# A Note on Operator Sampling and Fractional Fourier Transform

Authors: Qingyue Zhang
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

This paper presents that the kernel of the fractional Fourier transform (FRFT) satisfies the operator version of Kramer's Lemma (Hong and Pfander, 2010), which gives a new applicability of Kramer's Lemma. Moreover, we give a new sampling formulae for reconstructing the operators which are bandlimited in the FRFT sense.


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## Body

## 1. Introduction and Notations

Sampling theory for operators motivated by the operator identification problem in communications engineering has been developed during the last few years [1–4]. In [4], Hong and Pfander gave an operator version of Kramer's Lemma (see [4, Theorem 25]). But they did not give any explicit kernel satisfying the hypotheses in [4, Theorem 25] other than the Fourier kernel. In this paper, we present that the kernel of the fractional Fourier transform satisfies the hypotheses in [4, Theorem 25]. Therefore, we give a new applicability of Kramer's method.The FRFT—a generalization of the Fourier transform (FT)—has received much attention in recent years due to its numerous applications, including signal processing, quantum physics, communications, and optics [5–7]. Hong and Pfander studied the sampling theorem on the operators which are bandlimited in the FT sense (see [4]). In this paper, we generalize their results to bandlimited operators in the FRFT sense.Forf∈L2(ℝ), its FRFT is defined by(1.1)Fα(u)=Fα[f(x)](u)=∫Rf(x)Kα(u,x)dx,
where α∈ℝ, and the transform kernel is given by(1.2)Kα(u,x)={Aαe(i/2)(x2+u2)cotα-iuxcscαifα≠kπ,δ(x-u)ifα=2kπ,δ(x+u)ifα=(2k-1)π,
where δ(·) is Dirac distribution function over ℝ, Aα=(1-icotα)/2π, and k∈ℤ. The inverse FRFT is the FRFT at angle -α, given by(1.3)f(x)=F-α[Fα(u)]=∫RFα(u)Kα(u,x)¯du,
where the bar denotes the complex conjugation. Whenever α=π/2, (1.2) reduces to the FT. Through this paper, we assume that  α≠kπ.In FRFT domain, the function space with bandwidthΩ is defined by(1.4)FPWΩ={f∈L2(R):supp⁡Fαf⊆[-Ω2,Ω2]}.
For the sake of simplicity, when α=π/2, FPWΩ is written as PWΩ.In the following, we use the notation(1.5)A(F)≍B(F),F∈F
if there exist positive constants c and C such that cA(F)≤B(F)≤CA(F) for all objects F in the set ℱ.Letℍ be a Hilbert space and {fn:n∈ℤ} be a sequence in ℍ. The set {fn:n∈ℤ} is said to be a frame [8, 9] for ℍ if(1.6)‖f‖H2≍∑n∈Z|〈f,fn〉H|2,f∈H.LetΛ={λk:k∈ℤ}⊆ℝ with (λk<λk+1). Λ is a set sampling for FPWΩ if(1.7)‖f‖L22≍∑k∈Z|f(λk)|2,f∈FPWΩ.


