H -Valued Gabor Expansions in Weighted Amalgam Spaces

in the sense that the square partial sums of (3.8) converge to m_k in the norm of L^p(Q_1/β,H), cf. [101, 146]. Hence, for 1< p <∞ and 1≤q <∞ we can write the norm on S^p,q_˜v (H) as

kdkS_v^p,q_˜ (H)= X

k∈Z^d

Z

Q_1/βk X

n∈Z^d

d_kne^2πiβhn,xik^pHdxq/p

˜

v(k)^q1/q

.

The analysis and synthesis operators associated with theH-valued Gabor frame are defined as follows: Takeh∈Hsuch thatkhkH = 1. Letα, β >0, 1≤p, q≤ ∞and fixg,γ∈W_H(L^∞, L¹_w). Forf ∈W_H(L^p, L^q_v), define the analysis operator by

C_g,hf(k, n) = hf, M_βnT_αkgih

= Z

R^d

hf(x), M_βnT_αkg(x)iHhdx

= Z

R^dhf(x), T_αkg(x)iHhe^{−2πiβhn,xi}dx

= F(hf, T_αkgih)(βn) =F((h⊙T_αkg)f)(βn).

So

X

n∈Z^d

hf, M_βnT_αkgihe^2πiβhn,xi = X

n∈Z^d

F((h⊙T_αkg)f)(βn)e^2πiβhn,xi.

By using Poisson summation formula we get, X

n∈Z^d

hf, M_βnT_αkgihe^2πiβhn,xi = β^−d X

n∈Z^d

((h⊙T_αkg(x))f)(x−n

β) =m_k(x).

Letd= (d_kn)∈S_v^p,q_˜ (H). The synthesis operator is defined by R_g,hd(x) = X

k,n∈Z^d

hd_kn,hiHM_βnT_αkg(x)

= X

k∈Z^d

D X

n∈Z^d

d_kne^2πiβhn,xi,hE

HT_αkg(x) (3.9)

= X

k∈Z^d

hm_k(x),hiHT_αkg(x),

3.2. H-Valued Gabor Expansions in Weighted Amalgam Spaces 49

wherem_k(x) =P

n∈Z^dd_kne^2πiβhn,xi is 1/β-periodic.

From the above observations we obtain the analogue of the Walnut’s represen- tation for theH-valued Gabor frames in the following theorem.

Theorem 3.2.3. Let vbe anw-moderate weight onR^d. Letα, β >0and1≤p, q≤

∞ be given. Fixg,γ ∈W_H(L^∞, L¹_w) and h∈H withkhkH = 1. Then the following statements hold.

(i) The analysis operator C_g,hf = (hf, M_βnT_αkgih)_k,n

∈Z^d is a bounded mapping C_g,h : W_H(L^p, L^q_v) → S_v^p,q_˜ (H), Moreover, there exist unique functions m_k ∈ L^p(Q_1/β,H)which satisfy mˆ_k(n) =C_g,hf(k, n) for allk, n∈Z^d, and these are given explicitly by

m_k(x) = β^−d X

n∈Z^d

(h⊙T_αkg(x))f x−n

β

= β^−d X

n∈Z^d

(h⊙T_αk+ⁿ

βg(x))Tⁿ

βf(x). (3.10) The series on the right side of (3.10) converges unconditionally inL^p(Q_1/β,H) (unconditionally in the σ(L^∞(Q_1/β,H), L¹(Q_1/β,H)) topology ifp=∞).

(ii) Given d ∈ S_v^p,q_˜ (H), let m_k ∈ L^p(Q_α,H) be the unique functions satisfying ˆ

m_k(n) =d_kn for all k, n∈Z^d. Then the series R_g,hd= X

k∈Z^d

hm_k(·),hiHT_akg, (3.11)

converges unconditionally inW_H(L^p, L^q_v) (unconditionally in theσ(W_H(L^p, L^q_v), W_H(L^p

′

, L^q

′

1/v))topology if p=∞ or q =∞), and R_g,h is a bounded mapping R_g,h :S_v^p,q_˜ (H)→W_H(L^p, L^q_v).

