A Note on Operator Sampling and Fractional Fourier Transform

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. For f ∈ L2 R , its FRFT is defined by


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.
For f ∈ L 2 R , its FRFT is defined by where α ∈ R, and the transform kernel is given by where δ • is Dirac distribution function over R, A α 1 − i cot α /2π, and k ∈ Z.The inverse FRFT is the FRFT at angle −α, given by 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 bandwidth Ω is defined by For the sake of simplicity, when α π/2, FPW Ω is written as PW Ω .
In the following, we use the notation if there exist positive constants c and C such that cA F ≤ B F ≤ CA F for all objects F in the set F.
Let H be a Hilbert space and {f n : n ∈ Z} be a sequence in H.The set {f n : n ∈ Z} is said to be a frame 8, 9 for H if To prove Theorem 2.1, we need to introduce the following results.

The Properties of the Kernel of FRFT
Proof.Suppose that Λ is a set of sampling for FPW Ω .Then, for any On the other hand, suppose that

2.3
This completes our proof.
The following proposition gives a necessary and sufficient condition about where F denotes the FT operator, we have By Proposition 2.4, this completes the proof.
Proof of Theorem 2.1.By Lemmas 2.3 and 2.5, we immediately get the claim.

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 operators HS L 2 R consists of bounded linear operators on L 2 R which can be represented as integral operators of the form where V g f t, ν f, M ν T t g is the short-time Fourier transform of f with respect to the Gaussian g x e −πx 2 .An operator class O ⊆ HS L 2 R is identifiable if all H ∈ O extend to a domain containing a so-called identifier f ∈ S 0 R and The operator class O ⊆ HS L 2 R permits operator sampling if one can choose f in 3.3 with discrete support in R in the distributional sense.In that case, supp f is called sampling set for O.

3.4
The following theorem states that a bandlimited operator in FRFT sense permits operator sampling.Theorem 3.1.For Ω, T, T > 0 and 0 with F ϕ 1 on −Ω csc α/2, Ωcscα/2 and r ∈ L ∞ R with supp r ⊆ −T T , T and r = 1 on 0, T .Then OFPW T ,Ω permits operator sampling as , H ∈ OFPW T ,Ω , 3.5 and operator reconstruction is possible by means of the L 2 -convergent series

3.6
Before we give the proof of Theorem 3.1, the following two propositions are needed.

3.15
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.Let M, N ∈ N and f r denotes the rth derivative of f in the distributional sense.Then, , H ∈ OPW N,M2π .

3.16
and operator reconstruction is possible by means of the L 2 -convergent series where {ϕ j x − t − nN : 0 ≤ j ≤ MN, n ∈ N} is a Riesz basis for PW 2πM for each fixed t ∈ 0, N , and H j f x j r 0 j r −1 r Hf r j−r x .
For the operators which are bandlimited in FRFT sense.We have the following theorem.where

3.24
By the proof of Lemma 3.4, 3.18 holds.Moreover, by 3.21 we obtain MN−1 j 0 n∈Z j r 0 j r × e i/2 x 2 cotα j−r H r k∈Z δ kN t nN ϕ j x − t − nN .

3.25
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.

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 .

Theorem 3 . 5 . 2 ,H
Let M, N ∈ N and f r denotes the rth derivative of f in the distributional sense.Then, ∈ OFPW N,M2π sin α .
If H ∈ HS L 2 R , then the operator norm of H is defined by H HS : Let h H t, x κ H x, x−t .We call h H t, x the time-varying impulse response of H.
F α h H t, • u e − i/2 u 2 cot α iux csc α du and h H t, x ∈ OPW T ,Ω csc α.Let operator reconstruction is possible by means of the L 2 -convergent series n∈Z H k∈Z δ kT t nT ϕ x − t − nT .3.8 Proposition 3.3 see 11, Lemma 1 .Assume a signal f t ∈ FPW Ω .Let f t, x ∈ OFPW T ,Ω ⇐⇒ e i/2 x 2 cot α f t, x ∈ OP W T ,Ω csc α .n∈Z H k∈Z δ kT t nT ϕ x − t − nT .
− t − nN : 0 ≤ j ≤ MN, n ∈ N} is a Riesz basis for PW M2π for each fixed t ∈ 0, N R h H t, x f x − t dt.