Quaternionic fractional Fourier transform for Boehmians
UDC 517.9 We construct a Boehmian space of quaternion valued functions using the quaternionic fractional convolution. Applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the context of Boehmians and its properties are established.
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Institute of Mathematics, NAS of Ukraine
2020
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| author | Roopkumar, R. Roopkumar, R. |
| author_facet | Roopkumar, R. Roopkumar, R. |
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| description | UDC 517.9
We construct a Boehmian space of quaternion valued functions using the quaternionic fractional convolution. Applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the context of Boehmians and its properties are established. |
| doi_str_mv | 10.37863/umzh.v72i6.649 |
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DOI: 10.37863/umzh.v72i6.649
UDC 517.9
R. Roopkumar (Central Univ. Tamil Nadu, Thiruvarur, India)
QUATERNIONIC FRACTIONAL FOURIER TRANSFORM FOR BOEHMIANS
КВАТЕРНIОННЕ ДРОБОВЕ ПЕРЕТВОРЕННЯ ФУР’Є ДЛЯ БЬОМIАНIВ
We construct a Boehmian space of quaternion valued functions using the quaternionic fractional convolution. Applying
the convolution theorem, the quaternionic fractional Fourier transform is extended to the context of Boehmians and its
properties are established.
За допомогою кватернiонної дробової згортки побудовано бьомiанiв простiр функцiй iз значеннями у кватернiонах.
Застосовуючи теорему про згортку, ми поширюємо кватернiонне дробове перетворення Фур’є на бьомiанiв простiр
та встановлюємо його властивостi.
1. Introduction. It is well-known that the classical Fourier transform on square integrable functions
is of order 4. To introduce a generalization of Fourier transform with fractional order, the fractional
Fourier transform was introduced by Namias [26] and the works on fractional Fourier transform have
been developed with different objectives on pure and applied mathematics. In particular, in view of
classical analysis, properties, applications and generalizations of the fractional Fourier transform are
discussed (see [7, 17, 20 – 22, 25, 26, 29, 30, 35, 39, 40]). Following the introduction of the Fourier
transform of quaternion valued functions.
The fractional Fourier transform is discussed on quaternion valued functions on \BbbR 2 and their
properties including inversion formula and Parseval’s identity are derived in [13, 38]. Recently the
fractional Fourier transform on quaternion valued functions on \BbbR is introduced in [32] and all of its
properties including the inversion formula, Parsevel’s identity, convolution and product theorems are
proved. As the convolution theorem for quaternionic fractional Fourier transform in [32] is quite
analogous to that of fractional Fourier transform of complex valued functions in [39], in this paper,
following the techniques employed in [40], we extend the quaternionic fractional Fourier transform to
a suitable Boehmian space of quaternion valued functions. It is obvious to observe that the fractional
Fourier transform on Boehmians of quaternion valued functions is simultaneously generalizing the
theory of fractional Fourier transform on quaternion valued L2-functions [32] and the fractional
Fourier transform on Boehmians of complex valued functions [40].
To facilitate the reader, we recall the division algebra of quaternions, Lp-spaces of quaternion
valued functions, the theory of fractional Fourier transform in Section 2. The general theory of
Boehmians and the construction of two suitable Boehmian spaces are discussed in Section 3. The
last section is devoted to the definition and properties of the extended quaternionic fractional Fourier
transform.
2. Preliminaries. As usual, we denote by \BbbR and \BbbC , the sets of all real and complex numbers,
respectively. The set of all quaternions is defined by is defined by \BbbH =
\bigl\{
q1+jq2 : q1, q2 \in \BbbC
\bigr\}
, where
the j is an imaginary number other than the imaginary complex number i satisfying the properties
j2 = - 1 and jz = zj for all z \in \BbbC , where z is the complex conjugate of z. The addition and the
multiplication of two quaternions are explicitly given by
c\bigcirc R. ROOPKUMAR, 2020
812 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
QUATERNIONIC FRACTIONAL FOURIER TRANSFORM FOR BOEHMIANS 813
(p1 + jp2) + (q1 + jq2) = (p1 + q1) + j(p2 + q2),
(p1 + jp2)(q1 + jq2) = (p1q1 - \=p2q2) + j(\=p1q2 + p2q1).
