A Note on Operator Sampling and Fractional Fourier Transform

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.

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.

For 𝑓∈𝐿²(ℝ), its FRFT is defined by𝐹_𝛼(𝑢)=𝐹^𝛼𝑓(𝑥)(𝑢)=_ℝ𝑓(𝑥)𝐾_𝛼(𝑢,𝑥)𝑑𝑥, (1.1) where 𝛼∈ℝ, and the transform kernel is given by𝐾_𝛼()=𝐴𝑢,𝑥^𝛼𝑒^{(𝑖/2)(𝑥2+𝑢2)cot𝛼−𝑖𝑢𝑥csc𝛼}𝛿()𝛿()()if𝛼≠𝑘𝜋,𝑥−𝑢if𝛼=2𝑘𝜋,𝑥+𝑢if𝛼=2𝑘−1𝜋, (1.2) where 𝛿(⋅) is Dirac distribution function over ℝ, 𝐴_𝛼=(1−𝑖cot𝛼)/2𝜋, and 𝑘∈ℤ. The inverse FRFT is the FRFT at angle −𝛼, given by𝑓(𝑥)=𝐹^−𝛼𝐹_𝛼(𝑢)=_ℝ𝐹_𝛼(𝑢)𝐾_𝛼(𝑢,𝑥)𝑑𝑢, (1.3) 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 byFPW_Ω=𝑓∈𝐿²(ℝ)∶supp𝐹^𝛼−Ω𝑓⊆2,Ω2. (1.4) For the sake of simplicity, when 𝛼=𝜋/2, FPW_Ω is written as PW_Ω.

In the following, we use the notation𝐴(𝐹)≍𝐵(𝐹),𝐹∈ℱ (1.5) 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 if𝑓²_ℍ≍_𝑛∈ℤ⟨𝑓,𝑓_𝑛⟩_ℍ²,𝑓∈ℍ. (1.6)

Let Λ={𝜆_𝑘∶𝑘∈ℤ}⊆ℝ with (𝜆_𝑘<𝜆_𝑘+1). Λ is a set sampling for FPW_Ω if𝑓²_𝐿2≍_𝑘∈ℤ𝑓𝜆_𝑘²,𝑓∈FPW_Ω. (1.7)

2. The Properties of the Kernel of FRFT

In this section, we consider under what conditions {𝐾_𝛼(𝜆_𝑘+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 {𝐾_𝛼(𝜆_𝑘+𝑡,⋅)∶𝑘∈ℤ} to be a frame for 𝐿²[−Ω/2,Ω/2] for every 𝑡∈ℝ.

Theorem 2.1. For any 𝑡∈ℝ and sin𝛼>0. {𝐾_𝛼(𝜆_𝑘+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 Theorem 2.1, when taking appropriate 𝜆_𝑘, {𝐾_𝛼(𝜆_𝑘+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 Theorem 2.1, we need to introduce the following results.

Lemma 2.3. Λ={𝜆_𝑘∶𝑘∈ℤ} is a set of sampling for FPW_Ω if and only if {𝐾_𝛼(𝜆_𝑘+𝑡,⋅)∶𝑘∈ℤ} is a frame for 𝐿²[−Ω/2,Ω/2] for every 𝑡∈ℝ.

