On Generalized Space of Quaternions and its Application
to a Class of Mellin Transforms
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
Abstract
The Mellin transform is an important tool in mathematics and closely related to Fourier and
bi-lateral Laplace transforms. In this article we aim to investigate the Mellin transform in a
class of quaternions which are coordinates for rotations and orientations. We consider a set
of quaternions as a set of generalized functions. Then we present the characterized Mellin
transform for the provided set of quaternions. The attributive Mellin integral is one-one,
onto and continuous between the quaternion spaces. Further properties of the integral are
given on a quaternion context.
Keywords: Mellin transform; Rotations; Quaternions; Laplace transform; Boehmians.
1
Preliminaries, De…nitions and Notations
Quaternions were devised by William Hamilton in his extensions to vector algebras to satisfy the properties of division rings. Quaternions are four-element vectors that can be used
to encode rotations in a 3-dimension coordinate system. Compared to vector calculus,
quaternions have further advantages in physical laws as relativistic, classical and quantum
mechanics that can nicely be written using quaternions. The ultimate reason for this attentiveness is that quaternionic multiplication turns the 3-sphere of unit quaternions into a
group, acting by rotations of the 3-space of purely imaginary quaternions, by v ! qvq 1 :
One more reason for the renewed interest is the fact that the resulting substitution of matrices by quaternions speeds up considerably the numerical calculation of the composition of
rotations, their square roots, and other standard operations that must be performed when
controlling anything from aircrafts to robots. A more interesting application of the quaternionic formalism is to the motion of two spheres rolling on each other without slipping with
in…nite friction.
Numbers of the form 0 + 1 i + 2 j + 3 k; with 0 ; 1 ; 2 ; 3 2 R were called quaternions.
They were added, subtracted and multiplied according to the usual laws of arithmetic, except
for the commutative law of multiplication.
Quaternions are also written in the condensed notation q = 0 +w;
^ where w
^ = 1 i+ 2 j+ 3 k
is a vector in R3 called the pure part of the quaternion and 0 being its real part.
The addition rule for quaternions is component-wise addition: If p = 0 + w;
^ w
^ = 1i +
^
^
^ ; where
^+#
2 j + 3 k and q = 0 + #; # = 1 i + 2 j + 3 k; then p + q = ( 0 + 0 ) + w
^ =(
w
^+#
1
+
1) i
+(
2
+
2) j
+(
3
+
3 ) k:
The multiplication rule for quaternions is the same as for polynomials, extended by the
2
S.K.Q. Al-Omari
multiplicative properties of i; j; k given as
pq
=
( 0
+(
0
1 1
3 2
3
3 3)
2 2
1+
0
2+
0
+( 2 3
2) j + ( 1
2 3
2
+
2
0 2
2+
+
0
0 1) i
3+
0 3 ) k:
(1)
The set (Q; +; :) of quaternions with the base f1; i; j; kg and the identities i2 = j 2 = k 2 = 1;
ij = k; ji = k; j k = i; kj = i; ki = j; ik = j; supplied with the usual operations +
and : de…nes a non-commutative division ring.
The conjugate element of a quaternion q is given as q = 0 w:
^ The quantity 20 + 21 +
2
2
2
2 + 3 ; also denoted by N (q) = kqk ; is called the reduced norm of q: Clearly, q is nonnull
if N (q) 6= 0; in which case q 1 = Nq(q) is the multiplicative inverse of q. The norm is a
real-valued function and the norm of a product of two quaternions satis…es the property
2
2
2
kpqk = kpk kqk :
Division of a quaternion by a real-valued scalar is just componentwise division. The conjugate and magnitude of a product of two quaternions p and q satisfy the following properties
2
2
2
kpqk = kpk kqk ; (p ) = p; (p + q) = p + q :
(2)
To take possession of the complete account of quaternions we refer to [14; 15] and [19] :
The Mellin transform is a basic tool for analyzing the behavior of many important
functions in mathematics and mathematical physics, such as the zeta functions occurring in
number theory and in connection with various spectral problems. It is closely related to the
Fourier and bilateral Laplace transform, and is used in many diverse areas of mathematics,
including analytic number theory, the study of di¤erence equations, asymptotic expansions,
and the study of special functions. Many of the properties of the Mellin transform follow from
the properties of the bilateral (two-sided) Laplace transform and equivalently the Fourier
transform which can be related to the Mellin transform by the substitution u = e x :
The Mellin transform of a suitably restricted function ' over R was introduced by
Z 1
' (x) xy 1 dx;
(3)
'
^ (y) =
0
provided the integral exists.