(iii) The Walnut’s representation

R_γ,hC_g,hf =β^−d X

n∈Z^d

G_n Tⁿ

βf

, (3.12)

holds for f ∈ W_H(L^p, L^q_v), with the series on the right of (3.12) converging

absolutely in W_H(L^p, L^q_v), where G_n(x) = X

k∈Z^d

T_αkγ(x)⊙T_αk+ⁿ

βg(x)∈B(H). (3.13)

Remark 3.2.4. Ifg,γ ∈W_H(L^∞, L¹_w), then from the above theorem the Walnut’s representation of the frame operator on W_H(L^p, L^q_v) is

S_g,γf(x) =β^−d X

n∈Z^d

G_n(x) Tⁿ

βf(x)

. (3.14)

Since we deal with vector valued functions, obtaining the above expression is bit technical. Notice that in the Walnut’s representation for the scalar valued case (see [82]), the summation is taken over point-wise product of G_n(x) with Tⁿ

βf(x), whereas in our case G_n(x) is operating on the Hilbert space element Tⁿ

βf(x).

To prove Theorem 3.2.3 we need the following results. Since the proof of these results follows in a similar way as in scalar valued case we only state them without proof.

Lemma 3.2.5. ([82], Lemma 5.1) Let α, β >0 be given. Let K_αβ be the maximum number of _β¹Z^d-translates of Q_1/β required to cover any αZ^d-translate of Q_α, i.e., K_αβ = max

k∈Z^d

#{ℓ ∈ Z^d : |(_β^ℓ +Q_1/β) ∩(αk +Q_α)| > 0}. Then given 1 ≤ p ≤

∞, we have for any 1/β-periodic function m ∈ L^p(Q_1/β,H) and any k ∈ Z^d that kmkp,αk+Qα ≤K_αβ^1/pkmkp,Q1/β,where K_αβ^1/∞= 1.

Lemma 3.2.6. If g ∈W(L^∞(R^d, B(H)), L¹) and α >0 then ess sup

x∈R^d

X

n∈Z^d

kg(x−αn)kB(H)≤1 α + 1d

kgk_W(L^∞_(R^d_,B(H)),L¹_). (3.15)

Lemma 3.2.7. Ifg,γ∈W(L^∞, L¹), thenG_nare defined by (3.13) is inL^∞(R^d, B(H)), and

X

n∈Z^d

kG_nk_L^∞_(R^d_,B(H))≤1 α+ 1d

(2β+ 2)^dkgk_W(L^∞_,L¹₎kγk_W(L^∞_,L¹₎.

3.2. H-Valued Gabor Expansions in Weighted Amalgam Spaces 51

The next lemma is a weighted version of previous lemma that is useful in the Walnut’s representation of the frame operator (see [135], Lemma 2.2).

Lemma 3.2.8. Letwbe a submultiplicative weight, and let α, β >0 be given. Then there exists a constant C =C(α, β, w) >0 such that if g,γ∈W_H(L^∞, L¹_w) and the functions G_n are defined by (3.13), then

X

n∈Z^d

kG_nk_L^∞_(R^d_,B(H))wn β

≤C kgk_W

H(L^∞,L¹_w)kγk_W

H(L^∞,L¹_w).

Following lemma is an estimate on the effect of translations on the amalgam space norm.

Lemma 3.2.9. Let v be an w-moderate weight. Then for 1 ≤ p, q ≤ ∞, we have for each f ∈W_H(L^p, L^q_v) andn∈Z^d that

kT_αnfkW_H(L^p,L^q_v)≤C_vw(αn)kfkW_H(L^p,L^q_v).

The structural results about H-valued Gabor frames can be derived from the corresponding well-known results for scalar valued Gabor frames. Now we will present some important results onH-valued Gabor frames in the following remark.

Remark 3.2.10. The l-th Fourier coefficient of G_n is Gˆ_n(l) = α^−d

Z

Q_α

G_n(x)e^{−2πihl,x/αi}dx

= α^−d Z

Q_α

(X

k∈Z^d

T_akγ(x)⊙T_αk+ⁿ

βg(x))e^{−2πihl,x/αi}dx

= α^−d Z

R^d

(γ(x)⊙Tⁿ

βg(x))e^{−2πihl,x/αi}dx

= α^−d Z

R^d

γ(x)⊙Ml αTⁿ

βg(x)dx:=α^−d[γ, Ml αTⁿ

βg].