It is well-known that \BbbH is a skew-field but not a field with respect to the addition and multiplication
defined above. The conjugate and absolute value of a quaternion q = q1 + jq2 \in \BbbH are given by
qc = \=q1 - jq2 and | q| =
\sqrt{}
| q1| 2 + | q2| 2, respectively, where | qk| is the absolute value of the complex
number qk. The conjugate operator and absolute value operator on quaternions satisfy the following
properties:
(p+ q)c = pc + qc, (pq)c = qcpc, (qc)c = q \forall p, q \in \BbbH ,
and
qqc = | q| 2, | q| = | qc| , | p+ q| \leq | p| + | q| , | pq| = | p| | q| \forall p, q \in \BbbH .
If f = f1 + jf2 is a quaternion valued function, we define
\v f = \v f1 + j \v f2 and \~f(x) = \=f1 - j \v f2(x),
where \=f1(x) is the complex conjugate of f1(x) and \v f1(x) = f1( - x) for all x \in \BbbR . Let Lp(\BbbR ,\BbbH ) =
=
\bigl\{
f1 + jf2 : f1, f2 \in Lp(\BbbR ,\BbbC )
\bigr\}
, where Lp(\BbbR ,\BbbC ) is the Banach space of all complex valued
functions \varphi , satisfying
\int
\BbbR
| \varphi (x)| p dx < +\infty , p = 1, 2. As in the case of complex valued p-
integrable functions, the point-wise addition and point-wise scalar multiplication on Lp(\BbbR ,\BbbH ) are
defined by
(f + g)(x) = f(x) + g(x) and (qf)(x) = qf(x) \forall x \in \BbbR .
As \BbbH is not a field, we can say that Lp(\BbbR ,\BbbH ) is a left \BbbH -module, and it is equipped with the norm
defined as follows:
\| f\| p = \| f1 + jf2\| p =
\left( \int
\BbbR
| f1(x)| p + | f2(x)| pdx
\right) 1
p
, p = 1, 2.
It is obvious that
fn \rightarrow 0 in Lp(\BbbR ,\BbbH ) iff fn,1 \rightarrow 0, fn,2 \rightarrow 0 in Lp(\BbbR ,\BbbC ), (1)
where fn = fn,1 + jfn,2 for all n \in \BbbN , and, hence, Lp(\BbbR ,\BbbH ) is also complete with respect to the
metric induced by the norm. Furthermore, the norm on L2(\BbbR ,\BbbH ) is also obtained as \| f\| 2 =
\sqrt{}
\langle f, f\rangle ,
where
\langle f, g\rangle =
\int
\BbbR
f(x)(g(x))cdx \forall f, g \in L2(\BbbR ,\BbbH ).
Therefore, we say that L2(\BbbR ,\BbbH ) is an example of a left \BbbH -Hilbert space, as per the following
definition.
Definition 1. A nonempty set H is called a quaternion left Hilbert space if it is a left \BbbH -module
and there exists a function \langle \cdot , \cdot \rangle : H \times H \rightarrow \BbbH with the following properties:
(1) \langle u, v\rangle =
\bigl(
\langle v, u\rangle
\bigr) c \forall u, v \in H;
(2) \langle pu+ qv, w\rangle = p\langle u,w\rangle + q\langle v, w\rangle \forall p, q \in \BbbH , \forall u, v, w \in H;
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
814 R. ROOPKUMAR
(3) for each u \in H, \langle u, u\rangle is real and nonnegative;
(4) for u \in H, \langle u, u\rangle = 0 iff u = 0;
(5) every Cauchy sequence in the normed space
\bigl(
H, \| \cdot \| 2
\bigr)
converges in H, where \| u\| 2 =
=
\sqrt{}
\langle u, u\rangle for all u \in H.
Upto the author’s knowledge, the first paper in English, mentioning the definition of quaternionic
Hilbert space is [37].
In 1980’s the qunaternionic Fourier transform was introduced and applied independently by Som-
men, Earnest et al., and Delsuc. Then another modified version of the quaternioninc Fourier transform
was introduced by T. Ell in 1992, which is commonly used by many researchers in this field of re-
search. For more details on the history of quaternionic Fourier transform, refer the reader to [9].
After this various research papers on quternionic Fourier transforms were published. To mention a
few works, we refer to [10, 15 – 17].