The Mellin-type convolution product of two functions ' and is given by [1]
Z 1
1
d
'z (y) =
'( ) y 1
d :
(4)
0
The Mellin transform has a relationship with the convolution product given by [1; 17]
d ='
'z
^ ^;
(5)
where '
^ and ^ are the Mellin transforms of ' and ; respectively.
The Mellin transform was extended to distributions in [1] and to Boehmians in [2]. The
modi…ed Mellin transform was de…ned in [3] and represented in the space of generalized
functions in [18]. By combining Fourier and Mellin transforms, the Fourier-Mellin transform has many applications in digital signals, image processing and ship target recognition
by sonar system and radar signals as well. In addition, by combining Fourier and modi…ed Mellin transforms the Fourier-Modi…ed Mellin transform has diverse applications in
engineering and engineering mathematics as well; see [3; (16)] :
We organize this article as follows. In Section 2 we present a complex valued space
of
quaternions and establish a convolution theorem with an assigned convolution products.
In Section 3 and 4 we generate the spaces ßand ß̧
r of quaternions in a generalized sense.
In Section 5 we give the representative of the Mellin transform on the discussed spaces of
quaternions. Further results are also established in this article.
3
On generalized space of quaternions and its Application ...
2
The Space
Let q = 0 +
written as
1i
+
2j
+
3k
be a quaternion in Q. Then q and its conjugate q can be
q = u + j and q = u
where u = 0 + 1 i;
Remark 1 Let =
2
j;
(6)
= 2 + i 3 2 C; C is the …eld of complex numbers.
+ i 3 2 C. Then we have
j =
j and j
= j;
where
is the conjugate element of :
Definition 2 Let q1 and q2 be in Q. Then q1 = u1 +
we de…ne the operator ] : Q Q ! Q by
1j
(7)
and q2 = u2 +
2 j:
Hence, on Q
q1 ]q2 = q1 q2 :
(8)
In view of the products assigned to Q we by (7) get
q1 ]q2 = (u1 u2 +
1 2)
(u1
1 u2 ) j;
2
Following (6) ; every function ' : R ! Q can be written in terms of its components as
' = '1 + '2 j;
(9)
where '1 and '2 are complex-valued functions.
We give some auxilliary results we recall are needful to our investigation.
Lemma 3 Let q1 ; q2 ; q3 ; q4 = and q5 = be in Q and k1 ; k2 2 R: Then we have
(i) q1 ]q2 = (q2 ]q1 ) :
(ii) q1 ] (k1 q2 + k2 q3 ) = k1 q1 ]q2 + k2 q1 ]q3 :
(iii) q1 ] (q4 q2 + q5 q3 ) = (q1 ]q2 ) q4 + (q1 ]q3 ) q5 :
(iv) (q4 q1 + q5 q2 ) ]q3 = q4 (q1 ]q3 ) + q5 (q2 ]q3 ) :
Then, the mapping ] de…nes an inner product on Q by Lemma 3.
Definition 4 Let be the space of all functions ' = '1 +'2 j where '1 and '2 are complex
valued such that k'k < 1,
k'k =
where '1;2
2
2
Z
1
'1;2 (x)
2
1
2
(10)
dx
0
2
= j'1 j + j'2 j :
Indeed, if '1 and '2 are complex valued functions, '1 = 1 + 2 i and '2 = 3 + 4 i; then
ofcourse every function ' can written as ' = 1 + 2 i + 3 j + 4 k 2 :
Definition 5 Every sequence f'n g 2 is said to converge to ' 2 ; i.e. 'n ! ' in as
n ! 1; if 'n = '1n + '2n j, ' = '1 + '2 j and '1n ! '1 and '2n ! '2 as n ! 1:
De…ne as
Z 1
('
) (x) =
('] ) (x) dx:
(11)
0
Then ' ' = k'k :
Also, we can easily inspect that
Z 1
('
) (x) =
(('1
0
1
+ '2
2)
('1
2
'2
1 ) j) (x) dx;
4
S.K.Q. Al-Omari
Indeed the space ( ; ) is a Hilbert space.
In view of the preceding analysis, we state the following de…nition.
Definition 6 Let ' 2 ; ' = '1 + '2 j: Then we de…ne the Mellin transform of ' as
'
^='
^1 + '
^ 2 j;
where '
^ 1 and '
^ 2 are the Mellin transforms of '1 and '2 ; respectively.