Then Fourier series

G_n(x) =α^−dX

l∈Z^d

[γ, Ml αTⁿ

βg]e^2πihl,x/αi, (3.16)

is convergent in L²(Q_α, B(H)). By substituting this into Walnut’s representation, we obtain the expression

S_g,γf =β^−d X

n∈Z^d

G_n Tⁿ

βf

= (αβ)^−d X

n∈Z^d

X

l∈Z^d

[γ, Ml αTⁿ

βg]

Ml αTⁿ

βf ,

or in operator notation,

S_g,γ = (αβ)^−d X

n∈Z^d

X

l∈Z^d

[γ, Ml αTⁿ

βg]

Ml αTⁿ

β

. (3.17)

This is H-valued analogue of Janssen’s representation for the H-valued frame operator S_g,γ. Using Janssen’s representation we obtain the H-valued analogue of Wexler-Raz biorthogonality relation in the following theorem.

Theorem 3.2.11. (Wexler-Raz biorthogonality relation). Assume thatG(g, α, β), G(γ, α, β) are Bessel sequence in L²(R^d,H). Then the following conditions are equivalent:

(i) S_g,γ =S_γ,g =I on L²(R^d,H).

(ii) (αβ)^−d[γ, Ml αTⁿ

βg] =δ_l0δ_n0I_B(H) for l, n∈Z^d.

Proof. The implication (ii)⇒(i) is trivial consequence of Janssen’s representation.

For the converse (i)⇒ (ii), assume that S_g,γ =I. Let f,h∈L^∞(Q_1/β,H) and let l, m∈Z^d be arbitrary. Then

δ_lm[f,h] = δ_lm Z

R^d

f(x)⊙h(x)dx=δ_lm Z

R^d

S_g,γf(x)⊙h(x)dx

= Z

R^d

S_g,γTl

βf(x)⊙T^m

βh(x)dx

= β^−d Z

R^d

X

n∈Z^d

G_n(x) Tn+l

β f(x)

⊙T^m

βh(x)dx

= β^−d Z

R^d

G_m−l(x) T^m

βf(x)

⊙T^m

βh(x)dx

= β^−d Z

R^d

(T₋^m

βG_m−l(x))f(x)⊙h(x)dx

= β^−d[(T₋^m

βG_m−l)(f),h].

3.2. H-Valued Gabor Expansions in Weighted Amalgam Spaces 53

By density this identity extends tof,h∈L²(Q_1/β,H), so we conclude that β^−dG_m−l

x+m β

=δ_lmI_B(H),

for almost all x∈Q_1/β. Varying l, m∈Z^d it follows thatG_n(x) =β^dδ_n0I_B(H), for almost all x ∈ R^d. Therefore by (3.16) and uniqueness of Fourier coefficients we conclude that (αβ)^−d[γ, Ml

αTⁿ

βg] =δ_l0δ_n0I_B(H).

Now we will prove boundedness of the analysis and synthesis operator, and Walnut’s representation in the following proof.

Proof of Theorem 3.2.3. (i) Given thatg∈W_H(L^∞, L¹_w) and 1≤p, q≤ ∞.Letf ∈ W_H(L^p, L^q_v), which is a subspace of W(L¹, L^∞_1/w). First we show that the operators m_k given by (3.10) are well-defined. Observe that m_k is the 1/β-periodization of the integrable H-valued function (h⊙T_αkg)f and hence the series defining m_k converges inL¹(Q_α,H). To show that the periodization converges unconditionally in L^p(Q_1/β,H) (weakly ifp=∞) and to derive a useful estimate, fix any 1/β-periodic functionh^′∈L^p

′

(Q_1/β,H) and for each fixed k, we consider Z

Q_1/β|hX

n∈Z^d

(h⊙T_αk+ⁿ

βg(x))Tⁿ

βf(x),h^′(x)iH|dx

≤ Z

Q_1/β

X

n∈Z^d

|hhTⁿ

βf(x), T_αk+ⁿ

βg(x)iHh,h^′(x)iH|dx

= Z

R^d|hf(x), T_αkg(x)iHhh,h^′(x)iH|dx

= X

n∈Z^d

Z

Q_α|hf(x), T_αkg(x)iHhh,h^′(x)iH|T_αk+αnχ_Qα(x)dx

≤ X

n∈Z^d

kT_αkg·T_αk+αnχ_Qαk_L^∞_(R^d_,H)kf ·T_αk+αnχ_Qαkp× kh^′k_p^′_,αk+αn+Qαv(αk+αn−αn)

v(αk)

≤ X

n∈Z^d

kg·T_αnχ_Qαk_L^∞_(R^d_,H)kf·T_αk+αnχ_Qαkp×

K^1/p

′

αβ kh^′k_p^′_,Q1/βC_vv(αk+αn)w(αn) v(αk)

