Gen. Math. Notes, Vol. 8, No. 2, February 2012, pp.18-27
c
ISSN 2219-7184; Copyright ICSRS
Publication, 2012
www.i-csrs.org
Available free online at http://www.geman.in
Optical Fresnel-Wavelet Transforms for Certain
Space of Generalized Functions
S.K.Q. Al-Omari
Department of Applied Sciences, Faculty of Engineering Technology
Al-Balqa Applied University, Amman 11134, Jordan
E-mail: s.k.q.alomari@fet.edu.jo
(Received: 2-11-11/ Accepted: 14-2-12)
Abstract
A theory of the diffraction Fresnel transform is extended to certain spaces
of Schwartz distributions. The diffraction Fresnel transform is obtained as a
continuous function in the space of Boehmians. Convergence with respect to δ
and ∆ convergences is shown to be well defined .
Keywords: Fresnel Transform; Wavelet transform; Distribution space;
Boehmian space.
1
Introduction
Integral transforms play an important role in various fields of science. In optics,
several integral transforms are of great importance. Some of these transforms
are: the Fresnel transform [10, 12, 25, 26]; the fractional Fourier transform [5,
6, 11, 13, 18]; the linear canonical transform [22, 23]; the wavelet transform [20,
21]; the diffraction Fresnel transform [27,28] and, many others. The wavelet
transform is described in [20, 21] as
Ωf (µ, λ) =
√1
f
µR
(x) ψ ∗
x−λ
µ
dx
(1.1)
where ψ (x) is named as mother wavelet satisfying R dxψ (x) = 0. The parameters λ and µ are, respectively, the translate and dilate of w, whereas, w∗ is the
conjugate of w. The optical diffraction transform is described by the Fresnel
19
Optical Fresnel-Wavelet Transforms for Certain...
integration [27, 28]
F (x2 ) =
√ 1
f
2πiγ1 R
(x1 ) Kx2 (x1 ) dx1
(1.2)
where K (α1 , γ1 , γ2 , α2 ; x1 , x2 ) = exp 2γi 1 (α1 x21 − 2x1 x2 + α2 x22 ) is the transform kernel whose parameters α1 , γ1 , γ2 , α2 represent a ray transfer Matrix M,
in an optical system, with α1 α2 − γ1 γ2 = 1.
We consider the combined optical transform obtained jointly from (1.1)
and (1.2), named as the Fresnel-wavelet transform [10, Equ. (36)]
Fw (λ, µ, x2 ) =
√ 1
K
2πiγ1 R
(α1 , γ1 , γ2 , α2 ; x1 , x2 , λ, µ) f (x1 ) dx.
where
2
K (α1 , γ1 , γ2 , α2 ; x1 , x2 , λ, µ) = exp 2γi 1 α1 (xµ12−λ) −
2x2 (x1 −λ)
µ
+ α2 x22
(1.3)
is the transform kernel.
Parameters: α1 , γ1 , γ2 , and α2 appearing in the above expression are elements of a 2 × 2 matrix with unit determinant. Since the general single-mode
squeezing operator F in the generalized Fresnel transform is in wave optics,
applications of F is a faithful representation in the Fresnel-wavelet transform
[10]. Hence, the combined Fresnel-wavelet transform can be more conveniently
studied by the general single-more squeezed operation.
In the literature, it has not yet been reported that the Fresnel-wavelet
transform is extended to a space of generalized functions . Thus, we, in this
article, aim at extending the Fresnel-wavelet transform to certain generalized
function space ( Boehmian space). Such extension is mainly related to the fact
that the optical Fresnel-wavelet transform of a good function is certainly a C ∞
function.
We spread the article into five sections: In Section 2, we introduce the
notion of Boehmian spaces. In Section 3, we consider the Boehmian space
B? from [4]. Section 4 is devoted for a general construction of the space
BF w , where images of the extended Fresnel-wavelet transform lie. In the last
section, we establish that the optical Fresnel-wavelet transform of an arbitrary
Boehmian in B? is another Boehmian in BF w . Moreover, we discuss linearity
and continuity conditions with respect to certain types of convergence.
Let ε (R+ ) be the test function space of all C ∞ functions of arbitrary sup0
ports and ε (R+ ) be its strong duals of distributions of compact supports.
The kernel function K (α1 , γ1 , γ2 , α2 ; x1 , x2 , λ, µ) of the Fresnel-wavelet transform is clearly in ε (R+ ) . This leads to define the distributional transform on
the dual of distributions of compact support by the relation Fw (λ, µ, x2 ) =
0
√ 1
hf (x1 ) , K (α1 , γ1 , γ2 , α2 ; x1 , x2 , λ, µ)i , for every f ∈ ε (R+ ).
