also, we present a perturbation result and discuss the stability of the perturbation of a semi-continuous g-frame.

Date: August 28, 2018. 2010 Mathematics Subject Classification. Primary 42C15; Secondary 47B38, 42C40. Key words and phrases. g-frames; continuous g-frames; semi-continuous g-frames; perturbation; frame identity; stability.

1

2

ANIRUDHA PORIA

Throughout this paper, H and K are two Hilbert spaces; J is a countable index set; (X , µ) is a measure space with positive measure µ; {Kx }x∈X is a sequence of closed subspaces of K; L(H, Kx ) is the collection of all bounded linear operators from H into Kx ; if Kx = H for any x ∈ X , we denote L(H, Kx ) by L(H). We recall that a family {fj }j∈J in H is called a (discrete) frame for H, if there exist constants 0 < A ≤ B < ∞ such that Akf k2 ≤

X j∈J

|hf, fj i|2 ≤ Bkf k2 ,

∀f ∈ H.

The concept of the discrete frame was generalized to continuous frame by Kaiser [22] and independently by Ali et al. [1]. A family of vectors {ψx }x∈X ⊆ H is called a continuous frame for H with respect to (X , µ), if {ψx }x∈X is weakly-measurable, i.e., for any f ∈ H, x → hf, ψx i is a measurable function on X , and if there exist two constants A, B > 0 such that Z |hf, ψx i|2 dµ(x) ≤ Bkf k2 , ∀f ∈ H. Akf k2 ≤ X

Continuous frames have been widely applied in continuous wavelets transform [2] and the short-time Fourier transform [21]. We refer to [3, 16, 17] for more details on continuous frames. The notion of a discrete frame extended to g-frame by Sun [26], which generalized all the existing frames such as bounded quasi-projectors [15], frames of subspaces [7], pseudo-frames [23], oblique frames [9], etc. G-frames are natural generalizations of frames as members of a Hilbert space to bounded linear operators. Let {Kj : j ∈ J} ⊂ K be a sequence of Hilbert spaces. A family {Λj ∈ L(H, Kj ) : j ∈ J} is called a g-frame, for H with respect to {Kj : j ∈ J} if there are two constants A, B > 0 such that Akf k2 ≤

X j∈J

kΛj (f )k2 ≤ Bkf k2 ,

∀f ∈ H.

The continuous g-frames were proposed by Dehghan and Hasankhani Fard in [12], which are an extension of g-frames and continuous frames. A family {Λx ∈ L(H, Kx ) : x ∈ X } is called a continuous g-frame for H with respect to (X , µ), if {Λx : x ∈ X } is weaklymeasurable, i.e., for any f ∈ H, x → Λx (f ) is a measurable function on X , and if there exist two constants A, B > 0 such that Z kΛx (f )k2 dµ(x) ≤ Bkf k2 , Akf k2 ≤ X

∀f ∈ H.


3

Notice that if X is a countable set and µ is a counting measure, then the continuous g-frame is just the g-frame. By the Riesz representation theorem, for any Λ ∈ L(H, C), there exist a h ∈ H, such that Λ(f ) = hf, hi for all f ∈ H. Hence, if Kx = C for any x ∈ X , then the continuous g-frame is equivalent to the continuous frame. This paper is organized as follows. After the introduction, in Section 2, we introduce the semi-continuous g-frames in Hilbert spaces and construct an example using the Fourier transform of the Heisenberg group. Then we study the structure of semi-continuous g-frames using shift-invariant spaces. In Section 3, we first list some fundamental identities and inequalities of discrete frames just for the contrast to the main results of this section. Then we derive some important identities and inequalities of semi-continuous g-frames. Finally, in Section 4, we present a classical perturbation result and prove that semi-continuous g-frames are stable under small perturbations. 2. Semi-continuous g-frames Let {Kx,j : x ∈ X , j ∈ J} ⊂ K be a sequence of Hilbert spaces. Definition 2.1. A family {Λx,j ∈ L(H, Kx,j ) : x ∈ X , j ∈ J} is called a semi-continuous g-frame for H with respect to (X , µ), if {Λx,j : x ∈ X , j ∈ J} is weakly-measurable, i.e., for any f ∈ H and any j ∈ J, the function x → Λx,j (f ) is measurable on X , and if there exist two constants A, B > 0 such that Z X (2.1) Akf k2 ≤ kΛx,j (f )k2 dµ(x) ≤ Bkf k2 , X j∈J

∀f ∈ H.

If only the right-hand inequality of (2.1) is satisfied, we call {Λx,j : x ∈ X , j ∈ J} the semi-continuous g-Bessel sequence for H with respect to (X , µ) with Bessel bound B. Remark 2.2. If 0 < µ(X ) < ∞, and for any fixed x ∈ X , the sequence {Λx,j : j ∈ J} is a g-frame for H with respect to {Kx,j : j ∈ J}, then {Λx,j : x ∈ X , j ∈ J} is a semi-continuous g-frame for H with respect to (X , µ). Moreover if |J| < ∞, and for any fixed j ∈ J, the sequence {Λx,j : x ∈ X } is a continuous g-frame for H, then {Λx,j : x ∈ X , j ∈ J} is a semi-continuous g-frame for H with respect to (X , µ). In the following, we shall construct an example of such frames using the Fourier transform of the Heisenberg group.