Next, we recall the fractional Fourier transform on complex valued functions and quaternionic
fractional Fourier transform, respectively, from [6, 26, 32]. The fractional Fourier transform \scrF \alpha (f)
of a suitable complex valued function f on \BbbR is defined by
\scrF \alpha (f)(u) =
\int
\BbbR
f(t)K\alpha (t, u) dt \forall u \in \BbbR ,
where
K\alpha (t, u) =
\left\{
\sqrt{}
c(\alpha )
2\pi
eia(\alpha )(t
2+u2) - iutb(\alpha ), \alpha \not \in \pi \BbbZ ,
\delta (t - u), \alpha \in 2\pi \BbbZ ,
\delta (t+ u), \alpha + \pi \in 2\pi \BbbZ ,
a(\alpha ) =
\mathrm{c}\mathrm{o}\mathrm{t}\alpha
2
, b(\alpha ) = \mathrm{s}\mathrm{e}\mathrm{c}\alpha and c(\alpha ) =
\surd
1 - i \mathrm{c}\mathrm{o}\mathrm{t}\alpha and \alpha \in \BbbR \setminus \pi \BbbZ . Following the definition of
quaternionic Fourier transform in [14], the fractional Fourier transform of quaternion valued function
f \in L1(\BbbR ,\BbbH ) is defined in [32] by
\scrF \alpha (f) = \scrF \alpha (f1) + j\scrF \alpha (f2), where f = f1 + jf2. (2)
Then F\alpha is extended to L2(\BbbR ,\BbbH ) as a Hilbert space isomorphism, as in the case of classical Fourier
transform. Further, we also have F\alpha \circ F\beta = F\alpha +\beta on L2(\BbbR ,\BbbH ) and F\alpha is the identity operator if
\alpha = 0. Thus, F - 1
\alpha = F - \alpha .
Definition 2. For f \in L2(\BbbR ,\BbbH ) and g \in L1(\BbbR ,\BbbH ), define
(f \star \alpha g)(x) =
\bigl(
f1 \star g1 - \scrF - 2\alpha f2 \star g2
\bigr)
+ j
\bigl(
\scrF - 2\alpha f1 \star g2 + f2 \star g1
\bigr)
,
where \star is the convolution defined in [40] (Definition 1) as follows:
(f \star g)(x) =
c(\alpha )\surd
2\pi
eia(\alpha )x
2
( \~f \ast \~g)(x) and \~f(x) = eia(\alpha )x
2
f(x). (3)
Theorem 1 (convolution theorem, [32]). For f \in L2(\BbbR ,\BbbH ) and g \in L1(\BbbR ,\BbbH ), we have
\scrF \alpha (f \star \alpha g)(u) = \scrF \alpha (f)(u)\scrF \alpha (g)(u)e
- ia(\alpha )u2
for all u \in \BbbR .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
QUATERNIONIC FRACTIONAL FOURIER TRANSFORM FOR BOEHMIANS 815
3. Quaternionic fractional fourier transform on Boehmians. A class of generalized functions,
called Boehmian space with two convergences, was introduced by P. Mikusiński [23]. Since most of
the Boehmian spaces of functions are larger than dual spaces of suitable function spaces, a number of
papers on extensions of integral transforms in the context of Boehmian spaces have been published.
To mention a few recent papers, we refer to [8, 11, 12, 19, 27, 28, 31, 33, 34, 36]. A complete list of
papers on Boehmians is available in the following link http://mikusinski.cos.ucf.edu/boehmians.pdf.
In particular, Boehmians of quaternion valued functions were constructed, for the purpose of
extending quaternionic integral transforms, for example, quaternionic wavelet transform [1], quater-
nionic ridgelet transform [2], quaternionic Gabor transform [3], quaternionic Stockwell transform [4]
and quaternionic curvelet transform [5]. In this line, we provide an extension of the quaternionic
fractional Fourier transform to a Boehmian space of quaternion valued functions.