Proof. Suppose that Λ is a set of sampling for FPW_Ω. Then, for any 𝐹∈𝐿²[−Ω/2,Ω/2], there exists 𝑓∈FPW_Ω such that 𝐹=𝐹^𝛼𝑓. Since 𝐹(⋅)𝑒^{−𝑖(⋅)𝑡csc𝛼},𝐾_𝛼𝜆_𝑘+𝑡,⋅_𝐿2=𝐹(⋅)𝑒^{−𝑖(⋅)𝑡csc𝛼},𝐴_𝛼𝑒^{(𝑖/2)((𝜆𝑘+𝑡)2+(⋅)2)cot𝛼−𝑖(⋅)(𝜆𝑘+𝑡)csc𝛼}_𝐿2=𝑒^{−(𝑖/2)(2𝜆𝑘𝑡+𝑡2)}𝐹(⋅),𝐴_𝛼𝑒^{(𝑖/2)(𝜆2𝑘+(⋅)2)cot𝛼−𝑖(⋅)𝜆𝑘csc𝛼}_𝐿2=𝑒^{(−𝑖/2)(2𝜆𝑘𝑡+𝑡2)}𝐹(⋅),𝐾_𝛼⋅,𝜆_𝑘=𝑒^{−(𝑖/2)(2𝜆𝑘𝑡+𝑡2)}𝑓𝜆_𝑘, (2.1) we have 𝐹(⋅)𝑒^{−𝑖(⋅)𝑡csc𝛼2}_𝐿2=‖𝐹‖²_𝐿2=𝑓²_𝐿2≍_𝑘∈ℤ𝑓𝜆_𝑘²=_𝑘∈ℤ𝐹(⋅)𝑒^{−𝑖(⋅)𝑡csc𝛼},𝐾_𝛼𝜆_𝑘+𝑡,⋅². (2.2) Therewith, {𝐾_𝛼(𝜆_𝑘+𝑡,⋅)∶𝑘∈ℤ} is a frame for 𝐿²[−Ω/2,Ω/2] for any 𝑡∈ℝ.
On the other hand, suppose that {𝐾_𝛼(𝜆_𝑘+𝑡,⋅)∶𝑘∈ℤ} is a frame for 𝐿²[−Ω/2,Ω/2] for any 𝑡∈ℝ. Specifically, {𝐾_𝛼(𝜆_𝑘,⋅)∶𝑘∈ℤ} is also a frame for 𝐿²[−Ω/2,Ω/2]. Then, by (2.1), 𝑓²_𝐿2=‖𝐹‖²_𝐿2≍_𝑘∈ℤ𝐹(⋅),𝐾_𝛼𝜆_𝑘,⋅²=_𝑘∈ℤ𝑓𝜆_𝑘². (2.3) This completes our proof.

The following proposition gives a necessary and sufficient condition about Λ={𝜆_𝑘∶𝑘∈ℤ} is set of sampling for PW_Ω.

Proposition 2.4 (see [10, Lemma 3.5]). Λ={𝜆_𝑘∶𝑘∈ℤ} is set of sampling for PW_Ω if and only if {𝑒^{−𝑖𝜆𝑘𝜔}∶𝑘∈ℤ} is a frame for 𝐿²[−Ω/2,Ω/2].

Lemma 2.5. Λ={𝜆_𝑘∶𝑘∈ℤ} is set of sampling for FPW_Ω if and only if {𝑒^{−𝑖𝜆𝑘𝜔}∶𝑘∈ℤ} is a frame for 𝐿²[−(Ω/2)csc𝛼,(Ω/2)csc𝛼].

Proof. Since 𝐹^𝛼𝑓(𝑡)(𝑢)=√2𝜋𝑒^{(𝑖/2)𝑢2cot𝛼}𝔉𝑓(𝑡)𝑒^{(𝑖/2)𝑡2cot𝛼}(𝑢csc𝛼), (2.4) where 𝔉 denotes the FT operator, we have 𝑒^{(𝑖/2)(⋅)2cot𝛼}FPW_Ω=PW_Ωcsc𝛼. (2.5) 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.

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 operators HS(𝐿²(ℝ)) consists of bounded linear operators on 𝐿²(ℝ) which can be represented as integral operators of the form𝐻𝑓(𝑥)=_ℝ𝜅_𝐻(𝑥,𝑡)𝑓(𝑡)𝑑𝑡 (3.1) with kernel 𝜅_𝐻∈𝐿²(ℝ²). Let ℎ_𝐻(𝑡,𝑥)=𝜅_𝐻(𝑥,𝑥−𝑡). We call ℎ_𝐻(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∶=‖𝜅_𝐻‖_𝐿2=‖ℎ_𝐻‖_𝐿2.