4

ANIRUDHA PORIA

2.1. Heisenberg Group. The Heisenberg group H is a Lie group whose underlying manifold is R3 . We denote points in H by (p, q, t) with p, q, t ∈ R, and define the group operation by (2.2)

1 (p1 , q1 , t1 )(p2 , q2 , t2 ) = (p1 + p2 , q1 + q2 , t1 + t2 + (p1 q2 − q1 p2 )). 2

It is easy to verify that this is a group operation, with the origin 0 = (0, 0, 0) as the identity element. Notice that the inverse of (p, q, t) is given by (−p, −q, −t). The Haar measure on the group H = R3 is the usual Lebesgue measure. The irreducible representations of the Heisenberg group has been identified by all nonzero elements in R∗ (= R \ {0}) (see [14]). Indeed, for any λ ∈ R∗ , the associated irreducible representation ρλ of H is equivalent to Schr¨ odinger representation into the class of unitary operators on L2 (R), such that for any (p, q, t) ∈ H and f ∈ L2 (R), the operator ρλ (p, q, t) is defined by (2.3)

1

ρλ (p, q, t)f (x) = eiλt eiλ(px+ 2 (pq)) f (x + q).

It is easy to see that ρλ (p, q, t) is a unitary operator satisfying the homomorphism property: ρλ ((p1 , q1 , t1 )(p2 , q2 , t2 )) = ρλ (p1 , q1 , t1 )ρλ (p2 , q2 , t2 ). Thus each ρλ is a strongly continuous unitary representation of H. By Stone and von Neumann theorem ([14]), {ρλ : λ ∈ R∗ } are all the infinite dimensional irreducible unitary representations of H, whose set has non-zero Plancherel measure. The measure |λ|dλ is the b (∼ Plancherel measure on the dual space H = R∗ ) of H, and dλ is the Lebesgue measure on R∗ . For ϕ ∈ L2 (H) and λ ∈ R∗ , we denote ϕ(λ) b the operator-valued Fourier transform of ϕ

at a given irreducible representation ρλ , which is defined by (2.4)

ϕ(λ) b =

Z

ϕ(x)ρλ (x)dx.

H

The operator ϕ(λ) b is a unitary map on L2 (R) into L2 (R), such that for any f ∈ L2 (R) ϕ(λ)f b (y) =

Z

ϕ(x)ρλ (x)f (y)dx.

H

Therefore ϕ(λ) b belongs to L2 (R) ⊗ L2 (R). If ϕ ∈ L2 (H), ϕ(λ) b is actually a Hilbert−Schmidt

operator on L2 (R) and from the Plancherel theorem we have (2.5)

Z

H

|ϕ(x)|2 dx =

Z

R∗

2 kϕ(λ)k b H.S. |λ|dλ,


5

the norm k · kH.S. denotes the Hilbert−Schmidt norm in L2 (R) ⊗ L2 (R). The proof of the Plancherel theorem for the Heisenberg group can be found in [20], and for more general groups, see [14]. To construct our example of semi-continuous g-frames, we shall define another unitary operator as follows. Let Π := [0, 1] and L := ℓ2 (Z, L2 (R) ⊗ L2 (R)) be the Hilbert space of all sequences with values in the space L2 (R) ⊗ L2 (R), i.e., n o X L = {an }n∈Z : an ∈ L2 (R) ⊗ L2 (R) and kan k2H.S. < ∞ . n∈Z

1 Lemma 2.3. For any σ ∈ Π, let Tσ : L2 (H) → L given by Tσ f (j) = |σ + j| 2 fb(σ + j). Then P Tσ is well-defined and j∈Z |σ + j| kfb(σ + j)k2H.S. < ∞.

Proof. Let f ∈ L2 (H). Using Plancherel theorem and an application of periodization method, we obtain kf k2L2 (H) =

Z

R∗

kfb(λ)k2H.S. |λ|dλ

= =

Z

Z

X

σ∈Π j∈Z

X

σ∈Π j∈Z

|σ + j| kfb(σ + j)k2H.S. dσ kTσ f (j)k2H.S. dσ.

Hence, the result follows from the fact that f ∈ L2 (H).

Example 2.4. Consider X = Π and J = Z. For any σ ∈ Π and j ∈ Z, define Λσ,j : L2 (H) → L2 (R) ⊗ L2 (R) as Λσ,j (f ) = Tσ f (j). Then for every f ∈ L2 (H), using Lemma 2.3 we get Z

X

σ∈Π j∈Z

kΛσ,j (f )k2H.S. dσ

=

Z

X

σ∈Π j∈Z

=

Z

kTσ f (j)k2H.S. dσ

2 X 1

|σ + j| 2 fb(σ + j)

σ∈Π j∈Z

=

H.S.

dσ

kf k2L2(H) .