=C_vK^1/p

′

αβ kh^′k_p^′_,Q1/β 1 v(αk)

X

n∈Z^d

kg·T_αnχ_Qαk_L^∞_(R^d_,H)w(αn)×

kf·T_αk+αnχ_Qαkpv(αk+αn). (3.18) Taking the supremum in (3.18) over h^′ with unit norm we get,

km_kkp,Q_1/β ≤β^−dC_vK^1/p

′

αβ

1 v(αk)

X

n∈Z^d

kg·T_αnχ_Qαk_L^∞_(R^d_,H)w(αn)×

kf·T_αk+αnχ_Qαkpv(αk+αn), (3.19)

where K_αβ = max

k∈Z^d

#{ℓ ∈ Z^d : |

ℓ

β +Q_1/β

∩(αk +Q_α)| > 0}. This shows the convergence of the series definingm_k inL^p(Q_1/β,H). Note that,

ˆ

m_k(n) = β^d Z

Q_1/β

m_k(x)e^{−2πiβhn,xi}dx

= Z

Q_1/β

X

m∈Z^d

(h⊙T_αk+^m

βg(x))T^m

βf(x)e^{−2πiβhn,xi}dx

= Z

Q1/β

X

m∈Z^d

hT^m

βf(x), T_αk+^m

βg(x)iHhe^{−2πiβhn,x−m/βi}dx

= Z

R^dhf(x), T_αkg(x)iHhe^{−2πiβhn,xi}dx

= hf, M_βnT_αkgih=C_g,hf(k, n).

Finally, we show thatC_g,his a bounded mapping ofW_H(L^p, L^q_v) intoS^p,q_˜v (H). Given f ∈W_H(L^p, L^q_v), to show thatC_g,hf ∈S_v^p,q_˜ (H) it is enough to show that the sequence rgiven byr(k) =km_kkp,Q_1/β, k ∈Z^dlies inℓ^q_v˜.To do this, fix any sequencea∈ℓ^q

′

1/˜v. Then by using (3.19), we have

|hr, ai| ≤ X

k∈Z^d

km_kkp,Q_1/β|a(k)|

≤β^−dC_vK^1/p

′

αβ

X

n∈Z^d

kg·T_αnχ_Qαk_L^∞_(R^d_,H)w(αn)×

3.2. H-Valued Gabor Expansions in Weighted Amalgam Spaces 55 X

k∈Z^d

kf ·T_αk+αnχ_Qαkpv(αk+αn)|a(k)| 1 v(αk)

≤β^−dC_vK^1/p

′

αβ

X

n∈Z^d

kg·T_αnχ_Qαk_L^∞_(R^d_,H)w(αn)× X

k∈Z^d

kf·T_αk+αnχ_Qαk^qpv(αk+αn)^q1/q X

k∈Z^d

|a(k)|^q

′ 1

v(αk)^q

′

1/q^′

≤β^−dC_vK^1/p

′

αβ kgk_W

H(L^∞,L¹_w)kfk_WH(L^p_,L^q_v)kak

ℓ^q

′

1/˜v

. (3.20)

Since ℓ^q

′

1/˜v = (ℓ^q_v˜)^∗ when q <∞ and is a norm-fundamental subspace when q =∞, taking the supremum over sequences awith unit norm in (3.20) we get

kC_g,hfkS_v^p,q_˜ (H)=krkℓ^q_˜v ≤β^−dC_vK^1/p

′

αβ kgk_W

H(L^∞,L¹_w)kfkW_H(L^p,L^q_v). Hence C_g,h is a bounded mapping of W_H(L^p, L^q_v) into S^p,q_˜v (H). This proves (i).