The Feichtinger algebra is defined by𝑆₀(ℝ)=𝑓∈𝐿²(ℝ)∶𝑉_𝔤𝑓(𝑡,𝜈)∈𝐿¹ℝ², (3.2) where 𝑉_𝔤𝑓(𝑡,𝜈)=⟨𝑓,𝑀_𝜈𝑇_𝑡𝔤⟩ 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∈𝑆^′₀(ℝ) and‖𝐻‖_HS≍𝐻𝑓_𝐿2,𝐻∈𝑂. (3.3) 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.

For 𝑇,Ω>0, letOPW_𝑇,Ω=𝐻∈HS𝐿²(ℝ)∶supp𝔉ℎ_𝐻()⊆[]×−Ω𝑡,⋅0,𝑇2,Ω2,OFPW_𝑇,Ω=𝐻∈HS𝐿²(ℝ)∶supp𝐹^𝛼ℎ_𝐻()⊆[]×−Ω𝑡,⋅0,𝑇2,Ω2. (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<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 OFPW_𝑇′,Ω permits operator sampling as ‖𝐻‖_HS=√𝑇𝐻_𝑘∈ℤ𝛿_{𝑘𝑇𝐿2},𝐻∈OFPW_𝑇′,Ω, (3.5) and operator reconstruction is possible by means of the 𝐿²-convergent series ℎ_𝐻(𝑡,𝑥)=𝑒^{−(𝑖/2)𝑥2cot𝛼}𝑟(𝑡)𝑇_𝑛∈ℤ𝑒^{−(𝑖/2)(𝑡+𝑛𝑇)2cot𝛼}×𝐻_𝑘∈ℤ𝛿_𝑘𝑇(𝑡+𝑛𝑇)𝜑(𝑥−𝑡−𝑛𝑇). (3.6)

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

Proposition 3.2 (see [4, Theorem 8]). For Ω,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 OPW_𝑇′,Ω permits operator sampling as ‖𝐻‖_HS=√𝑇𝐻_𝑘∈ℤ𝛿_{𝑘𝑇𝐿2},𝐻∈OPW_𝑇′,Ω, (3.7) and operator reconstruction is possible by means of the 𝐿²-convergent series ℎ_𝐻(𝑡,𝑥)=𝑟(𝑡)𝑇_𝑛∈ℤ𝐻_𝑘∈ℤ𝛿_𝑘𝑇(𝑡+𝑛𝑇)𝜑(𝑥−𝑡−𝑛𝑇). (3.8)

Proposition 3.3 (see [11, Lemma 1]). Assume a signal 𝑓(𝑡)∈FPW_Ω. Let 𝑔(𝑡)=^Ω/2_−Ω/2𝐹^𝛼𝑓(𝑡)(𝑢)𝑒^{−(𝑖/2)𝑢2cot𝛼+𝑖𝑢𝑡csc𝛼}𝑑𝑢. (3.9) Then 𝑔(𝑡)∈PW_Ωcsc𝛼.