Let us recall the abstract construction of Boehmian space from the literature [23, 24]. Let G be a
complex topological vector space, (S, \cdot ) be a commutative semigroup, \circ : G\times S \rightarrow G be satisfying
the following conditions:
1) (a+ b) \circ s = a \circ s+ b \circ s \forall a, b \in G, \forall s \in S ;
2) (\kappa a) \circ s = \kappa (a \circ s) \forall \kappa \in \BbbC ,\forall a \in G, \forall s \in S ;
3) a \circ (s \cdot t) = (a \circ s) \circ t \forall a \in G, \forall s, t \in S ;
4) if an \rightarrow a as n \rightarrow \infty in G and s \in S, then an \circ s \rightarrow a \circ s as n \rightarrow \infty ,
and \Delta be a collection of sequences from S with the following properties:
1) if (sn), (tn) \in \Delta , then (sn \cdot tn) \in \Delta ;
2) if an \rightarrow a in G as n \rightarrow \infty and (sn) \in \Delta , then an \circ sn \rightarrow a as n \rightarrow \infty in G.
A pair of sequences
\bigl(
(an), (sn)
\bigr)
with an \in G for all n \in \BbbN and (sn) \in \Delta is called a quotient if
an \circ sm = am \circ sn for all m,n \in \BbbN , and it is denoted by
(an)
(sn)
. An equivalence relation \sim is defined
on the collection of all quotients by
(an)
(sn)
\sim (bn)
(tn)
if an \circ tm = bm \circ sn \forall m,n \in \BbbN .
Every equivalence class induced by \sim is called a Boehmian and the collection of all Boehmians
B = B(G, (S, \cdot ), \circ ,\Delta ) is a vector space over \BbbC with respect to the addition and scalar multiplication
defined as follows:\biggl[
(an)
(sn)
\biggr]
+
\biggl[
(bn)
(tn)
\biggr]
=
\biggl[
(an \circ tn + bn \circ sn)
(sn \cdot tn)
\biggr]
, c
\biggl[
(an)
(sn)
\biggr]
=
\biggl[
(san)
(sn)
\biggr]
.
Every element a \in G can be uniquely identified as a member of B by
\biggl[
(a \circ tn)
(tn)
\biggr]
, where (tn) \in \Delta
is arbitrary and the operation \circ is also extended as
\biggl[
(an)
(sn)
\biggr]
\circ v =
\biggl[
(an \circ v)
(sn)
\biggr]
and
\biggl[
(an)
(sn)
\biggr]
\circ
\biggl[
(bn)
(tn)
\biggr]
=
=
\biggl[
(an \circ tn + bn \circ sn)
(sn \cdot tn)
\biggr]
, whenever v, bn \in S for all n \in N.
Definition 3 (\delta -convergence). A sequence (Bn) of Boehmians is said to \delta -converge to B in B
if there exist an,k, ak \in G, n, k \in \BbbN and (sk) \in \Delta such that Bn =
\biggl[
(an,k)
(sk)
\biggr]
, B =
\biggl[
(ak)
(sk)
\biggr]
and for
each k \in \BbbN , an,k \rightarrow ak as n \rightarrow \infty in G. In this case, we write Bn
\delta \rightarrow B in B as n \rightarrow \infty .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
816 R. ROOPKUMAR
Definition 4 (\Delta -convergence). A sequence (Bn) of Boehmians is said to \Delta -converge to B in
B if there exist an \in G for all n \in \BbbN and (sn) \in \Delta such that (Bn - B) \circ sn =
\biggl[
(an \circ sk)
(sk)
\biggr]
and
an \rightarrow 0 as n \rightarrow \infty in G. In this case, we write Bn
\Delta \rightarrow B in B as n \rightarrow \infty .
It should be noted that although G is mentioned as a complex vector space in the abstract
construction of the Boehmian space B(G, (S, \cdot ), \circ ,\Delta ), this constraint could be relaxed by taking G
as a left-module over quaternions.