## 2. The Properties of the Kernel of FRFT

In this section, we consider under what conditions{Kα(λk+t,·):k∈ℤ} is a frame for L2[-Ω/2,Ω/2] for every t∈ℝ. The following theorem gives a necessary and sufficient condition for {Kα(λk+t,·):k∈ℤ} to be a frame for L2[-Ω/2,Ω/2] for every t∈ℝ.Theorem 2.1.
For anyt∈ℝ and sinα>0. {Kα(λk+t,·):k∈ℤ} is a frame for L2[-Ω/2,Ω/2] if and only if {e-iλkω:k∈ℤ} is a frame for L2[-(Ω/2)cscα,(Ω/2)cscα].Remark 2.2.
By Theorem2.1, when taking appropriate λk, {Kα(λk+t,·):k∈ℤ} is a frame for each t∈ℝ. Therefore, we give a kernel satisfying the hypotheses in [4, Theorem 25], which gives a new applicability of Kramer's Lemma.To prove Theorem2.1, we need to introduce the following results.Lemma 2.3.
Λ={λk:k∈ℤ} is a set of sampling for FPWΩ if and only if {Kα(λk+t,·):k∈ℤ} is a frame for L2[-Ω/2,Ω/2] for every t∈ℝ.Proof.
Suppose thatΛ is a set of sampling for FPWΩ. Then, for any F∈L2[-Ω/2,Ω/2], there exists f∈FPWΩ such that F=Fαf. Since
(2.1)〈F(⋅)e-i(⋅)tcscα,Kα(λk+t,⋅)〉L2=〈F(⋅)e-i(⋅)tcscα,Aαe(i/2)((λk+t)2+(⋅)2)cotα-i(⋅)(λk+t)cscα〉L2=e-(i/2)(2λkt+t2)〈F(⋅),Aαe(i/2)(λk2+(⋅)2)cotα-i(⋅)λkcscα〉L2=e(-i/2)(2λkt+t2)〈F(⋅),Kα(⋅,λk)〉=e-(i/2)(2λkt+t2)f(λk),
we have
(2.2)‖F(⋅)e-i(⋅)tcscα‖L22=‖F‖L22=‖f‖L22≍∑k∈Z|f(λk)|2=∑k∈Z|〈F(⋅)e-i(⋅)tcscα,Kα(λk+t,⋅)〉|2.
Therewith, {Kα(λk+t,·):k∈ℤ} is a frame for L2[-Ω/2,Ω/2] for any t∈ℝ.
On the other hand, suppose that{Kα(λk+t,·):k∈ℤ} is a frame for L2[-Ω/2,Ω/2] for any t∈ℝ. Specifically, {Kα(λk,·):k∈ℤ} is also a frame for L2[-Ω/2,Ω/2]. Then, by (2.1),
(2.3)‖f‖L22=‖F‖L22≍∑k∈Z|〈F(⋅),Kα(λk,⋅)〉|2=∑k∈Z|f(λk)|2.
This completes our proof.The following proposition gives a necessary and sufficient condition aboutΛ={λk:k∈ℤ} is set of sampling for PWΩ.Proposition 2.4 (see [10, Lemma 3.5]).
Λ={λk:k∈ℤ} is set of sampling for PWΩ if and only if {e-iλkω:k∈ℤ} is a frame for L2[-Ω/2,Ω/2].Lemma 2.5.
Λ={λk:k∈ℤ} is set of sampling for FPWΩ if and only if {e-iλkω:k∈ℤ} is a frame for L2[-(Ω/2)cscα,(Ω/2)cscα].Proof.
Since(2.4)Fα[f(t)](u)=2πe(i/2)u2cotαF[f(t)e(i/2)t2cotα](ucscα),
where 𝔉 denotes the FT operator, we have
(2.5)e(i/2)(⋅)2cotαFPWΩ=PWΩcscα.
By Proposition 2.4, this completes the proof.Proof of Theorem2.1.
By Lemmas2.3 and 2.5, we immediately get the claim.