2πiγ1
20
2
S.K.Q. Al-Omari
General Boehmian Spaces
Let G be a linear space and H be a subspace of G. Assume to each pair of
elements f, g ∈ G and ψ, φ ∈ H, is assigned the product f • g such that:
φ • ψ ∈ H and φ • ψ = ψ • φ, ∀φ, ψ ∈ H, (f • φ) • ψ = f • (φ • ψ) ,and
(f + g) • φ = f • φ + g • φ , k (f • φ) = (kf ) • φ, k ∈ C. A family of sequences
∆ from H, is said to be delta sequence if for each f, g ∈ G, (ψn ) , (δn ) ∈ ∆,
the following should satisfy: f • δn = g • δn (n = 1, 2, ...) , implies f = g, and
(φn • ψn ) ∈ ∆. Let O be a class of pair of sequences
O = ((fn ) , (φn )) : (fn ) ⊆ GN , (φn ) ∈ ∆ ,
for each n ∈ N. An element ((fn ) , (φn )) ∈ O is said to be a quotient of sequences, denoted by φfnn if fi •φj = fj •φi , ∀i, j ∈ N. Two quotients of sequences
fn
and ψgnn are equivalent, φfnn ∼ ψgnn , if fi • ψj = gj • φi , ∀i, j ∈ N. The relation ∼
φn
is an equivalent relation on O and hence, splits hO into
i equivalence classes. The
fn
fn
equivalence class containing φn is denoted by φn .These equivalence classes
are called Boehmians and the space of all Boehmians is denoted by B• . The
sum hof two
byh a iscalarh is defined
in a natural
i Boehmians
h i
h and multiplication
i
i
(fn •ψn )+(gn •φn )
fn
gn
fn
fn
way φn + ψn =
and α φn = α φn , α ∈ C. The opφn •ψn
h
i
h i h i
gn
fn •gn
fn
eration • and the differentiation are defined by φn • ψn = φn •ψn and
h i
h α i
Dα φfnn = Dφnfn . The relationship between the notion of convergence and
the product • are given by:
1−If fn → f as n → ∞ in G and, φ ∈ H is any fixed element, thenfn •
φ → f • φ, as n → ∞ in G.
2−If fn → f as n → ∞ in G and (δn ) ∈ ∆, then fn • δn → f as n → ∞
in G. In B• two types of convergence:
δ
δ−convergence : Let (βn ) ∈ B• then βn → β, if there is (δn ) ∈ ∆,
(βn • δn ) , (β • δn ) ∈ G, ∀k, n ∈ N,and (βn • δk ) → (β • δk ) as n → ∞,in G, ∀
k ∈ N.
∆
∆−convergence : (βn ) in B• is ∆−convergent to β in B• , βn → β, if
there is (δn ) ∈ ∆ such that (βn − β) • δn ∈ G, ∀n ∈ N, and (βn − β) • δn → 0
as n → ∞ in G. For further analysis, see [1-4, 8, 14, 15, 17].
3
The Boehmian Space B?
Let f and g be C ∞ functions , over R+ . Then the convolution between f and
g is defined by [4, Equ.3.2]
(f . g) (x) =R+ f xy −1 φ (y) y −1 dy,
(3.1)
Optical Fresnel-Wavelet Transforms for Certain...
21
where x is a non-negative real number.
In the rest of investigations, it is more convenient to use the noation ?
instead of the used one, .. Further, we retain likewise notations and the results
established in [4].
Let D = D (R+ ), be the Schwartz’ space of all C ∞ complex-valued functions which are compactly supported in R+ . Then, we recall the following
definition [4]
Definition 3.1. Let S = φ ∈ D (R+ ) : φ ≥ 0 and R+ φ = 1 and ∆ be the
set of all delta sequences φn , n = 0, 1, 2, ..., from S, such that supp φn → 0 as
n → ∞. Then, (φn ) ∈ ∆ if and only if (φn ) ∈ D (R+ ) , and
∆1 R+ φn = 1, ∀n ∈ N;
∆2 φn ≥ 0, ∀n ∈ N;
∆3 inf { > 0 : suppφn ⊆ (0, )} → 0,as n → ∞.
The following are proved in [4]
Lemma 3.2. Let f ∈ C ∞ (R+ ) and φ ∈ S, then f ? φ ∈ C ∞ (R+ ) .
Lemma 3.3. Let f, g ∈ C ∞ (R+ ) , φ, ψ ∈ S and, α ∈ C( The set of complex
numbers ) . Then, the following are true
(1) (f + g) ? φ = f ? φ + g ? φ.