Proof of Theorem 3.1. Due to (2.4), we have 𝑓(𝑡,𝑥)∈OFPW_𝑇′,Ω⟺𝑒^{(𝑖/2)𝑥2cot𝛼}𝑓(𝑡,𝑥)∈𝑂𝑃𝑊_{𝑇′,Ωcsc𝛼}. (3.10) By the proof of Proposition 3.2, ‖𝐻‖_HS=√𝑇𝐻_𝑘∈ℤ𝛿_{𝑘𝑇𝐿2},𝐻∈OFPW_𝑇′,Ω. (3.11) Applying Proposition 3.3, we obtain ℎ_𝐻(𝑡,𝑥)=𝐴_𝛼𝑒^{−(𝑖/2)𝑥2cot𝛼}ℎ_𝐻(𝑡,𝑥), (3.12) where ℎ_𝐻(𝑡,𝑥)=∫^Ω/2_−Ω/2𝐹^𝛼[ℎ_𝐻(𝑡,⋅)](𝑢)𝑒^{−(𝑖/2)𝑢2cot𝛼+𝑖𝑢𝑥csc𝛼}𝑑𝑢andℎ_𝐻(𝑡,𝑥)∈OPW_{𝑇′,Ωcsc𝛼.}Let𝐻𝑓(𝑥)=_ℝℎ_𝐻(𝑡,𝑥)𝑓(𝑥−𝑡). (3.13) Using Proposition 3.2 again, ℎ_𝐻(𝑡,𝑥)=𝑟(𝑡)𝑇_𝑛∈ℤ𝐻_𝑘∈ℤ𝛿_𝑘𝑇(𝑡+𝑛𝑇)𝜑(𝑥−𝑡−𝑛𝑇). (3.14) Therewith, ℎ_𝐻(𝑡,𝑥)=𝐴_𝛼𝑒^{−(𝑖/2)𝑥2cot𝛼}𝑟(𝑡)𝑇_𝑛∈ℤ𝐻_𝑘∈ℤ𝛿_𝑘𝑇(𝑡+𝑛𝑇)𝜑(𝑥−𝑡−𝑛𝑇)=𝐴_𝛼𝑒^{−(𝑖/2)𝑥2cot𝛼}𝑟(𝑡)𝑇_𝑛∈ℤℎ_𝐻(𝑡,𝑡+𝑛𝑇)𝜑(𝑥−𝑡−𝑛𝑇)=𝑒^{−(𝑖/2)𝑥2cot𝛼}𝑟(𝑡)𝑇_𝑛∈ℤ𝑒^{−(𝑖/2)(𝑡+𝑛𝑇)2cot𝛼}ℎ_𝐻(𝑡,𝑡+𝑛𝑇)𝜑(𝑥−𝑡−𝑛𝑇)=𝑒^{−(𝑖/2)𝑥2cot𝛼}𝑟(𝑡)𝑇_𝑛∈ℤ𝑒^{−(𝑖/2)(𝑡+𝑛𝑇)2cot𝛼}×𝐻_𝑘∈ℤ𝛿_𝑘𝑇(𝑡+𝑛𝑇)𝜑(𝑥−𝑡−𝑛𝑇). (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‖²_HS≍^𝑀𝑁−1_𝑗=0^𝑗_𝑟=0𝑗𝑟(−1)^𝑟𝐻_𝑘∈ℤ𝛿^(𝑟)_𝑘𝑁^{(𝑗−𝑟)2},𝐻∈OPW_{𝑁,𝑀2𝜋}. (3.16) and operator reconstruction is possible by means of the 𝐿²-convergent series ℎ_𝐻(𝑡,𝑥)=^𝑀𝑁−1_{𝑗=0𝑛∈ℤ}𝐻_{𝑗𝑘∈ℤ}𝛿_𝑘𝑁(𝑡+𝑛𝑁)𝜑_𝑗(𝑥−𝑡−𝑛𝑁), (3.17) 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_𝑗𝑓(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.

Theorem 3.5. Let 𝑀,𝑁∈ℕ and 𝑓^(𝑟) denotes the 𝑟th derivative of 𝑓 in the distributional sense. Then, ‖𝐻‖²_HS≍^𝑀𝑁−1_𝑗=0^𝑗_𝑟=0𝑗𝑟𝑒^{(𝑖/2)𝑥2cot𝛼𝑟(𝑗−𝑟)}_𝑝=0𝑟𝑝(−1)^𝑝𝐻_𝑘∈ℤ𝛿^(𝑟)_𝑘𝑁^𝑟−𝑝2,𝐻∈OFPW_{𝑁,𝑀2𝜋sin𝛼}. (3.18) and operator reconstruction is possible by means of the 𝐿²-convergent series ℎ_𝐻(𝑡,𝑥)=𝑒^{−(𝑖/2)𝑥2cot𝛼𝑀𝑁−1}_𝑗=0^𝑗_{𝑛∈ℤ𝑟=0}𝑗𝑟×𝑒^{(𝑖/2)𝑥2cot𝛼(𝑗−𝑟)}𝐻_{𝑟𝑘∈ℤ}𝛿_𝑘𝑁(𝑡+𝑛𝑁)𝜑_𝑗(𝑥−𝑡−𝑛𝑁), (3.19) where {𝜑_𝑗(𝑥−𝑡−𝑛𝑁)∶0≤𝑗≤𝑀𝑁,𝑛∈ℕ} is a Riesz basis for PW_𝑀2𝜋 for each fixed 𝑡∈[0,𝑁] and 𝐻_𝑗𝑓(𝑥)=(∑^𝑗_𝑟=0^𝑗_𝑟(−1)^𝑟(𝐻𝑓^(𝑟))^𝑗−𝑟)(𝑥).