## 3. Sampling of Operators Related to FRFT

In this section, a new sampling formulae for operator is proposed. First we introduce some definitions and notations about sampling of operator.The class of Hilbert-Schmidt operatorsHS(L2(ℝ)) consists of bounded linear operators on L2(ℝ) which can be represented as integral operators of the form(3.1)Hf(x)=∫RκH(x,t)f(t)dt
with kernel κH∈L2(ℝ2). Let hH(t,x)=κH(x,x-t). We call hH(t,x) the time-varying impulse response of H. If H∈HS(L2(ℝ)), then the operator norm of H is defined by ∥H∥HS∶=∥κH∥L2=∥hH∥L2.The Feichtinger algebra is defined by(3.2)S0(R)={f∈L2(R):Vgf(t,ν)∈L1(R2)},
where V𝔤f(t,ν)=〈f,MνTt𝔤〉 is the short-time Fourier transform of f with respect to the Gaussian 𝔤(x)=e-πx2. An operator class O⊆HS(L2(ℝ)) is identifiable if all H∈O extend to a domain containing a so-called identifier f∈S0′(ℝ) and(3.3)‖H‖HS≍‖Hf‖L2,H∈O.
The operator class O⊆HS(L2(ℝ)) permits operator sampling if one can choose f in (3.3) with discrete support in ℝ in the distributional sense. In that case, suppf is called sampling set for O.ForT,Ω>0, let(3.4)OPWT,Ω={H∈HS(L2(R)):supp⁡F(hH(t,⋅))⊆[0,T]×[-Ω2,Ω2]},OFPWT,Ω={H∈HS(L2(R)):supp⁡Fα(hH(t,⋅))⊆[0,T]×[-Ω2,Ω2]}.The following theorem states that a bandlimited operator in FRFT sense permits operator sampling.Theorem 3.1.
ForΩ,T,T>0 and 0<T′Ω≤TΩ≤2πsinα, choose φ∈PW([-(2π/T-(Ωcscα/2)),(2π/T-(Ωcscα/2))]) with 𝔉(φ)=1 on [-Ωcscα/2,Ωcscα/2] and r∈L∞(ℝ) with suppr⊆[-T+T′,T] and r = 1 on [0,T]. Then OFPWT′,Ω permits operator sampling as
(3.5)‖H‖HS=T‖H∑k∈ZδkT‖L2,H∈OFPWT′,Ω,
and operator reconstruction is possible by means of the L2-convergent series
(3.6)hH(t,x)=e-(i/2)x2cotαr(t)T∑n∈Ze-(i/2)(t+nT)2cotα×(H∑k∈ZδkT)(t+nT)φ(x-t-nT).Before we give the proof of Theorem3.1, the following two propositions are needed.Proposition 3.2 (see [4, Theorem 8]).
ForΩ,T,T>0 and 0<T′Ω≤TΩ≤2π, choose φ∈PW([-(2π/T-Ω/2),(2π/T-Ω/2)]) with 𝔉(φ)=1 on [-Ω/2,Ω/2] and r∈L∞(ℝ) with suppr⊆[-T+T′,T] and r = 1 on [0,T]. Then OPWT′,Ω permits operator sampling as
(3.7)‖H‖HS=T‖H∑k∈ZδkT‖L2,H∈OPWT′,Ω,
and operator reconstruction is possible by means of the L2-convergent series
(3.8)hH(t,x)=r(t)T∑n∈Z(H∑k∈ZδkT)(t+nT)φ(x-t-nT).Proposition 3.3 (see [11, Lemma 1]).
Assume a signalf(t)∈FPWΩ. Let
(3.9)g(t)=∫-Ω/2Ω/2Fα[f(t)](u)e-(i/2)u2cotα+iutcscαdu.
Then g(t)∈PWΩcscα.Proof of Theorem3.1.
Due to (2.4), we have
(3.10)f(t,x)∈OFPWT′,Ω⟺e(i/2)x2cotαf(t,x)∈OPWT′,Ωcscα.
By the proof of Proposition 3.2,
(3.11)‖H‖HS=T‖H∑k∈ZδkT‖L2,H∈OFPWT′,Ω.
Applying Proposition 3.3, we obtain
(3.12)hH(t,x)=Aαe-(i/2)x2cotαh¯H(t,x),
where h¯H(t,x)=∫-Ω/2Ω/2Fα[hH(t,·)](u)e-(i/2)u2cotα+iuxcscαduandh¯H(t,x)∈OPWT′,Ωcscα.Let(3.13)H¯f(x)=∫Rh¯H(t,x)f(x-t).
Using Proposition 3.2 again,
(3.14)h¯H(t,x)=r(t)T∑n∈Z(H¯∑k∈ZδkT)(t+nT)φ(x-t-nT).
Therewith,
(3.15)hH(t,x)=Aαe-(i/2)x2cotαr(t)T∑n∈Z(H¯∑k∈ZδkT)(t+nT)φ(x-t-nT)=Aαe-(i/2)x2cotαr(t)T∑n∈Zh¯H(t,t+nT)φ(x-t-nT)=e-(i/2)x2cotαr(t)T∑n∈Ze-(i/2)(t+nT)2cotαhH(t,t+nT)φ(x-t-nT)=e-(i/2)x2cotαr(t)T∑n∈Ze-(i/2)(t+nT)2cotα×(H∑k∈ZδkT)(t+nT)φ(x-t-nT).Next, we give an important multichannel operator sampling theorem, namely, derivative operator sampling. Checking the proof of [4, Theorem 32], we have following lemma.Lemma 3.4.
