Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2011, Article ID 303460, 9 pages
doi:10.1155/2011/303460
Research Article
A Note on Operator Sampling and Fractional
Fourier Transform
Qingyue Zhang
Department of Mathematics and LPMC, Nankai University, Tianjin 300071, China
Correspondence should be addressed to Qingyue Zhang, jczhangqingyue@163.com
Received 28 June 2011; Accepted 12 July 2011
Academic Editor: Blas M. Vinagre
Copyright q 2011 Qingyue Zhang. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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 by
Fα u F fx u α
R
fxKα u, xdx,
1.1
2
Mathematical Problems in Engineering
where α ∈ R, and the transform kernel is given by
⎧
2
2
⎪
Aα ei/2x u cot α−iux csc α
⎪
⎪
⎨
Kα u, x δx − u
⎪
⎪
⎪
⎩
δx u
if α /
kπ,
if α 2kπ,
1.2
if α 2k − 1π,
where δ· is Dirac distribution function over R, Aα inverse FRFT is the FRFT at angle −α, given by
1 − i cot α/2π, and k ∈ Z. The
−α
fx F Fα u R
Fα uKα u, xdu,
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 by
FPWΩ Ω Ω
f ∈ L2 R : supp F α f ⊆ − ,
.
2 2
1.4
For the sake of simplicity, when α π/2, FPWΩ is written as PWΩ .
In the following, we use the notation
AF BF,
F∈F
1.5
if there exist positive constants c and C such that cAF ≤ BF ≤ CAF for all objects F in
the set F.
Let H 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
2 f f, fn 2 ,
H
H
f ∈ H.
1.6
n∈Z
Let Λ {λk : k ∈ Z} ⊆ R with λk < λk1 . Λ is a set sampling for FPWΩ if
2
f 2 fλk 2 ,
L
f ∈ FPWΩ .
1.7
k∈Z
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 any t ∈ R and sin α > 0. {Kα λk t, · : k ∈ Z} is a frame for L2 −Ω/2, Ω/2 if
and only if {e−iλk ω : k ∈ Z} is a frame for L2 −Ω/2csc α, Ω/2csc α.
Mathematical Problems in Engineering
3
Remark 2.2. By Theorem 2.1, 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 Theorem 2.1, 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 that Λ is a set of sampling for FPWΩ . Then, for any F ∈ L2 −Ω/2, Ω/2, there
exists f ∈ FPWΩ such that F F α f. Since
F·e−i·t csc α , Kα λk t, ·
L2
2
2
F·e−i·t csc α , Aα ei/2λk t · cot α−i·λk tcsc α
2
2
2
e−i/22λk tt F·, Aα ei/2λk · cot α−i·λk csc α
L2
L2
2.1
e−i/22λk tt F·, Kα ·, λk e−i/22λk tt fλk ,
2
2
we have
2
2
fλk 2
F·e−i·tcscα 2 F2L2 f L2 L
k∈Z
2
F·e−i·tcscα , Kα λk t, · .
2.2
k∈Z
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
f 2 F2 2 |F·, Kα λk , ·|2
L
L
k∈Z
fλk 2 .
2.3
k∈Z
This completes our proof.
The following proposition gives a necessary and sufficient condition about Λ {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−iλk ω : 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−iλk ω : k ∈ Z} is a frame
for L2 −Ω/2csc α, Ω/2csc α.
4
Mathematical Problems in Engineering
Proof. Since
√
2
2
F α ft u 2πei/2u cot α F ftei/2t cot α u csc α,
2.4
where F denotes the FT operator, we have
2
ei/2· cot α FPWΩ PWΩ csc α .
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 HSL2 R consists of bounded linear operators
2
on L R which can be represented as integral operators of the form
Hfx R
κH x, tftdt
3.1
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 ∈ HSL2 R, then the operator norm of H is defined by HHS :
κH L2 hH L2 .
The Feichtinger algebra is defined by
S0 R f ∈ L2 R : Vg ft, ν ∈ L1 R2 ,
3.2
where Vg ft, ν f, Mν Tt g is the short-time Fourier transform of f with respect to the
2
Gaussian gx e−πx . An operator class O ⊆ HSL2 R is identifiable if all H ∈ O extend to
a domain containing a so-called identifier f ∈ S
0 R and
HHS Hf L2 ,
H ∈ O.
3.3
The operator class O ⊆ HSL2 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.
For T, Ω > 0, let
OPWT,Ω
OFPWT,Ω
Ω Ω
2
H ∈ HS L R : supp FhH t, · ⊆ 0, T × − ,
,
2 2
Ω Ω
2
α
.
H ∈ HS L R : supp F hH t, · ⊆ 0, T × − ,
2 2
3.4
Mathematical Problems in Engineering
5
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 ≤ T Ω ≤ 2π sin α, choose ϕ ∈ PW−2π/T −
Ω csc α/2, 2π/T − Ω csc α/2 with Fϕ 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 OFPWT ,Ω permits operator sampling as
HHS
√ T H δkT , H ∈ OFPWT ,Ω ,
k∈Z 2
3.5
L
and operator reconstruction is possible by means of the L2 -convergent series
hH t, x e−i/2x
×
H
2
cot α
rtT
2
e−i/2tnT cot α
n∈Z
δkT
3.6
t nT ϕx − t − nT .
k∈Z
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 ≤ T Ω ≤ 2π, choose ϕ ∈
PW−2π/T − Ω/2, 2π/T − Ω/2 with Fϕ 1 on −Ω/2, Ω/2 and r ∈ L∞ R with
supp r ⊆ −T T , T and r = 1 on 0, T . Then OPWT ,Ω permits operator sampling as
HHS
√ T H δkT ,
k∈Z 2
H ∈ OPWT ,Ω ,
3.7
L
and operator reconstruction is possible by means of the L2 -convergent series
hH t, x rtT
H
n∈Z
δkT
t nT ϕx − t − nT .