(2) (αf ? φ) = α (f ? φ) .
(3) φ ? ψ = ψ ? φ.
(4) f ? (φ ? ψ) = (f ? φ) ? ψ.
Theorem 3.4. If lim fn = f , in C ∞ (R+ ) , and φ ∈ S , then
n→∞
lim fn ? φ = f ? φ in C ∞ (R+ ) .
n→∞
Lemma 3.5. Let fn → f , in C ∞ (R+ ) , and (δn ) ∈ ∆. Then,fn ? δn → f
in C ∞ (R+ ) .
Theorem 3.6 . Given (φn ) , (ψn ) ∈ ∆. Then, (φn ? ψn ) ∈ ∆..
After this sequence of results, the desired Boehmian space B? was constructed in [4].
In B? , it is needful to have the following definition:
h i h i
Definition 3.7. Let fδnn , φgnn ∈ B? . Then, the convolution of two
Boehmians is defined as
h i h i h
i
fn
gn
fn ?gn
?
=
, for all n ∈ N.
(3.2)
δn
φn
δn ?φn
Equ.(3.2) is well-defined by Theorem 3.6 and Lemma 3.2.
22
S.K.Q. Al-Omari
Differentiation is defined by
D
α
h
fn
φn
i
=
h
D α fn
φn
i
.
Addition and scalar multiplication is defined in B? as
h i h i h
i
h i h
i
(fn ?ψn )+(gn ?φn )
fn
gn
fn
fn
+ ψn =
andα φn = α φn , α ∈ C.
φn
φn ?ψn
4
The Boehmian Space BFw
3
3
Let S R+
, be the space of rapidly decreasing functions on
R+ = R+ ×R∞+ ×R+
3
[19, 7]. Then the Fresnel-wavelet
transform of f ∈ S R+ is indeed a C (R+ )
∞
3
(R+ ) .
function. Let f ∈ S R+ and ψ ∈ C
3
We define a mapping ⊗ : S R+ → C ∞ (R+ ) by
Z
(f ⊗ ψ) (λ, µ, x2 ) =
f λt−1 , µt−1 , x2 ψ (t) dt.
(4.1)
R+
Following theorem is very needful
3
Lemma 4.1. Let f ∈ S R+
and ψ ∈ C ∞ (R+ ) then
3
f ⊗ ψ ∈ S R+
.
Proof. To show f ⊗ ψ ∈ S, we establish the following three relations
Dλ (f ⊗ ψ) (λ, µ, x2 ) = (Dλ f ⊗ ψ) (λ, µ, x2 ) ;
(4.2)
Dµ (f ⊗ ψ) (λ, µ, x2 ) = (Dλ f ⊗ ψ) (λ, µ, x2 ) ;
(4.3)
Dx2 (f ⊗ ψ) (λ, µ, x2 ) = (Dx2 f ⊗ ψ) (λ, µ, x2 ) .
(4.4)
and
To establish (4.2), let µ0 , x20 > 0 be fixed and, λ0 vary over R+ then
Dλ (f ⊗ ψ) (λ0 , µ0 , x20 ) =
f (λt−1 ,µ0 t−1 ,x20 )−f (λ0 t−1 ,µ0 t−1 ,x20 )
ψ (t) dt
lim
λ−λ0
λ→λ0 R+
=R+ Dλ f (λ0 t−1 , µ0 t−1 , x20 ) ψ (t) dt
= (Dλ f ⊗ ψ) (λ0 , µ0 , x20 ) .
Thus,
Dλ (f ⊗ ψ) (λ0 , µ0 , x20 ) = (Dλ f ⊗ ψ) (λ0 , µ0 , x20 ) .
23
Optical Fresnel-Wavelet Transforms for Certain...
Proof of (4.3) and (4.4) is analogous. Induction on the partial differention
with respect to λ,µ and x2 yields
Dλk (f ⊗ ψ) = Dλk f ⊗ ψ, Dµk (f ⊗ ψ) = Dµk ⊗ ψ and Dxk2 (f ⊗ ψ) = Dxk2 f ⊗ ψ.
(5.5)
Hence, using the topology of S we have
kf ∗ ψkS ≤ kψkL1 kf kS
3
Lemma 4.2 f ⊗ ψn → f for every f ∈ S R+
and (ψn ) ∈ ∆.
(1)
Proof. Using (4.2) − (4.4) , mean value theorem and ∆3 we write
i k
λ Dλ (f ⊗ ψn − f ) (λ, µ, x2 ) = λi Dλk f ⊗ ψn − Dλk f (λ, µ, x2 ) .