Therefor {Λσ,j : σ ∈ Π, j ∈ Z} is a semi-continuous g-frame with frame bounds A = B = 1. Corollary 2.5. Let 0 < µ(X ) < ∞. For any fixed σ ∈ X , let {Λσ,j : j ∈ J} be a g-frame for L2 (H). Then {Λσ,j : σ ∈ X , j ∈ J} is a semi-continuous g-frame for L2 (H) with respect to (X , µ) with unified frame bounds multiplied by µ(X ).

6

ANIRUDHA PORIA

Proof. Since {Λσ,j : j ∈ J} be a g-frame for L2 (H), there exist constants A, B > 0 such that Akf k2L2 (H) ≤

X j∈J

kΛσ,j (f )k2 ≤ Bkf k2L2 (H) ,

∀f ∈ L2 (H).

Taking integral from all sides of the preceding inequality, we obtain Z X 2 Aµ(X )kf kL2 (H) ≤ kΛσ,j (f )k2 dµ(σ) ≤ Bµ(X )kf k2L2 (H) , X j∈J

∀f ∈ L2 (H).

Hence, the result follows.

Now, we shall define shift-invariant spaces and give an example. Definition 2.6. Let Γ be a countable subset of H. A subspace V ⊂ L2 (H) is called Γinvariant if Lγ φ ∈ V for all γ ∈ Γ and all φ ∈ V, where Lγ φ(w) = φ(γ −1 w), w ∈ H. If Γ is a discrete subset of H, then V is called shift-invariant. Example 2.7. Let φ ∈ L2 (H) and Γ be a lattice. Then the space hφiΓ generated by Γ-shifts of φ is a shift-invariant space. Before we prove the main result of this section, we first need the following. Let T : L2 (H) → L2 (Π, L). Then for any σ ∈ Π and j ∈ Z, T f (σ)(j) ∈ L2 (R) ⊗ L2 (R). By Lemma 2.3, it is clear that T f (σ) = Tσ f . Let Γ = Γ1 Γ0 = xz ∈ H : x ∈ Γ1 , z ∈ Γ0 ,

where Γ1 be any discrete subset of H and Γ0 be the lattice of integral points in Z. Then for y ∈ H and σ ∈ Π, define the unitary operator ρ˜σ (y) : L → L by (˜ ρσ (y)h)j = ρσ+j (y) ◦ hj ,

h ∈ L,

where ρσ+j (y) ◦ hj denotes function composition. Also, define ρ˜(y) : L2 (Π, L) → L2 (Π, L) by (˜ ρ(y)a)(σ) = ρ˜σ (y)a(σ),

a ∈ L2 (Π, L).

Note that if γ ∈ Γ0 , then (˜ ρ(γ)a)(σ) = e2πihσ,γi a(σ) for all a ∈ L2 (Π, L). Further, the mapping T is unitary, and for each y ∈ H, we have T (Ly φ)(σ) = (˜ ρ(y)T φ)(σ). Proofs of these results and a more detailed study of these operators can be found in ([10], Section 3). Fix a discrete subset Γ of H of the form Γ1 Γ0 . Let V ⊂ L2 (H) be a countable set.


7

Define E(V) = {Lγ φ : γ ∈ Γ, φ ∈ V} and put S = span E(V). Let R be the range function associated with S. Motivated by the results of Currey et al. [10], we obtain the following. Lemma 2.8. Let f ∈ L2 (H), Γ ⊆ H and V ⊂ L2 (H). Then Z X X |hT f (σ), T (Lk φ)(σ)i|2 dσ. |hf, Lγ φi|2 = Π φ∈V,k∈Γ 1

φ∈V,γ∈Γ

Proof. Let f ∈ L2 (H). Since kT f k = kf k, we have X

2

φ∈V,γ∈Γ

|hf, Lγ φi| =

X

φ∈V,γ∈Γ

|hT f, T (Lγ φ)i|

2

X

=

φ∈V,γ∈Γ

X

=

φ∈V,γ∈Γ

Putting γ = kl, with k ∈ Γ1 , l ∈ Γ0 , we get

Z 2 hT f (σ), T (Lγ φ)(σ)i dσ Π

Z 2 hT f (σ), (˜ ρ(γ)T φ)(σ)i dσ . Π

(˜ ρ(kl)T φ)(σ) = ρ˜σ (kl)T φ(σ) = ρ˜σ (k)˜ ρσ (l)T φ(σ) = e2πihσ,li ρ˜σ (k)T φ(σ). Thus X

φ∈V,γ∈Γ

Z 2 hT f (σ), (˜ = ρ (γ)T φ)(σ)i dσ Π

X

φ∈V,(k,l)∈Γ

Z 2 hT f (σ), ρ˜σ (k)T φ(σ)ie−2πihσ,li dσ . Π

For each k, define Fk (σ) = hT f (σ), ρ˜σ (k)T φ(σ)i. Then Fk is integrable with square summable Fourier coefficients, hence Fk ∈ L2 (Π). Using Fourier inversion formula we obtain 2 Z X X hT f (σ), ρ˜σ (k)T φ(σ)ie−2πihσ,li dσ = |Fˆk (l)|2 φ∈V,(k,l)∈Γ

Π

φ∈V,(k,l)∈Γ

X

=

φ∈V,k∈Γ1

X

=

φ∈V,k∈Γ1

kFk k2

Z

Π

|Fk (σ)|2 dσ.