Theorem 7 Let ';
2 ; ' = '1 + '2 j and
=
1
+
2 j:
Then we have
d (y) = U '
']
^ ; ^ (y) ;
where
U '
^ ; ^ (y) = '
^1 ^1
'
^ 2 ^ 2 (y) + '
^1 ^1 + '
^ 2 ^ 1 (y) j;
(12)
^ and ^ are the Mellin transforms of and ; respectively.
i
i
Proof of this theorem follows from (8). Details are thus avoided.
3
The Boehmian Space ß
Quaternions have become a common part of mathematics and physics culture, but quaternions nowhere discussed in a generalized sense. In what follows we generate spaces of
generalized functions named as Boehmians in a quaternion concept. The complete account
of Boehmian spaces can be obtained from the cited papers [2] ; [4-12] and [14; 16; 18] : Let D
be the space of test functions of bounded support over (0; 1) and f n g 2 D be a sequence
of D such that (26)-(28) are satis…ed
Z 1
(13)
n ( ) d = 1; n 2 N;
0
supp
Then f
.
ng
Z
n
1
0
j
n
( )
( )j d
(
n;
k; n 2 N; k 2 R;
n) ;
n;
n
(14)
! 1 as n ! 1:
(15)
is said to be delta sequence. The collection of all delta sequences is denoted by
Theorem 8 Let ';
2 ; ' = '1 + '2 j;
=
1
+
2j
and
2 D: Then we have
(' + ) ] = '] + ] :
Theorem 9 k ('] ) = (k') ] = '] (k ) = ('] ) k; k 2 R:
Proofs of Theorem 8 and 9 are straightforward. Proofs are therefore deleted.
Theorem 10 Let 'n f'n = '1n + '2n jg and ' f' = '1 + '2 jg be in
'n ! ' as n ! 1: Then for every 2 D we have
'n ] ! '] in
for every n 2 N and
as n ! 1:
Proof of the theorem follows from simple integration. Hence we omit the deatails.
Theorem 11 Let 'n ! ' in
Let f n g 2 : Then we have
as n ! 1, where ' = '1 + '2 j; 'n = '1n + '2n j are in
'n ]
n
! ' as n ! 1:
:
5
On generalized space of quaternions and its Application ...
Proof On aid of the hypothesis of the theorem we write
Z 1
2
2
j'1n ] n (x) '1 (x)j + j'2n ] n (x)
k'n ] n (x) ' (x)k
=
Z 01
Z 1
2
i:e:
=
j'1n ] n (x) '1 (x)j dx +
j'2n ]
i:e:
Z
=
+
0
1
0
1
1
'1n x
n
( )d
'1 (x)
0
Z
1
1
'2n x
1
n ( )d
Z
2
'2 (x)j
2
(x)
n
dx
'2 (x)j dx
2
1
n
dx
( )d
0
'2 (x)
0
Z
2
1
dx:
n ( )d
0
By the parity of (13) we get
k'n ]
n (x)
2
' (x)k
Z
=
Z
1
0
+
Z
1
0
2
1
1
'1n x
1
'1 (x)
0
Z
2
1
1
'2n x
1
'2 (x)
+
Z
Z
1
0
+
Z
1
0
Z
1
0
Z
( )d
dx:
n
( )j d dx
n
n
'2n x
1
2
1
'2 (x) j
n
n
'1n x
Z
n
0
By aid of Jensens inequality we regulate the above equation to have
Z 1Z n
2
2
1
'1n x 1
'1 (x) j
k'n ] n (x) ' (x)k
0
dx
n ( )d
1 2
1
n
n
'2n x
1
2
+ j'1 (x)j
1 2
n
n
( )j d dx
j
2
+ j'2 (x)j
n
j
( )j d dx
n
( )j d dx
By taking into account of (15) we by simple computation have
2
k'n ]
n
(x)
' (x)k ! 0 as n ! 1:
'n ]
n
(x) ! ' (x) as n ! 1 in
Hence,
:
This …nishes the proof of the theorem.
The Boehmian space ßis therefore described where every typical element in ßwill be written
as
f'n g
:
f ng
Di¤erentiation in ßis given as
Dk
f'n g
=
f ng
"
D k 'n
f ng
#
; k 2 N; Dk =
dk
:
dxk
Addition in ßis given as
f'n g
fgn g
f'n g ] f
+
=
f ng
f ng
f
ng
+ fgn g ] f
g
]
f ng
n
ng
:
6
S.K.Q. Al-Omari
Scalar multiplication in ßis given as
k1
convergence in ßis given as :
(
f'n g
fk1 'n g
=
; k1 2 C:
f ng
f ng
n
n] n) ;
!
in ß
; if there exists a delta sequence f
( ]
n)
ng
such that
2ß
; for every k; n 2 N;
and
(
n] k )
! ( ] k ) as n ! 1 in ß
; for every k 2 N:
The equivalent statement for convergence is given as :
and f k g 2
n !# as (n ! 1) in ßif and only if there is 'n;k ; 'k 2
"
'n;k
f'k g
; =
and for each k 2 N; 'n;k ! 'k as n ! 1 in :
f kg
f kg
convergence in ßis given as:
n
!