In a problem of extending an integral transform to the context of Boehmians, the crux is proving
the convolution theorem for the integral transform, and constructing the suitable Boehmian space(s)
by proving the auxiliary results of their constructions. If this part is done properly, the definition of
the extended integral transform and its properties will be obtained by straightforward arguments. In
this paper, we shall first construct the Boehmian space
B2
\star \alpha = B
\bigl(
L2(\BbbR ,\BbbH ), (L1(\BbbR ,\BbbC ), \star ) \star \alpha ,\Delta \alpha
\bigr)
,
where \Delta \alpha is the collection of all sequences (\delta n) from L1(\BbbR ,\BbbC ), called delta sequences, satisfying
the following conditions:
1)
\int
\BbbR
\delta n(t)e
- ia(\alpha )t2 dt = 1 \forall n \in \BbbN ;
2)
\int
\BbbR
| \delta n(t)| dt \leq M \forall n \in \BbbN , for some M > 0;
3) support of \delta n(t) \rightarrow \{ 0\} as n \rightarrow \infty ; that is, given \varepsilon > 0, there exists N \in \BbbN such that
support of \delta n \subseteq ( - \varepsilon , \varepsilon ) \forall n \geq N,
and \star \alpha , \star are as defined in Definition 2. We observe that \star \alpha is a binary operation on L1(\BbbR ,\BbbH ), and
it is not commutative, as the multiplication on \BbbH -is not commutative. So, we choose (L1(\BbbR ,\BbbC ), \star )
as the commutative semigroup, from which the delta sequences are to be taken. It is interesting to
note that \star \alpha coincides with \star , whenever f and g are complex valued functions such that f \star g exists.
Now we prove the auxiliary results required to construct the Boehmian space B2
\star \alpha .
Lemma 1. If f \in L2(\BbbR ,\BbbH ) and g \in L1(\BbbR ,\BbbH ), then f \ast \alpha g \in L2(\BbbR ,\BbbH ).
Proof. If f = f1 + jf2, g = g1 + jg2, then f1, f2 \in L2(\BbbR ,\BbbC ), g1, g2 \in L1(\BbbR ,\BbbC ) and, hence,
\scrF - 2\alpha f1,\scrF - 2\alpha f2 \in L2(\BbbR ,\BbbC ). Therefore, by using the fact that
\| \mu \star \lambda \| 2 = \| \mu \ast \lambda \| 2 \leq \| \mu \| 2\| \lambda \| 1, whenever \mu \in L2(\BbbR ,\BbbC ) and \lambda \in L1(\BbbR ,\BbbC ),
we get f1 \star g1,\scrF - 2\alpha f2 \star g2, f2 \star g1,\scrF - 2\alpha f1 \star g1 \in L2(\BbbR ,\BbbC ). Thus, we have f \ast \alpha g \in L2(\BbbR ,\BbbH ).
Lemma 2. If f, g \in L2(\BbbR ,\BbbH ), h \in L1(\BbbR ,\BbbH ), and q \in \BbbH , then (f + g) \star \alpha h = f \star \alpha h+ g \star \alpha h
and (qf) \star \alpha h = q(f \star \alpha h).
Proof. We observe that using (\mu + \nu ) \ast \lambda = \mu \ast \lambda + \nu \ast \lambda , we can prove that
(\mu + \nu ) \star \lambda = \mu \star \lambda + \nu \star \lambda \forall \mu , \nu \in L2(\BbbR ,\BbbC ), \lambda \in L1(\BbbR ,\BbbC ).
Therefore, for f = f1 + jf2, g = g1 + jg2, h = h1 + jh2, then
(f + g) \star \alpha h = [f1 + g1] \star h1 - \scrF - 2\alpha [f2 + g2] \star h2+
+j(\scrF - 2\alpha [f1 + g1] \star h2 + [f2 + g2] \star h1) =
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
QUATERNIONIC FRACTIONAL FOURIER TRANSFORM FOR BOEHMIANS 817
= [f1 + g1] \star h1 - [\scrF - 2\alpha f2 + \scrF - 2\alpha g2] \star h2+
+j
\bigl(
[\scrF - 2\alpha f1 + \scrF - 2\alpha g1] \star h2 + [f2 + g2] \star h1
\bigr)
=
(since \scrF - 2\alpha and complex conjugation are linear)
= f1 \star h1 + g1 \star h1 - \scrF - 2\alpha f2 \star h2 - \scrF - 2\alpha g2 \star h2+
+j(\scrF - 2\alpha f1 \star h1 + \scrF - 2\alpha g1 \star h2 + f2 \star h1 + g2 \star h1) =
=
\bigl[
f1 \star h1 - \scrF - 2\alpha f2 \star h2 + j(\scrF - 2\alpha f1 \star h2 + f2 \star h1)
\bigr]
+
+
\bigl[
g1 \star h1 - \scrF - 2\alpha g2 \star h2 + j(\scrF - 2\alpha g1 \star h2 + g2 \star h1)
\bigr]
=
= f \star \alpha h+ g \star \alpha h.