Again, by substituting Fk (σ) = hT f (σ), ρ˜σ (k)T φ(σ)i in the above we get X X Z X Z 2 2 |hf, Lγ φi| = |Fk (σ)| dσ = |hT f (σ), ρ˜σ (k)T φ(σ)i|2 dσ φ∈V,γ∈Γ

φ∈V,k∈Γ1

(2.6)

Π

φ∈V,k∈Γ1

=

Z

X

Π

Π φ∈V,k∈Γ 1

|hT f (σ), T (Lk φ)(σ)i|2 dσ.

This completes the proof.

Now we are in a position to prove our main result of this section. Theorem 2.9. If {Tσ (Lk φ) : k ∈ Γ1 , φ ∈ V} is a frame for its spanned vector space for almost every σ ∈ Π. Then {Lγ φ : γ ∈ Γ, φ ∈ V} is also a frame for its spanned vector space.

8

ANIRUDHA PORIA

Proof. Suppose that f ∈ S, then T f (σ) ∈ R(σ) holds for a.e. σ. Since for a.e. σ ∈ Π, {Tσ (Lk φ) : k ∈ Γ1 , φ ∈ V} is a frame for its spanned vector space, there exist 0 < A ≤ B < ∞ such that 2

AkT f (σ)k ≤

Z

X

Π φ∈V,k∈Γ 1

|hT f (σ), T (Lk φ)(σ)i|2 dσ ≤ BkT f (σ)k2 .

holds for a.e. σ. Integrating over Π yields Z kT f (σ)k2 dσ Akf k2 = AkT f k2 = A

≤

Π

Z

X

Π φ∈V,k∈Γ 1

≤ B

Z

Π

|hT f (σ), T (Lk φ)(σ)i|2 dσ

kT f (σ)k2 dσ = Bkf k2 .

Using (2.6) we obtain Akf k2 ≤

X

φ∈V,γ∈Γ

|hf, Lγ φi|2 ≤ Bkf k2 .

Hence, we have the desired result.

Remark 2.10. Notice that the family {Tσ (Lk φ) : k ∈ Γ1 , φ ∈ V} constitutes a frame for the space which consists of all functions of the form Tσ f for every f ∈ L2 (H). Similarly, the above result can be extended for semi-continuous g-frames using the Riesz representation theorem. 3. Identities and inequalities for semi-continuous g-frames Let {Λx,j ∈ L(H, Kx,j ) : x ∈ X , j ∈ J} be a semi-continuous g-frame for H with respect to (X , µ). Then we define the semi-continuous g-frame operator S as follows: Z X S : H → H, Sf = Λ∗x,j Λx,j f dµ(x), X j∈J

where Λ∗x,j is the adjoint of Λx,j . It is easy to show that S is a bounded, invertible, self-adjoint and positive operator. Therefore for any f ∈ H, we have the following reconstructions: Z X −1 f = SS f = Λ∗x,j Λx,j S −1 f dµ(x), X j∈J

f = S −1 Sf =

Z X X j∈J

S −1 Λ∗x,j Λx,j f dµ(x).

˜ x,j = Λx,j S −1 . Then {Λ ˜ x,j : x ∈ X , j ∈ J} is also a semi-continuous g-frame with Denote Λ frame bounds

1 1 B, A,

which we call the canonical dual frame of {Λx,j : x ∈ X , j ∈ J}. A


9

semi-continuous g-frame {Gx,j ∈ L(H, Kx,j ) : x ∈ X , j ∈ J} is called an alternate dual frame of {Λx,j : x ∈ X , j ∈ J} if for all f ∈ H, the following identity holds: Z X Z X ∗ (3.1) f= Λ∗x,j Gx,j f dµ(x) = Gx,j Λx,j f dµ(x). X j∈J

X j∈J

A semi-continuous g-frame {Λx,j : x ∈ X , j ∈ J} is called a Parseval semi-continuous gframe, if the frame bounds A = B = 1. For any X1 ⊂ X , we denote X1c = X \ X1 , and define the following operator: SX 1 f =

Z

X

X1 j∈J

Λ∗x,j Λx,j f dµ(x).

In [4], the authors proved a longstanding conjecture of the signal processing community: a signal can be reconstructed without information about the phase. While working on efficient algorithms for signal reconstruction, Balan et al. [5] discovered a remarkable new identity for Parseval discrete frames, given in the following form. Theorem 3.1. Let {fj }j∈J be a Parseval frame for H, then for every K ⊂ J and every f ∈ H, we have

X

j∈K

X

2

X

2 X

2

hf, fj ifj |hf, fj i| − hf, fj ifj = |hf, fj i| −

. 2

j∈K c

j∈K c

j∈K

Theorem 3.2. If {fj }j∈J be a Parseval frame for H, then for every K ⊂ J and every f ∈ H, we have

X

j∈K

X

2

3 2

hf, fj ifj |hf, fj i| +

≥ 4 kf k . c 2

j∈K

In fact, the identity appears in Theorem 3.1 was obtained in [5] as a particular case of the following result for general frames. Theorem 3.3. Let {fj }j∈J be a frame for H with canonical dual frame {f˜j }j∈J . Then for every K ⊂ J and every f ∈ H, we have X

j∈K

|hf, fj i|2 −

X j∈J

|hSK f, f˜j i|2 =

X

j∈K c

|hf, fj i|2 −

X j∈J

|hSK c f, f˜j i|2 .