3.8
k∈Z
Proposition 3.3 see 11, Lemma 1. Assume a signal ft ∈ FPWΩ . Let
gt Ω/2
−Ω/2
2
F α ft ue−i/2u cot αiut csc α du.
3.9
Then gt ∈ PWΩ csc α .
Proof of Theorem 3.1. Due to 2.4, we have
ft, x ∈ OFPWT ,Ω ⇐⇒ ei/2x
2
cot α
ft, x ∈ OP WT ,Ω csc α .
3.10
6
Mathematical Problems in Engineering
By the proof of Proposition 3.2,
√ T H δkT ,
k∈Z 2
HHS
H ∈ OFPWT ,Ω .
3.11
L
Applying Proposition 3.3, we obtain
hH t, x Aα e−i/2x
where hH t, x Ω/2
2
cot α
3.12
hH t, x,
F α hH t, ·ue−i/2u cot αiux csc α du and hH t, x ∈ OPWT ,Ω csc α. Let
2
−Ω/2
Hfx R
hH t, xfx − t.
3.13
Using Proposition 3.2 again,
hH t, x rtT
H
n∈Z
δkT
t nT ϕx − t − nT .
3.14
k∈Z
Therewith,
hH t, x Aα e
−i/2x2 cot α
rtT
H
n∈Z
Aα e−i/2x
2
cot α
rtT
δkT
t nT ϕx − t − nT k∈Z
hH t, t nT ϕx − t − nT n∈Z
e−i/2x
2
cot α
rtT
2
e−i/2tnT cot α hH t, t nT ϕx − t − nT n∈Z
e−i/2x
×
H
2
cot α
rtT
2
e−i/2tnT cot α
n∈Z
δkT
3.15
t nT ϕx − t − nT .
k∈Z
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,
H2HS j
MN−1
j0
j
−1r
r0 r
j−r 2
r
,
H δkN
k∈Z
H ∈ OPWN,M2π .
3.16
Mathematical Problems in Engineering
7
and operator reconstruction is possible by means of the L2 -convergent series
hH t, x MN−1
Hj
j0 n∈Z
δkN t nNϕj x − t − nN,
3.17
k∈Z
where {ϕj x − t − nN : 0 ≤ j ≤ MN, n ∈ N} is a Riesz basis for PW2πM for each fixed t ∈ 0, N,
j j−r
and Hj fx r0 rj −1r Hf 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,
H2HS j j
MN−1
j0
r0 r
ei/2x
2
r
j−r r
cot α
−1p H
p
p0
r
δkN
k∈Z
r−p 2
,
3.18
H ∈ OFPWN,M2π sin α .
and operator reconstruction is possible by means of the L2 -convergent series
hH t, x e
−i/2x2 cot α
j
j
MN−1
j0 n∈Z r0
×
ei/2x
2
cot α
j−r
Hr
r
3.19
δkN t nNϕj x − t − nN,
k∈Z
where {ϕj x − t − nN : 0 ≤ j ≤ MN, n ∈ N} is a Riesz basis for PWM2π for each fixed t ∈ 0, N
j j−r
and Hj fx r0 rj −1r Hf r x.
Proof. Due to 2.4, we have
ft, x ∈ OFPWN,M2π sin α ⇐⇒ ei/2x
2
cot α
ft, x ∈ OPWN,M2π .
3.20
By Proposition 3.3, we obtain
hH t, x Aα e−i/2x
where hH t, x Ω/2
−Ω/2
2
cot α
hH t, x,
3.21
F α hH t, ·ue−i/2u cot αiux csc α du. Put
2
Hfx R
hH t, xfx − tdt.
3.22
8
Mathematical Problems in Engineering
By Lemma 3.4, we have
hH t, x MN−1
Hj
j0 n∈Z
δkN t nNϕj x − t − nN,
3.23
k∈Z
where
H j fx j
j
r
r0
r
−1 Hf
r
j−r
x R
∂j
hH t, xfx − tdt
∂xj
∂j i/2x2 cot α
fx − tdt
e
h
x
t,
H
j
R ∂x
j
1 j i/2x2 cot α j−r
∂r
e
h t, xfx − tdt
x
r H
Aα r0 r
R ∂x
j
1 j i/2x2 cot α j−r
e
xHr fx
Aα r0 r
j
1 j i/2x2 cot α j−r
e
x
Aα r0 r
⎛ ⎞
r
r−p
r
⎠x.
×⎝
−1p Hf p
p
p0
1
Aα
3.24
By the proof of Lemma 3.4, 3.18 holds. Moreover, by 3.21 we obtain
hH t, x Aα e
−i/2x2 cotα
MN−1
Hj
j0 n∈Z
e
−i/2x2 cotα
j
MN−1
×
e
i/2x2 cotα
j−r
δkN t nNϕj x − t − nN
k∈Z
j
j0 n∈Z r0
Hr
3.25
r
δkN t nNϕj x − t − nN.
k∈Z
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
Mathematical Problems in Engineering
9
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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