LetM,N∈ℕ and f(r) denotes the rth derivative of f in the distributional sense. Then,
(3.16)‖H‖HS2≍∑j=0MN-1‖∑r=0j(jr)(-1)r(H∑k∈ZδkN(r))(j-r)‖2,H∈OPWN,M2π.
and operator reconstruction is possible by means of the L2-convergent series
(3.17)hH(t,x)=∑j=0MN-1∑n∈Z(Hj∑k∈ZδkN)(t+nN)φj(x-t-nN),
where {φj(x-t-nN):0≤j≤MN,n∈ℕ} is a Riesz basis for PW2πM for each fixed t∈[0,N], and Hjf(x)=(∑r=0j(jr)(-1)r(Hf(r))j-r)(x).For the operators which are bandlimited in FRFT sense. We have the following theorem.Theorem 3.5.
LetM,N∈ℕ and f(r) denotes the rth derivative of f in the distributional sense. Then,
(3.18)‖H‖HS2≍∑j=0MN-1‖∑r=0j(jr)(e(i/2)x2cotα)(j-r)∑p=0r(rp)(-1)p(H∑k∈ZδkN(r))r-p‖2,H∈OFPWN,M2πsinα.
and operator reconstruction is possible by means of the L2-convergent series
(3.19)hH(t,x)=e-(i/2)x2cotα∑j=0MN-1∑n∈Z∑r=0j(jr)×((e(i/2)x2cotα)(j-r)Hr∑k∈ZδkN)(t+nN)φj(x-t-nN),
where {φj(x-t-nN):0≤j≤MN,n∈ℕ} is a Riesz basis for PWM2π for each fixed t∈[0,N] and Hjf(x)=(∑r=0j(jr)(-1)r(Hf(r))j-r)(x).Proof.
Due to (2.4), we have
(3.20)f(t,x)∈OFPWN,M2πsinα⟺e(i/2)x2cotαf(t,x)∈OPWN,M2π.
By Proposition 3.3, we obtain
(3.21)hH(t,x)=Aαe-(i/2)x2cotαh¯H(t,x),
where h¯H(t,x)=∫-Ω/2Ω/2Fα[hH(t,·)](u)e-(i/2)u2cotα+iuxcscαdu. Put
(3.22)H¯f(x)=∫Rh¯H(t,x)f(x-t)dt.
By Lemma 3.4, we have
(3.23)hH¯(t,x)=∑j=0MN-1∑n∈Z(H¯j∑k∈ZδkN)(t+nN)φj(x-t-nN),
where
(3.24)H¯jf(x)=(∑r=0j(jr)(-1)r(H¯f(r))j-r)(x)=∫R∂j∂xjh¯H(t,x)f(x-t)dt=1Aα∫R∂j∂xj(e(i/2)x2cotαhH(t,x))f(x-t)dt=1Aα∑r=0j(jr)(e(i/2)x2cotα)(j-r)(x)∫R∂r∂xrhH(t,x)f(x-t)dt=1Aα∑r=0j(jr)(e(i/2)x2cotα)(j-r)(x)Hrf(x)=1Aα∑r=0j(jr)(e(i/2)x2cotα)(j-r)(x)×(∑p=0r(rp)(-1)p(Hf(p))r-p)(x).
By the proof of Lemma 3.4, (3.18) holds. Moreover, by (3.21) we obtain
(3.25)hH(t,x)=Aαe-(i/2)x2cotα∑j=0MN-1∑n∈Z(H¯j∑k∈ZδkN)(t+nN)φj(x-t-nN)=e-(i/2)x2cotα∑j=0MN-1∑n∈Z∑r=0j(jr)×((e(i/2)x2cotα)(j-r)Hr∑k∈ZδkN)(t+nN)φj(x-t-nN).Remark 3.6.
After careful development of pertinent tools, one can formulate extensions of results in this paper to the linear canonical transform case (see [11–13]). We have presented the FRFT case because of its simplicity and applicability.


## 4. Conclusion

Kramer's Lemma is very important in the proofs of a number of sampling theorems. In [4, Theorem 25], Hong and Pfander proved an operator sampling version of Kramer's Lemma. But they did not give any explicit kernel satisfying the hypotheses in [4, Theorem 25] other than the Fourier kernel. In this paper, we find that the kernel of the fractional Fourier transform satisfies the hypotheses in [4, Theorem 25]. This observation gives a new applicability of Kramer's method. Moreover, we give a new sampling formulae for reconstructing the operators which are bandlimited in the FRFT sense. This is an extension of some results in [4].



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