Motivated by these interesting results, the authors in [18, 29] generalized Theorems 3.1 and 3.2 to canonical and alternate dual frames. In this section, we investigate the above mentioned results for semi-continuous g-frames and derive some important identities and inequalities of these frames. We first state a simple result on operators.

10

ANIRUDHA PORIA

Lemma 3.4. [29] If P, Q ∈ L(H) satisfying P + Q = I, then P − P ∗ P = Q∗ − Q∗ Q. Proof. We compute P − P ∗ P = (I − P ∗ )P = Q∗ (I − Q) = Q∗ − Q∗ Q.

Theorem 3.5. Let {Λx,j : x ∈ X , j ∈ J} be a Parseval semi-continuous g-frame for H with respect to (X , µ). Then for every X1 ⊂ X and every f ∈ H, we have

Z X

2 Z X

2 ∗

kΛx,j f k dµ(x) − Λx,j Λx,j f dµ(x)

X1 j∈J

Z

=

(3.2)

X1 j∈J

Z X

2 kΛx,j f k dµ(x) −

c

X1 j∈J

X

X1c j∈J

Λ∗x,j Λx,j f

2

dµ(x)

.

Proof. Since {Λx,j : x ∈ X , j ∈ J} is a Parseval semi-continuous g-frame, the corresponding frame operator S = I, and hence SX1 + SX1c = I. Note that SX1c is a self-adjoint operator, ∗ and therefore SX c = SX c . Applying Lemma 3.4 to the operators SX1 and SX c , we obtain 1 1 1

that for every f ∈ H ∗ ∗ ∗ c f, f i − hSX c SX c f, f i S f, f i = hSX hSX1 f, f i − hSX 1 X1 1 1 1

(3.3)

⇒

hSX1 f, f i − kSX1 f k2 = hSX1c f, f i − kSX1c f k2 .

We have hSX1 f, f i =

*Z

X

X1 j∈J

Λ∗x,j Λx,j f

dµ(x), f

(3.4)

+

= =

Z

Z

X hΛx,j f, Λx,j f i dµ(x)

X1 j∈J

X

X1 j∈J

kΛx,j f k2 dµ(x).

Similarly (3.5)

hSX1c f, f i =

Z

X1c

X j∈J

kΛx,j f k2 dµ(x).

Using equations (3.4) and (3.5) in (3.3), we obtain the desired result.

Now we generalize Theorem 3.1 to dual semi-continuous g-frames. We first need the following lemma. Lemma 3.6. [24] Let P, Q ∈ L(H) be two self-adjoint operators such that P + Q = I. Then for any λ ∈ [0, 1] and every f ∈ H we have kP f k2 + 2λhQf, f i = kQf k2 + 2(1 − λ)hP f, f i + (2λ − 1)kf k2 ≥ (1 − (λ − 1)2 )kf k2 .


11

Theorem 3.7. Let {Λx,j : x ∈ X , j ∈ J} be a semi-continuous g-frame for H with respect ˜ x,j : x ∈ X , j ∈ J} be the canonical dual frame of {Λx,j : x ∈ X , j ∈ J}. to (X , µ) and {Λ Then for any λ ∈ [0, 1], for every X1 ⊂ X and every f ∈ H, we have Z X Z X 2 ˜ kΛx,j SX1 f k dµ(x) + kΛx,j f k2 dµ(x) X1c j∈J

X j∈J

=

Z X X j∈J

˜ x,j SX c f k2 dµ(x) + kΛ 1 2

≥ (2λ − λ )

Z

X

X1 j∈J

Z

X

X1 j∈J

kΛx,j f k2 dµ(x)

2

2

kΛx,j f k dµ(x) + (1 − λ )

Z

X

X1c j∈J

kΛx,j f k2 dµ(x).

Proof. Let S be the frame operator for {Λx,j : x ∈ X , j ∈ J}. Since SX1 + SX1c = S, it follows that S −1/2 SX1 S −1/2 + S −1/2 SX1c S −1/2 = I. Considering P = S −1/2 SX1 S −1/2 , Q = S −1/2 SX1c S −1/2 , and S 1/2 f instead of f in Lemma 3.6, we obtain kS −1/2 SX1 f k2 + 2λhS −1/2 SX1c f, S 1/2 f i = kS −1/2 SX1c f k2 + 2(1 − λ)hS −1/2 SX1 f, S 1/2 f i + (2λ − 1)kS 1/2 f k2 ≥ (1 − (λ − 1)2 )kS 1/2 f k2 ⇒ hS −1 SX1 f, SX1 f i + hSX1c f, f i = hS −1 SX1c f, SX1c f i + hSX1 f, f i (3.6)

≥ (2λ − λ2 )hSX1 f, f i + (1 − λ2 )hSX1c f, f i.