(ii) Let 1≤p, q <∞.Givend∈S_v^p,q_˜ (H), we haveP

k∈Z^dkm_kk^q_p,Q1/βv(k)˜ ^q <∞. That means for everyε >0, there exists a finite setF₀ such that

X

k /∈F

km_kk^q_p,Q1/βv(k)˜ ^q < ε^q, ∀finiteF ⊃F₀. (3.21)

Since 1/v is also anw-moderate weight, for any fixh^′ ∈W_H(L^p

′

, L^q

′

1/v), we have X

k /∈F

|hhm_k(·),hiHT_αkg,h^′i|

≤X

k /∈F

Z

R^d|hhm_k(x),hiHT_αkg(x),h^′(x)iH|dx

=X

k /∈F

X

n∈Z^d

Z

Q_α|hm_k(x),hiHhT_αkg(x),h^′(x)iH|T_αn+αkχ_Qα(x)dx

≤X

k /∈F

X

n∈Z^d

kT_αkg·T_αn+αkχ_Qαk_L^∞_(R^d_,H)km_kkp,αn+αk+Q_α× kh^′·T_αn+αkχ_Qαk_p^′ v(αk)

v(αn+αk−αn)

≤ X

n∈Z^d

kg·T_αnχ_Qαk_L^∞_(R^d_,H)× X

k /∈F

K_αβ^1/pkm_kkp,Q_1/βkh^′·T_αn+αkχ_Qαk_p^′ C_vv(αk)w(αn) v(αn+αk)

≤C_vK_αβ^1/p X

n∈Z^d

kg·T_αnχ_Qαk_L^∞_(R^d_,H)w(αn) X

k /∈F

km_kk^q_p,Q1/βv(αk)^q1/q

× X

k∈Z^d

kh^′·T_αn+αkχ_Qαk^q

′

p^′

1 v(αn+αk)^q

′

1/q^′

. (3.22)

Combining (3.21) and (3.22), we get X

k /∈F

|hhm_k(·),hiHT_αkg,h^′i| ≤ǫC_vK_αβ^1/pkgk_W

H(L^∞,L¹_w)kh^′k

W_H(L^p

′

,L^q

′

1/v). Taking the supremum over allh^′ of unit norm, we see that

R_g,hd= X

k∈Z^d

hm_k(·),hiHT_akg

converges unconditionally. Now replacing F by Z^din (3.22), we get

|hR_g,hd,h^′i| ≤ X

k∈Z^d

|hhm_k(·),hiHT_αkg,h^′i|

≤ C_vK_αβ^1/pkgk_W

H(L^∞,L¹_w)kdk_Sv^p,q_{˜ (H)}kh^′k

W_H(L^p

′

,L^q

′

1/v). (3.23) Since W_H(L^p

′

, L^q

′

1/v) is the dual space of W_H(L^p, L^q_v), taking the supremum over all h^′ of unit norm in (3.23) shows that

kR_g,hdkW_H(L^p,L^q_v) = sup{|hR_g,hd,h^′i|:kh^′k

W_H(L^p

′

,L^q

′

1/v)= 1}

≤ C_vK_αβ^1/pkgk_W

H(L^∞,L¹_w)kdkS_v^p,q_˜ (H), (3.24) This completes the proof for the case 1 ≤ p, q < ∞. Since W_H(L^p

′

, L^q

′

1/v) is the K¨othe dual of W_H(L^p, L^q_v) a similar argument as in (3.22), (3.23) will imply the

3.2. H-Valued Gabor Expansions in Weighted Amalgam Spaces 57

convergence of R_g,h in the weak topology whenp =∞ or q =∞ and the estimate kR_g,hdkW_H(L^p,L^q_v) as in (3.24). This proves part (ii).

(iii) Now we show the frame operator R_γ,hC_g,h admits Walnut’s representation on W_H(L^p, L^q_v). Given g,γ ∈ W_H(L^∞, L¹_w) and 1 ≤ p, q ≤ ∞. Notice that for a w-moderate weightv andf ∈W_H(L^p, L^q_v), 1≤p, q≤ ∞and for each n∈Z^d,α >0 we have

kT_αnfkW_H(L^p,L^q_v)≤C_vw(αn)kfkW_H(L^p,L^q_v). (3.25) Replacing α by 1/β in (3.25) we get,

kTⁿ

βfkW_H(L^p,L^q_v) ≤C_vwn β

kfkW_H(L^p,L^q_v).