Hence,
(4.1) , we get
i using
λ Dk (f ⊗ ψn − f ) (λ, µ, x2 ) ≤
λ
i k
−1
−1
R+ λ Dλ (f (λt , µt , κ2 ) − f (λ, µ, x2 )) ψ (t) dt.
Hence the above expression approaches 0 as n → ∞.
It
icank be similarly proved that
µ Dµ (f ⊗ ψn − f ) (λ, µ, x2 ) and xi2 Dxk (f ⊗ ψn − f ) (λ, µ, x2 ) approach
2
0 as n → ∞.
This completes the proof of the lemma.
3
and ψ ∈ C ∞ (R+ ) .
Lemma 4.3 fn ⊗ ψ → f ⊗ ψ for every fn , f ∈ S R+
Proof. Employing (4.1)-(4.4) the lemma can easily be established in a
manner similar to that of above Lemma . The Boehmian space BF w (S, ⊗, ∆)
is therefore established. Operations such as addition, scalar multiplication,
Differentiation and the operation ⊗ between two Boehmians in BF w can be
defined similarly as done in the previous section.
5
Fresnel-Wavelet Transform of Boehmians
Following is lemma suggesting a new definition for the Fresnel-wavelet transform of a Boehmian in the space B? .
3
Lemma 5.1 Given f ∈ S R+
and ψ ∈ C ∞ (R+ ) then
Fw (f ? ψ) (λ, µ, x2 ) = f ⊗ Fw ψ,
Proof. The Fresnel-Wavelet transform is written in the form
Z
0
Fw (f (x1 )) (λ, µ, x2 ) =
f (x1 ) Kλ,µ,x2 (x1 ) dx1
R+
(5.1)
24
S.K.Q. Al-Omari
0
where Kλ,µ,x2 (x1 ) = K (α1 , γ1 , γ2 , α2 ; x1 , x2 , λ, µ) and
(x1 −λ)2
(x1 −λ)2
2
K (α1 , γ1 , γ2 , α2 ; x1 , x2 , λ, µ) = exp α1 µ − 2x2 µ + α2 x2 .
Hence
Z
0
Fw (f ? ψ) (λ, µ, x2 ) =
(f ? ψ) (x1 ) Kλ,µ,x2 (x1 ) (x1 ) dx1
R+
Z Z
0
−1
−1
f x1 y
ψ (y) y dy Kλ,µ,x2 (x1 ) dx1 .
=
R+
R+
The substitution x1 = yt implies
Z
Z
f (t)
Fw (f ? ψ) (λ, µ, x2 ) =
R+
R+
0
ψ (y) Kλt−1 ,µt−1 ,x2 (y) dy dt
= (Fw ψ ⊗ f ) (λ, µ, x2 ) .
This completes the proof. Hence, we define the Fresnel-wavelet transform
of a Boehmian in B? as
h i h
i
fn
Fw fn
S ∂ n = ∂n .
(5.2)
3
in the space BF w S R+
, ⊗, ∆ .
The definition, in (5.2) , is well defined. For, if fδnn ∼ fδnn in B? then
fn ? δm = gm ? δm . Applying the Fresnel-wavelet transform and
h Theorem
i h 5.1i
Fw fn
Fw fn
Fw gn
imply Fw fn ⊗δm = Fw gm ⊗δn ,. Hence δn ∼ δn . Therefore δn = Fwδngn
in BF w .
Theorem 5.2. The S : B? → BF w is linear.
Proof. is obvious.
Theorem 5.3: The S : B? → BF w is continuous with respect to ∆ convergence.
h
i
∆
fv ?δi
Proof. If βv → β in B? then (βv → β) ? δv = δi for some δi ∈ ∆, fn ∈
∞
3
C (R+ ) and fv → 0 as v → ∞. Thus Fw fv → 0 in S R+
since fv → 0 as
∆
v → ∞. Hence we conclude Fw βv → Fw β as v → ∞. This completes the proof
of te theorem.
Theorem 5.4. S : B? → BF w is continuous with respect to the δ convergence.
25
Optical Fresnel-Wavelet Transforms for Certain...
δ
Proof. Let βv → β as v → ∞ in B? then using [15] there can be found
fv,,j , fj such that
fv,j → fj , as v → ∞,
(5.3)
where
h
fv,j
δj
i
= βv and
h i
fj
δj
=β
Applying the Fresnel-wavelet transform on (5.3) we get
Fw fv,j → Fw fj as v → ∞.
Thus
h
Fw fv,j
δj
i
→
h
Fw fj
δj
i
.