If \tau \in \BbbC , then using (\tau \mu ) \ast \nu = \tau (\mu \ast \nu ), we can prove that
(\tau \mu ) \star \alpha \nu = \tau (\mu \star \alpha \nu ). (4)
Next, by a direct computation, we have
j(f \star \alpha h) = j[f1 \star h1 - \scrF - 2\alpha f2 \star h2] - (\scrF - 2\alpha f1 \star h2 + f2 \star h1) =
= j[f1 \star h1] - j[\scrF - 2\alpha f2 \star h2] - \scrF - 2\alpha f1 \star h2 - f2 \star h1
and
(jf) \star \alpha h = ( - f2 + jf1) \star \alpha (h1 + jh2) =
= - f2 \star h1 - \scrF - 2\alpha f2 \star h2 - \scrF - 2\alpha f1 \star h2 + j(f1 \star h1),
which implies that
j(f \star \alpha h) = (jf) \star h. (5)
Finally, for q = q1 + jq2 \in \BbbH ,
(qf) \star \alpha h = (q1f + jq2f) \star \alpha h =
(since multiplication is distributive over addition in \BbbH )
= [(q1f) \star \alpha h] + [(jq2f) \star \alpha h] (by the first assertion of this lemma) =
= q1(f \star \alpha h) + jq2(f \star \alpha h) (by (4) and (5)) =
= (q1 + jq2)(f \star \alpha h) = q(f \star \alpha h).
Lemma 2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
818 R. ROOPKUMAR
Lemma 3. If fn \rightarrow f in L2(\BbbR ,\BbbH ) as n \rightarrow \infty and h \in L1(\BbbR ,\BbbH ), then fn \star \alpha h \rightarrow f \star \alpha h in
L2(\BbbR ,\BbbH ) as n \rightarrow \infty .
Proof. First, we observe that
\| \mu \star \nu \| 2 = \| \mu \ast \nu \| 2 \leq \| \mu \| 2\| \nu \| 1 \forall \mu \in L2(\BbbR ,\BbbC ), \nu \in L1(\BbbR ,\BbbC ),
which implies that
\mu n \star \nu \rightarrow \mu \star \nu in L2(\BbbR ,\BbbC ) whenever \mu n \rightarrow \mu in L2(\BbbR ,\BbbC ) as n \rightarrow \infty . (6)
If fn = fn,1+jfn,2 for all \forall n \in \BbbN and f = f1+jf2, then by (1), we have fn,r - fr \rightarrow 0 as n \rightarrow \infty
for r \in \{ 1, 2\} . Therefore, by (6), we get that (fn,r - fr) \star hs \rightarrow 0 as n \rightarrow \infty for r, s \in \{ 1, 2\} . By
using (6) and continuity of \scrF - 2\alpha , we obtain that
(fn,1 - f1) \star h1 \rightarrow 0 as n \rightarrow \infty ,
\scrF - 2\alpha (fn,2 - f2) \star h2 \rightarrow 0 as n \rightarrow \infty ,
(\scrF - 2\alpha (fn,1 - f1) \star h2 \rightarrow 0 as n \rightarrow \infty ,
(fn,2 - f2) \star h1 \rightarrow 0 as n \rightarrow \infty ,
and, hence, again using (1), we have\bigl[
(fn,1 - f1) \star h1 - \scrF - 2\alpha (fn,2 - f2) \star h2
\bigr]
+ j
\bigl[
(\scrF - 2\alpha (fn,1 - f1) \star h2 + (fn,2 - f2) \star h1
\bigr]
\rightarrow 0
as n \rightarrow \infty . Thus, fn \star \alpha h \rightarrow f \star \alpha h in L2(\BbbR ,\BbbH ) as n \rightarrow \infty .
Lemma 4. If f \in L2(\BbbR ,\BbbH ) and (\delta n) \in \Delta \alpha , then f \star \alpha \delta n \rightarrow f in L2(\BbbR ,\BbbH ) as n \rightarrow \infty .