Therefore, for f ∈W_H(L^p, L^q_v) consider X

n∈Z^d

kG_n Tⁿ

βf

kW_H(L^p,L^q_v)

≤ X

n∈Z^d

kG_nk_L^∞_(R^d_,B(H))kTⁿ

βfkW_H(L^p,L^q_v)

≤ C_vkfk_WH(L^p_,L^q_v) X

n∈Z^d

kG_nk_L^∞_(R^d_,B(H))wn β

≤ CC_vkfk_WH(L^p_,L^q_v)kgk_W

H(L^∞,L¹_w)kγk_W

H(L^∞,L¹_w), by Lemma 3.2.8. Therefore the series P

n∈Z^dG_n Tⁿ

βf

converges absolutely in W_H(L^p, L^q_v). Now for fixed f ∈ W_H(L^p, L^q_v), define m_k such that c_g,hf(k, n) =

ˆ

m_k(n). Forh^′∈W_H(L^p

′

, L^q

′

1/v) we have hR_γ,hC_g,hf,h^′i

= X

k∈Z^d

hhm_k(·),hiHT_akγ,h^′i

= X

k∈Z^d

Z

R^dhhm_k(x),hiHT_akγ(x),h^′(x)iHdx

= β^−d X

k∈Z^d

Z

R^d

X

n∈Z^d

hh(h⊙T_αk+ⁿ

βg(x))Tⁿ

βf(x),hiHT_akγ(x),h^′(x)iHdx

= β^−d X

k∈Z^d

Z

R^d

X

n∈Z^d

hhhTⁿ

βf(x), T_αk+ⁿ

βg(x)iHh,hiHT_akγ(x),h^′(x)iHdx

= β^−d X

k∈Z^d

Z

R^d

X

n∈Z^d

hhTⁿ

βf(x), T_αk+ⁿ

βg(x)iHT_akγ(x),h^′(x)iHdx

= β^−d X

n∈Z^d

Z

R^d

X

k∈Z^d

h(T_akγ(x)⊙T_αk+ⁿ

βg(x))Tⁿ

βf(x),h^′(x)iHdx

= β^−d X

n∈Z^d

Z

R^dhG_n(x) Tⁿ

βf(x)

,h^′(x)iHdx

= β^−d X

n∈Z^d

hG_n Tⁿ

βf ,h^′i,

from which (3.12) follows. The interchanges of integration and summation can be

justified by Lemma 3.2.8 and Fubini’s theorem.

Using the expression form_kin (3.8), the synthesis operatorR_g,hcan be expressed as the iterated sum as

X

k∈Z^d

hX

n∈Z^d

d_kne^2πiβhn,xi,hiHT_αkg = X

k∈Z^d

X

n∈Z^d

hd_kn,hiHM_βnT_αkg.

We show that this series of partial sums converge to R_g,hdin theW_H(L^p, L^q_v) norm in the following proposition.

Proposition 3.2.12. Let v be an w-moderate weight on R^d. Let α, β > 0 and 1 < p < ∞, 1 ≤ q <∞ be given. Assume that g,γ ∈ W_H(L^∞, L¹_w) are such that G(g, α, β) is a Gabor frame for L²(R^d,H) with dual window γ and take h∈Hwith unit norm. Then the following statements hold.

(i) If d∈S_˜v^p,q(H),then the partial sums S_K,Nd= X

kkk^∞≤K

X

knk^∞≤N

hd_kn,hiHM_βnT_αkg, K, N >0,

converge to R_g,hdin the norm of W_H(L^p, L^q_v), i.e., for each ε > 0 there exist

3.2. H-Valued Gabor Expansions in Weighted Amalgam Spaces 59

K₀, N₀ >0 such that

∀K ≥K₀, ∀N ≥N₀, kR_g,hd−S_K,Ndk_WH(L^p_,L^q_v)< ε.

(ii) If f ∈W_H(L^p, L^q_v), then the partial sums S_K,N(R_γ,hC_g,hf) = X

kkk^∞≤K

X

knk^∞≤N

hf, M_βnT_αkgiM_βnT_αkγ

of the Gabor expansion of f converge to f in the norm of W_H(L^p, L^q_v) and for 1≤p, q≤ ∞, in the σ(W_H(L^p, L^q_v), W_H(L^p

′

, L^q

′

1/v))-topology.

Proof. The idea of the proof is similar to Proposition 4.6 of [82] with appropriate modifications.

The Walnut’s representation for superframe operator and the multi-window Ga- bor frame operator can be obtained by choosing an appropriate Hilbert space in Theorem 3.2.3. However, we list out some consequences of Theorem 3.2.3 and Proposition 3.2.12 in the following remark:

Remark 3.2.13. (i) If H= C then the rank one operator x⊙y turns out to the point-wise product xy and all the above results for the H-valued Gabor frame viz.

Walnut’s representation of H-valued Gabor frame operator, convergence of Gabor expansions, etc., coincides with the results for the scalar valued Gabor frames (see [67, 82, 135, 137]).