Hence the theorem.
References
[1] S.K.Q. Al-Omari, D. Loonker, P.K. Banerji and S.L. Kalla, Fourier
sine(cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces, Integ. Trans. Spl. Funct. 19(6) (2008),
453–462.
[2] S.K.Q. Al-Omari, The generalized Stieltjes and Fourier transforms of certain spaces of generalized functions, Jord. J. Math. Stat., 2(2) (2009),
55-66.
[3] S.K.Q. Al-Omari , On the distributional Mellin transformation and its
extension to Boehmian spaces, Int. J. Contemp. Math. Sciences, 6(17)
(2011), 801-810.
[4] S.K.Q. Al-Omari , A Mellin transform for a space of Lebesgue integrable
Boehmians, Int. J. Contemp. Math. Sciences, (32) (2011), 1597-1606.
[5] A.W. Lohmann, Image rotation, Wigner rotation, and the fractional
Fourier transform, J. Opt. Soc. Am. A, 10(1993), 2181-2186.
[6] A.C. McBride and F.H. Kerr, On Namias’s fractional Fourier transforms
IMA J. Appl. Math., 39(1987), 159-175.
[7] P.K. Banerji, S.K. Alomari and L. Debnath, Tempered distributional Fourier sine(cosine) transform, Integ. Trans. Spl. Funct., 17(11)
(2006),759-768.
[8] T.K. Boehme, The support of Mikusinski operators, Tran.Amer. Math.
Soc., 176(1973), 319-334.
26
S.K.Q. Al-Omari
[9] D. Mendlovic and H.M. Ozakatas, Fractional Fourier transforms and their
optical implementation:I, J. Opt. Soc. Am. A, 10(1993), 1875–1881.
[10] Hong-Yi Fan and Hai-Liang Lu, Wave-function transformations by generalSU(1,1) single-mode squeezing and analogy to Fresnel transformations
in wave optics, Optics Communications, 258(2006), 51-58.
[11] H.M. Ozakatas and D. Mendlovic,Fractional Fourier transforms and their
optical implementation. II, J. Opt. Soc. Am. A, 10(1993) 2522-2531.
[12] L. Mertz, Transformations in Optics, Wiley, New York, (1965).
[13] L.M. Bernardo and O.D. Soaares, Fractional Fourier transforms and optical systems, Opt.Commun., 110(1994), 517–522.
[14] P. Mikusinski, Fourier transform for integrable Boehmians, Rocky Mountain J. Math., 17(3) (1987), 577-582.
[15] P. Mikusinski, Convergence of Boehmians, Japan, J. Math, 9(1)(1983),
169-179. .
[16] R.S. Pathak, Integral Transforms of Generalized Functions and Their Applications, Gordon and Breach Science Publishers, Australia, Canada, India, Japan, (1997).
[17] R. Roopkumar, Mellin transform for Boehmians, Bull. Institute of Math.,
Academica Sinica, 4(1) (2009), 75-96.
[18] V. Namias, The fractional order Fourier transform and its application to
quantum mechanics, J. Inst. Maths. Appl., 25(1980), 241-265.
[19] A.H. Zemanian, Generalized Integral Transformation, Dover Publications,
Inc., New York (1987), First published by Interscience Publishers, New
York (1968).
[20] M. Holschneider, Wavelets, An Analysis Tool, Oxford, Clarendon Press,
(1995).
[21] M. Holschneider, Wavelet analysis on the circle, J. Math. Phys., 31(1)
(1990), 39-44.
[22] R.G. Campos and J. Figueroa, A fast algorithm for the linear canonical
transform, Signal Processing, 91(2011), 1444-1447.
[23] M. Moshinsky and C. Quense, Linear canonical transformations and their
unitary representations, J. Math. Phys., 12(1971),1772-1783.
Optical Fresnel-Wavelet Transforms for Certain...
27
[24] B. Rong Wu, S. Ito,Y. Kamimura, and Y.Yamada, , Zero-filling Technique in Fresnel Transform Image Reconstruction for MR Image Denoising, Progress In Electromagnetics Research Symposium 2005, Hangzhou,
China, 22-26.
[25] J.W. Goodman, Introductionto Fourier Optics, McGraw-Hill, New York,
(1972).
[26] A.W. Lohmann, Image rotation, Wigner rotation, and the fractional
Fourier transform, J. Opt. Soc. Am. a, 10(1993), 2181-2186.
[27] H.M. Ozakatas and D. Mendlovic, Fractional Fourier transforms and their
optical implementation:II, J. Opt. Soc. A, 10(1993), 2522–2531.