Proof. Due to (2.4), we have 𝑓(𝑡,𝑥)∈OFPW_{𝑁,𝑀2𝜋sin𝛼}⟺𝑒^{(𝑖/2)𝑥2cot𝛼}𝑓(𝑡,𝑥)∈OPW_{𝑁,𝑀2𝜋}. (3.20) By Proposition 3.3, we obtain ℎ_𝐻(𝑡,𝑥)=𝐴_𝛼𝑒^{−(𝑖/2)𝑥2cot𝛼}ℎ_𝐻(𝑡,𝑥), (3.21) where ℎ_𝐻(𝑡,𝑥)=∫^Ω/2_−Ω/2𝐹^𝛼[ℎ_𝐻(𝑡,⋅)](𝑢)𝑒^{−(𝑖/2)𝑢2cot𝛼+𝑖𝑢𝑥csc𝛼}𝑑𝑢. Put 𝐻𝑓(𝑥)=_ℝℎ_𝐻(𝑡,𝑥)𝑓(𝑥−𝑡)𝑑𝑡. (3.22) By Lemma 3.4, we have ℎ_𝐻(𝑡,𝑥)=^𝑀𝑁−1_{𝑗=0𝑛∈ℤ}𝐻_{𝑗𝑘∈ℤ}𝛿_𝑘𝑁(𝑡+𝑛𝑁)𝜑_𝑗(𝑥−𝑡−𝑛𝑁), (3.23) where 𝐻_𝑗𝑓(𝑥)=^𝑗_𝑟=0𝑗𝑟(−1)^𝑟𝐻𝑓^{(𝑟)𝑗−𝑟}(𝑥)=_ℝ𝜕^𝑗𝜕𝑥^𝑗ℎ_𝐻()𝑓()=1𝑡,𝑥𝑥−𝑡𝑑𝑡𝐴_𝛼ℝ𝜕^𝑗𝜕𝑥^𝑗𝑒^{(𝑖/2)𝑥2cot𝛼}ℎ_𝐻(𝑓()=1𝑡,𝑥)𝑥−𝑡𝑑𝑡𝐴_𝛼^𝑗_𝑟=0𝑗𝑟𝑒^{(𝑖/2)𝑥2cot𝛼(𝑗−𝑟)}(𝑥)_ℝ𝜕^𝑟𝜕𝑥^𝑟ℎ_𝐻()𝑓()=1𝑡,𝑥𝑥−𝑡𝑑𝑡𝐴_𝛼^𝑗_𝑟=0𝑗𝑟𝑒^{(𝑖/2)𝑥2cot𝛼(𝑗−𝑟)}(𝑥)𝐻_𝑟𝑓(𝑥)=1𝐴_𝛼^𝑗_𝑟=0𝑗𝑟𝑒^{(𝑖/2)𝑥2cot𝛼(𝑗−𝑟)}(𝑥)×^𝑟_𝑝=0𝑟𝑝(−1)^𝑝𝐻𝑓^{(𝑝)𝑟−𝑝}(𝑥). (3.24) By the proof of Lemma 3.4, (3.18) holds. Moreover, by (3.21) we obtain ℎ_𝐻(𝑡,𝑥)=𝐴_𝛼𝑒^{−(𝑖/2)𝑥2cot𝛼𝑀𝑁−1}_{𝑗=0𝑛∈ℤ}𝐻_{𝑗𝑘∈ℤ}𝛿_𝑘𝑁(𝑡+𝑛𝑁)𝜑_𝑗(𝑥−𝑡−𝑛𝑁)=𝑒^{−(𝑖/2)𝑥2cot𝛼𝑀𝑁−1}_𝑗=0^𝑗_{𝑛∈ℤ𝑟=0}𝑗𝑟×𝑒^{(𝑖/2)𝑥2cot𝛼(𝑗−𝑟)}𝐻_{𝑟𝑘∈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.

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].