(ii) If H = Cⁿ then the H-valued Gabor frame is the super Gabor frame (see [84]). The Gabor expansions for the Gabor super-frames on vector valued amalgam spaces also converge by Proposition 3.2.12.

(iii) Let (H₁,h·,·i1),(H₂,h·,·i2),· · · ,(H_r,h·,·ir) be r Hilbert spaces. If H = Lr

i=1H_i (i.eH is the direct sum ofr Hilbert spaces) then His also a Hilbert space with respect to the inner product hx, yi = Pr

i=1hx_i, y_ii where x = ⊕^ri=1x_i, y =

⊕^ri=1y_i, x, y ∈H, x_i, y_i ∈H_i, i= 1,2· · · , r.Iff :R^d→Hthen f(x) =f₁(x)M

f₂(x)M

· · ·M

f_r(x) with f_i(x)∈H_i, i= 1,2· · · , r.

Note that f ∈ W_H(L^p, L^q_w) ⇔ f_i ∈ W_Hi(L^p, L^q_w) for all i = 1,2,· · · , r. The frame operator of the Gabor system on W_H(L^p, L^q_w) with respect to a single lattice Λ = αZ×βZ is given by

S_g,γf = β^−d X

n∈Z^d

G_n Tⁿ

βf

=β^−d X

n∈Z^d

X

k∈Z^d

T_αkγ(x)⊙T_αk+ⁿ

βg(x) Tⁿ

βf

= β^−d X

n∈Z^d

X

k∈Z^d

T_αkM^r

i=1

γ_i(x)

⊙T_αk+ⁿ

β

M^r

i=1

g_i(x) Tⁿ

β

M^r

i=1

f_i

= β^−d X

n∈Z^d

X

k∈Z^d

Xr i=1

T_αkγ_i(x)⊙T_αk+ⁿ

βg_i(x) Tⁿ

βf_i

= Xr

i=1

S_gi,γ

if_i, where g(x) = Lr

i=1g_i(x),γ(x) = Lr

i=1γ_i(x) and S_gi,γi is the frame operator of the Gabor system on W_Hi(L^p, L^q_w). If H₁ = H₂· · · = H_r = C then the H−valued Gabor frame turns out to be a Gabor superframe.

(iv) Let Λ = Λ¹×...×Λ^r be the Cartesian product of separable lattices Λⁱ = α_iZ^d ×β_iZ^d and let g₁, ...,g_r,γ₁, ...,γ_r ∈ W_Hi(L^∞, L¹_w). Suppose the collection Gi(g_i, α_i, β_i) is a frame forL²(R^d,H_i) with the corresponding frame operatorS_g^Λ_iⁱ_,γi. As in the previous set up (as in (iii)) we show that frame operator associated with the Gabor system onW_H(L^p, L^q_w) is the sum of frame operator associated with the Gabor systems on W_Hi(L^p, L^q_w). In this case we consider cartesian product of separable lattices Λⁱ =α_iZ^d×β_iZ^d, i= 1,2,· · · , rinstead of a single latticeαZ^d×βZ^d. For 1≤ p, q≤ ∞the spaceS_˜v^p,q(H) turns out to beS_v^p,q_˜ (H₁)×S_˜v^p,q(H₂)× · · · ×S_v^p,q_˜ (H_r) with the normkdkS_v^p,q_˜ (H)=k(d¹, d²· · ·, d^r)kS_v^p,q_˜ (H) =Pr

i=1kdⁱkS_v^p,q_˜ (H_r),whereS_v˜^p,q(H_i)is defined as in Definition 3.2.2 with respect to the lattice Λⁱ and the Hilbert space H_i. For x ∈ R^d, k, n ∈ Z^d, α = (α₁, α₂· · · , α_r) and β = (β₁, β₂· · · , β_r) with α_i > 0, β_i > 0 the translation operator T_αk and the modulation operator M_βn of f(x) = Lr

i=1f_i(x) are defined as T_αkf(x) = Lr

i=1T_αikf_i(x) and M_βnf(x) = Lr

i=1M_βinf_i(x). The frame operator of the Gabor system on W_H(L^p, L^q_w) is given by

S_g,γ^Λ f = β^−d X

n∈Z^d

G_n Tⁿ

βf

=β^−d X

n∈Z^d

X

k∈Z^d

T_αkγ(x)⊙T_αk+ⁿ

βg(x) Tⁿ

βf