We have hS −1 SX1 f, SX1 f i = =

hSS −1 SX1 f, S −1 SX1 f i *Z + X −1 ∗ −1 Λx,j Λx,j S SX1 f dµ(x), S SX1 f X j∈J

=

Z X X j∈J

=

Z X X j∈J

(3.7)

=

Z X X j∈J

hΛx,j S −1 SX1 f, Λx,j S −1 SX1 f i dµ(x) ˜ x,j SX1 f i dµ(x) ˜ x,j SX1 f, Λ hΛ ˜ x,j SX1 f k2 dµ(x). kΛ

Similarly (3.8)

(3.9)

hS −1 SX1c f, SX1c f i =

hS

X1c

f, f i =

Z

Z X X j∈J

X

X1c j∈J

˜ x,j SX c f k2 dµ(x). kΛ 1

kΛx,j f k2 dµ(x).

12

ANIRUDHA PORIA

hSX1 f, f i =

(3.10)

Z

X

kΛx,j f k2 dµ(x).

X1 j∈J

Using equations (3.7)–(3.10) in the inequality (3.6), we obtain the desired result.

Lemma 3.8. [24] If P, Q ∈ L(H) satisfy P + Q = I, then for any λ ∈ [0, 1] and every f ∈ H we have P ∗ P + λ(Q∗ + Q) = Q∗ Q + (1 − λ)(P ∗ + P ) + (2λ − 1)I ≥ (1 − (λ − 1)2 )I. Theorem 3.9. Let {Λx,j : x ∈ X , j ∈ J} be a semi-continuous g-frame for H with respect to (X , µ) and {Gx,j : x ∈ X , j ∈ J} be an alternate dual frame of {Λx,j : x ∈ X , j ∈ J}. Then for any λ ∈ [0, 1], for every X1 ⊂ X and every f ∈ H, we have

2 Z X Z X

∗

Re hGx,j f, Λx,j f idµ(x) + Λx,j Gx,j f dµ(x)

X1c j∈J

= Re

Z

X1 j∈J 2

X1 j∈J

Z X

hGx,j f, Λx,j f idµ(x) +

≥ (2λ − λ )Re

Z

X

X1 j∈J

X

X1c j∈J

Λ∗x,j Gx,j f

2

dµ(x)

Z 2 hGx,j f, Λx,j f idµ(x) + (1 − λ )Re

X

X1c j∈J

hGx,j f, Λx,j f idµ(x) .

Proof. For X1 ⊂ X and f ∈ H, define the operator FX1 by Z X (3.11) FX1 f = Λ∗x,j Gx,j f dµ(x). X1 j∈J

Then FX1 ∈ L(H). By (3.1), we have FX1 + FX1c = I. By Lemma 3.8, we get (1 − (λ − 1)2 )kf k2 ≤ hFX∗1 FX1 f, f i + λh(FX∗1c + FX1c )f, f i = hFX∗1c FX1c f, f i + (1 − λ)h(FX∗1 + FX1 )f, f i + (2λ − 1)kf k2 ⇒ (2λ − λ2 )Re(hIf, f i) ≤ kFX1 f k2 + λ(hFX1c f, f i + hFX1c f, f i) = kFX1c f k2 + (1 − λ)(hFX1 f, f i + hFX1 f, f i) + (2λ − 1)kf k2 ⇒ (2λ − λ2 )Re(hFX1 f, f i) + (1 − λ2 )Re(hFX1c f, f i) ≤ kFX1 f k2 + Re(hFX1c f, f i) (3.12)

= kFX1c f k2 + Re(hFX1 f, f i).

We have (3.13) hFX1 f, f i = (3.14)

*Z

X

X1 j∈J

Λ∗x,j Gx,j f

hFX1c f, f i =

Z

dµ(x), f X

X1c j∈J

+

=

Z

X

X1 j∈J

hGx,j f, Λx,j f idµ(x).

hGx,j f, Λx,j f idµ(x).

Using equations (3.13), (3.14) and (3.11) in (3.12), we obtain the desired inequality.


13

Next we give a generalization of the above theorem to a more general form that does not involve the real parts of the complex numbers. Theorem 3.10. Let {Λx,j : x ∈ X , j ∈ J} be a semi-continuous g-frame for H with respect to (X , µ) and {Gx,j : x ∈ X , j ∈ J} be an alternate dual frame of {Λx,j : x ∈ X , j ∈ J}. Then for every X1 ⊂ X and every f ∈ H, we have Z X Z


X1c j∈J

=

Z

X

X1 j∈J

Z X


X1 j∈J

2

Λ∗x,j Gx,j f dµ(x)

X

X1c j∈J

Λ∗x,j Gx,j f

2

dµ(x)

.

Proof. For X1 ⊂ X and f ∈ H, we define the operator FX1 as in Theorem 3.9. Therefore, we have FX1 + FX1c = I. By Lemma 3.4, we have Z X Z


X1c j∈J

= hFX1c f, f i +

hFX∗1 FX1 f, f i

=

X

X1 j∈J ∗ hFX1 f, f i +

= hFX1 f, f i + kFX1c f k2 Z X Z

hGx,j f, Λx,j f idµ(x) + =

X1 j∈J

Λ∗x,j Gx,j f

hFX∗1c FX1c f, f i

X

X1c j∈J

Hence the relation stated in the theorem holds.