Further, if fn \rightarrow f in L2(\BbbR ,\BbbH ) as n \rightarrow \infty , then fn \star \alpha \delta n \rightarrow f in L2(\BbbR ,\BbbH ) as n \rightarrow \infty .
Proof. From [40], it is well-known that the above lemma is true if f and fn are complex valued
functions. Assuming this fact, and using the continuity of \scrF - 2\alpha , one can prove the above lemma, as
in the proof of the previous lemma.
In the next section, we shall define the extended quaternionic fractional Fourier transform on B2
\star \alpha .
As the codomain of the extended quaternionic fractional Fourier transform, we introduce another
Boehmian space
B2
\odot \alpha
= B(L2(\BbbR ,\BbbH ), L1(\BbbR ,\BbbC ),\odot \alpha , \^\Delta \alpha ),
where (f \odot \alpha g)(x) = f(x)g(x)eia(\alpha )x
2
for all x \in \BbbR , and \^\Delta \alpha =
\bigl\{
(\scrF \alpha (\delta n)) : (\delta n) \in \Delta \alpha
\bigr\}
. By
the convolution theorem for quaternionic fractional Fourier transform (Theorem 1), it is clear that
\scrF \alpha (f \star \alpha g) = \scrF \alpha (f) \odot \alpha \scrF \alpha (g). Therefore, all the auxiliary results for constructing this Boehmian
space could be obtained by applying the convolution theorem for quaternionic fractional Fourier
transform in the corresponding results for the construction of B2
\star \alpha .
4. Extended quaternionic fractional Fourier transform. For a given Boehmian B \in B2
\star \alpha , we
define the extended quaternionic fractional Fourier transform of B by the Boehmian
\biggl[
(\scrF \alpha (fn))
(\scrF \alpha (\delta n))
\biggr]
\in
\in B2
\odot \alpha
, where
(fn)
(\delta n)
is an arbitrary representative of B. Since
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QUATERNIONIC FRACTIONAL FOURIER TRANSFORM FOR BOEHMIANS 819
fn \star \alpha \delta m = fm \star \alpha \delta n \forall m,n \in \BbbN ,
applying the convolution theorem for quaternionic fractional Fourier transform, we get
\scrF \alpha (fn)\odot \alpha \scrF \alpha (\delta m) = \scrF \alpha (fm)\odot \alpha \scrF \alpha (\delta n) \forall m,n \in \BbbN .
Thus,
(\scrF \alpha (fn))
(\scrF \alpha (\delta n))
represents a Boehmian in B2
\odot \alpha
. Further, if
(gn)
(\varepsilon n)
is another representative of B, by
applying the convolution theorem, we get that
(\scrF \alpha (fn))
(\scrF \alpha (\delta n))
is equivalent to
(\scrF \alpha (gn))
(\scrF \alpha (\varepsilon n))
, and, hence, the
extended quaternionic fractional Fourier transform F\alpha : B2
\star \alpha \rightarrow B2
\odot \alpha
is well defined. As the proofs
of the following theorems are similar to that of any integral transform on Boehmians satisfying the
convolution theorem as in Theorem 1, we prefer to omit the details. For example, we refer to [18].
Theorem 2. The extended quaternionic fractional Fourier transform F\alpha on B2
\star \alpha is
1) consistent with the quaternionic fractional Fourier transform on L2(\BbbR ,\BbbH ) as defined in (2);
2) a \BbbH -linear map;
3) an injective map;
4) a continuous map with respect to \delta -convergence as well as \Delta -convergence;
5) F\alpha (B\ast C) = (F\alpha B)\odot \alpha (F\alpha C) \forall B,C \in B2
\star \alpha with C =
\biggl[
(gn)
(\varepsilon n)
\biggr]
and gn \in L1(\BbbR ,\BbbC ) \forall n \in \BbbN .
Theorem 3. Let X =
\biggl[
(\Phi n)
(\scrF \alpha (\delta n))
\biggr]
\in B2
\odot \alpha
. Then X is in the range of F\alpha : B2
\star \alpha \rightarrow B2
\odot \alpha
iff
there \Phi n belongs to the range of \scrF \alpha : L2(\BbbR ,\BbbH ) \rightarrow L2(\BbbR ,\BbbH ) for all n \in \BbbN .