2

dµ(x)

2

Λ∗x,j Gx,j f dµ(x)

.

4. Stability of semi-continuous g-frames The stability of frames is important in practice, so it has received much attentions and is, therefore, studied widely by many authors (see [8, 25, 27]). In this section, we study the stability of semi-continuous g-frames. The following is a fundamental result in the study of the stability of frames. Proposition 4.1. ([6], Theorem 2) Let {fi }∞ i=1 be a frame for some Hilbert space H with bounds A, B. Let {gi }∞ i=1 ⊆ H and assume that there exist constants λ1 , λ2 , µ ≥ 0 such that max(λ1 + (4.1)

√µ , λ2 ) A

< 1 and

#1/2 " n n n n

X

X

X X

2 |ci | ci g i + µ ci fi + λ2 ci (fi − gi ) ≤ λ1

i=1

i=1

i=1

i=1

for all c1 , ..., cn (n ∈ N). Then {gi }∞ i=1 is a frame for H with bounds !2 !2 λ1 + λ2 + √µA λ1 + λ2 + √µB A 1− , B 1+ . 1 + λ2 1 − λ2

14

ANIRUDHA PORIA

Similar to discrete frames, semi-continuous g-frames are stable under small perturbations. The stability of semi-continuous g-frames is discussed in the following theorem. Theorem 4.2. Let {Λx,j : x ∈ X , j ∈ J} be a semi-continuous g-frame for H with respect to (X , µ), with frame bounds A and B. Suppose that Γx,j ∈ L(H, Kx,j ) for any x ∈ X , j ∈ J and there exist constants λ1 , λ2 , µ ≥ 0 such that max(λ1 +

√µ , λ2 ) A

< 1 and the following

condition is satisfied  1/2 Z X  k(Λx,j − Γx,j )f k2 dµ(x) X j∈J

  1/2 1/2 Z X Z X ≤ λ1  kΛx,j (f )k2 dµ(x) + λ2  kΓx,j (f )k2 dµ(x) + µkf k,

(4.2)

X j∈J

X j∈J

for all f ∈ H. Then {Γx,j : x ∈ X , j ∈ J} is a semi-continuous g-frame for H with respect to (X , µ), with frame bounds (4.3)

A 1−

λ1 + λ2 + 1 + λ2

√µ A

!2

, B

1+

√µ B

λ1 + λ2 + 1 − λ2

!2

.

Proof. Notice that Z X X j∈J

kΛx,j (f )k2 dµ(x) ≤ Bkf k2 .

From (4.2) we see that   1/2 1/2 Z X Z X √ 2 2  k(Λx,j − Γx,j )f k dµ(x) kΓx,j (f )k dµ(x) . ≤ λ1 B + µ kf k+λ2  X j∈J

X j∈J

Using the triangle inequality, we get  1/2 Z X  k(Λx,j − Γx,j )f k2 dµ(x) X j∈J

 1/2  1/2 Z X Z X ≥ kΓx,j (f )k2 dµ(x) −  kΛx,j (f )k2 dµ(x) . X j∈J

Hence



(1 − λ2 ) 

Z X X j∈J

X j∈J

1/2

kΓx,j (f )k2 dµ(x)

 1/2 Z X √ √ µ kf k. kΛx,j (f )k2 dµ(x) ≤ B 1 + λ1 + √ ≤ λ1 B + µ kf k +  B X j∈J


Therefore

Z X X j∈J

2

kΓx,j (f )k dµ(x) ≤ B

1+

λ1 + λ2 +

√µ B

1 − λ2

!2

kf k2 .

!2

kf k2 .

15

Similarly, we can prove that Z X X j∈J

2

kΓx,j (f )k dµ(x) ≥ A 1 −

λ1 + λ2 + 1 + λ2

√µ A

This completes the proof.

Remark 4.3. In general, the inequality (4.2) does not imply that {Γx,j : x ∈ X , j ∈ J} is a semi-continuous g-frame regardless how small the parameters λ1 , λ2 , µ are. A counterexample for g-frames can be found in [27], and an example can be constructed similarly for semicontinuous g-frames. Corollary 4.4. Let {Λx,j : x ∈ X , j ∈ J} be a semi-continuous g-frame for H with respect to (X , µ), with frame bounds A, B, and let {Γx,j : x ∈ X , j ∈ J} be a sequence in L(H, Kx,j ) for any x ∈ X , j ∈ J. Assume that there exists a constant 0 < M < A such that Z X k(Λx,j − Γx,j )f k2 dµ(x) ≤ M kf k2 , ∀f ∈ H, X j∈J

then {Γx,j : x ∈ X , j ∈ J} is a semi-continuous g-frame for H with respect to (X , µ), with bounds A[1 − (M/A)1/2 ]2 and B[1 + (M/B)1/2 ]2 . Proof. Let λ1 = λ2 = 0 and µ =

p √ √ M . Since M < A, µ/ A = M/A < 1. So, by Theorem

4.2, {Γx,j : x ∈ X , j ∈ J} is a semi-continuous g-frame for H with respect to (X , µ), with bounds A[1 − (M/A)1/2 ]2 and B[1 + (M/B)1/2 ]2 .