5. Conclusion. In this paper, we extended the one-dimensional quaternionic fractional Fourier
transform to a suitable space of Boemians and obtained its properties consistency, continuity, linearity,
etc. The quaternionic linear canonical transform [17] generalizes the fractional Fourier transform and
quaternionic Fourier transform simultaneously by providing a suitable generalized kernel. Likewise,
the present work is also generalizing the quaternionic fractional Fourier transform of functions and
the Fourier transform on Boehmians simultaneously.
In the application point of view, playing the role of identity for the usual convolution, the Dirac’s
delta distribution \delta is useful to find the system waiting function g of a filter, in signal processing.
The output of an input signal f passing through a filter with system waiting function g can be simply
calculated by f \ast g. This tool is useful for the signals represented in frequency domain by the classical
Fourier transform. If a signal is quaternion valued and it is represented in frequency domain by the
fractional Fourier transform, then we need we need a suitable identity for \star \alpha to find the system
waiting function of a filter.
Let B =
\biggl[
(fn)
(\delta n)
\biggr]
\in B2
\star \alpha and (\varepsilon n), (\lambda n) \in \Delta \alpha be arbitrary. From the definition of Boehmians,
one can easily observe that\biggl[
(\varepsilon n)
(\varepsilon n)
\biggr]
=
\biggl[
(\lambda n)
(\lambda n)
\biggr]
\in B2
\star \alpha and B =
\biggl[
(fn)
(\delta n)
\biggr]
=
\biggl[
(fn \star \alpha \varepsilon n)
(\delta n \star \varepsilon n)
\biggr]
= B \star \alpha
\biggl[
(\varepsilon n)
(\varepsilon n)
\biggr]
.
Therefore, if we denote I =
\biggl[
(\varepsilon n)
(\varepsilon n)
\biggr]
, then B \star \alpha I = B is anlogous to f \ast \delta = f. So this work
may be helpful for the people working on signal processing those who need a theory of quaternionic
fractional Fourier transform which is applicable on the ``identity” for \star \alpha .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 6
820 R. ROOPKUMAR
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after revision — 17.09.19
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| id | umjimathkievua-article-649 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:03:34Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/19/1be2a3a515a420e853db9be71c5ec719.pdf |
| spelling | umjimathkievua-article-6492022-03-26T11:01:46Z Quaternionic fractional Fourier transform for Boehmians Quaternionic fractional Fourier transform for Boehmians Roopkumar, R. Roopkumar, R. дробове перетворення Фур’є бьомiани згортка функції у кватернiонах Fractional Fourier transform quaternion valued functions convolution Boehmians UDC 517.9 We construct a Boehmian space of quaternion valued functions using the quaternionic fractional convolution. Applying the convolution theorem, the quaternionic fractional Fourier transform is extended to the context of Boehmians and its properties are established. УДК 517.9 За допомогою кватернiонної дробової згортки побудовано бьомiанiв простiр функцiй iз значеннями у кватернiонах.&nbsp;Застосовуючи теорему про згортку, ми поширюємо кватернiонне дробове перетворення Фур’є на бьомiанiв простiр та встановлюємо його властивостi. Institute of Mathematics, NAS of Ukraine 2020-06-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/649 10.37863/umzh.v72i6.649 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 6 (2020); 812-821 Український математичний журнал; Том 72 № 6 (2020); 812-821 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/649/8717 |
| spellingShingle | Roopkumar, R. Roopkumar, R. Quaternionic fractional Fourier transform for Boehmians |
| title | Quaternionic fractional Fourier transform for Boehmians |
| title_alt | Quaternionic fractional Fourier transform for Boehmians |
| title_full | Quaternionic fractional Fourier transform for Boehmians |
| title_fullStr | Quaternionic fractional Fourier transform for Boehmians |
| title_full_unstemmed | Quaternionic fractional Fourier transform for Boehmians |
| title_short | Quaternionic fractional Fourier transform for Boehmians |
| title_sort | quaternionic fractional fourier transform for boehmians |
| topic_facet | дробове перетворення Фур’є бьомiани згортка функції у кватернiонах Fractional Fourier transform quaternion valued functions convolution Boehmians |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/649 |
| work_keys_str_mv | AT roopkumarr quaternionicfractionalfouriertransformforboehmians AT roopkumarr quaternionicfractionalfouriertransformforboehmians |