Acknowledgments The author is deeply indebted to Dr. Azita Mayeli for several valuable comments and suggestions. The author is grateful to the United States-India Educational Foundation for providing the Fulbright-Nehru Doctoral Research Fellowship, and Department of Mathematics and Computer Science, the Graduate Center, City University of New York, New York, USA for its kind hospitality during the period of this work. He would also like to express his gratitude to the Norbert Wiener Center for Harmonic Analysis and Applications at the University of Maryland, College Park for its kind support.

16

ANIRUDHA PORIA

References [1] S.T. Ali, J.P. Antoine, and J.P. Gazeau. Continuous frames in Hilbert spaces. Ann. Phys., 222(1):1–37, 1993. [2] S.T. Ali, J.P. Antoine, and J.P. Gazeau. Coherent States, Wavelets and Their Generalizations. SpringerVerlag, New York, 2000. [3] A. Askari-Hemmat, M.A. Dehghan, and M. Radjabalipour. Generalized frames and their redundancy. Proc. Amer. Math. Soc., 129(4):1143–1147, 2001. [4] R. Balan, P.G. Casazza, and D. Edidin. On signal reconstruction without phase. Appl. Comput. Harmon. Anal., 20(3):345–356, 2006. [5] R. Balan, P.G. Casazza, D. Edidin, and G. Kutyniok. A new identity for Parseval frames. Proc. Amer. Math. Soc., 135(4):1007–1015, 2007. [6] P.G. Casazza and O. Christensen. Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl., 3(5):543–557, 1997. [7] P.G. Casazza and G. Kutyniok. Frames of subspaces. Contemp. Math., 345:87–114, 2004. [8] O. Christensen. An Introduction to Frames and Riesz Bases. Birkh¨ auser, Boston, 2003. [9] O. Christensen and Y.C. Eldar. Oblique dual frames and shift-invariant spaces. Appl. Comput. Harmon. Anal., 17(1):48–68, 2004. [10] B. Currey, A. Mayeli and V. Oussa. Characterization of shift-invariant spaces on a class of nilpotent Lie groups with applications. J. Fourier Anal. Appl., 20:384–400, 2014. [11] I. Daubechies, A. Grossmann, and Y. Meyer. Painless nonorthogonal expansions. J. Math. Phys., 27(5):1271–1283, 1986. [12] M.A. Dehghan and M.A. Hasankhani Fard. G-continuous frames and coorbit spaces. Acta Math. Acad. Paedagog. Nyh´ azi (NS), 24:373–383, 2008. [13] R.J. Duffin and A.C. Schaeffer. A class of nonharmonic Fourier series. Trans. Amer. Math. Soc., 72(2):341–366, 1952. [14] G.B. Folland. A Course in Abstract Harmonic Analysis. CRC Press, Boca Raton, Florida, 1995. [15] M. Foranasier. Quasi-orthogonal decompositions of structured frames. J. Math. Anal. Appl., 289(1):180– 199, 2004. [16] M. Foranasier and H. Rauhut. Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl., 11(3):245–287, 2005. [17] J.P. Gabardo and D. Han. Frames associated with measurable spaces. Adv. Comput. Math., 18(3):127– 147, 2003. [18] P. G˘ avrut¸a. On some identities and inequalities for frames in Hilbert spaces. J. Math. Anal. Appl., 321(1):469–478, 2006. [19] L. G˘ avrut¸a. Frames for operators. Appl. Comput. Harmon. Anal., 32(1):139–144, 2012. [20] D. Geller. Fourier analysis on the Heisenberg group. Proc. Natl. Acad. Sci. U.S.A., 74(4):1328–1331, 1977. [21] K. Gr¨ ochenig. Foundations of Time-Frequency Analysis. Birkh¨ auser, Boston, 2001. [22] G. Kaiser. A Friendly Guide to Wavelets. Birkh¨ auser, Boston, 1994. [23] S. Li and H. Ogawa. Pseudoframes for subspaces with applications. J. Fourier Anal. Appl., 10(4):409– 431, 2004. [24] A. Poria. Some identities and inequalities for Hilbert–Schmidt frames. Mediterr. J. Math., 14(2): Art. 59, 14 pp., 2017. [25] A. Poria. Approximation of the inverse frame operator and stability of Hilbert–Schmidt frames. Mediterr. J. Math., 14(4): Art. 153, 22 pp., 2017. [26] W. Sun. G-frames and g-Riesz bases. J. Math. Anal. Appl., 322(1):437–452, 2006. [27] W. Sun. Stability of g-frames. J. Math. Anal. Appl., 326(2):858–868, 2007. [28] R.M. Young. An Introduction to Non-Harmonic Fourier Series. Academic Press, New Work, 1980. [29] X. Zhu and G. Wu. A note on some equalities for frames in Hilbert spaces. Appl. Math. Lett., 23(7):788– 790, 2010. Department of Mathematics, School of Engineering and Applied Sciences, Bennett University, Greater Noida 201310, India. E-mail address: [email protected]