(
for all n 2 N; and
(
in ßif there exists f
)]
n
n
)]
n
n
ng
2
such that
such that
2ß
,
! 0 in ß
as n ! 1:
4
The Boehmian Space ß̧
r
n
n o
Denote by A¾ = fw : w = '
^ ; for some ' 2 g ; E¾= f n g : f n g = ^n ; for some f
n
o
and Ŗ =
: = ^; for some 2 D .
ng
2
We state the following de…nition.
Definition 12 Let w 2 A¾ and 2 Ŗ: Then we de…ne (w ? ) ( ) = U (w; ) ( ) :
Theorem 13 Let w 2 A¾ and 2 Ŗ: Then we have w ? 2 A:
¾
Proof For each w 2 A¾ and 2 Ŗ there are ' 2 and 2 D such that w = '
^ and
Hence by De…nition 12 and (12) we get
(w ? ) ( )
= ^.
= U (w; ) ( )
= U '
^; ^ ( )
=
d ( ):
']
But since f ] 2 ; it follows w ? 2 A:
¾
This completes the proof of the theorem.
Theorem 14 The mapping ? : A¾ Ŗ ! A¾ obeys the following identities:
(i) Let 1 and 2 2 Ŗ: Then we have 1 ? 2 2 Ŗ:
(ii) Let 1 and 2 2 Ŗ: Then we have 1 ? 2 = 2 ? 1 :
(iii) Let w1 ; w2 2 A¾ and 2 Ŗ: Then we have (w1 + w2 ) ? = w1 ? + w2 ? :
(vi) Let w 2 A¾ and 1 ; 2 2 Ŗ: Then we have w ? ( 1 ? 2 ) = (w ? 1 ) ? 2 :
Proof of this theorem can be obtained by similar computation to that of [4] and [18] : We
prefer to delete the details.
o
7
On generalized space of quaternions and its Application ...
Theorem 15 Let w1 ; w2 2 A¾ and f n g 2 E¾ be such that w1 ? n = w2 ?
have w1 = w2 :
Proof Assume for every n 2 N; w1 ? n = w2 ? n : Then we have
(w1
2 N. Then we
(16)
n o
Let '1 ; '2 2 and f n g 2
such that '
^ 1 = w1 ; '
^ 2 = w2 and f n g = ^n : From (16)
and De…nition 12 and Theorem 11 we get
0 = '\
'2 ?
1
n
w2 ) ?
n; n
n
= 0 for all n 2 N:
= ('1 \
'2 ) ]
! ('\
'2 )
1
n
as n ! 1:
Hence
'
^1
'
^ 2 ! 0 as n ! 1:
Therefore w1 ! w2 as n ! 1 .
This completes the proof of the theorem.
Theorem 16 Let wn ! w as n ! 1 in A¾ and
2 Ŗ: Then we have
wn ? ! w ? as n ! 1:
Proof Let wn ; w 2 A¾ for all n 2 N and 2 Ŗ: Then there are f'n g ; ' 2
that
f^
'n g = fwn g ; '
^ = w and ^ = :
and
2 D such
Hence we simply write
w ? = (('n\
') ] ) ! 0 as n ! 1:
wn ?
This …nishes the proof of the theorem.
Theorem 17 Let wn ! w as n ! 1 in A¾ and f n g 2 E:
¾ Then we have
wn ?
n
! w as n ! 1:
Proof of of this theorem is similar to that of Theorem 16. We prefer not to add details.
Thus the theorem is completely proved.
The Boehmian space ß̧
r is de…ned where every typical element in ß̧
r will be written as
fwn g
:
f ng
Di¤erentiation in ß̧
r is given by
D
k
fwn g
=
f ng
"
Dk wn
f ng
#
; Dk =
dk
:
dxk
Addition in ß̧
r is given by
fwn g ? f n g + fwn g ? f n g
fwn g
fwn g
+
=
:
f ng
f ng
f ng ? f ng
Scalar multiplication in ß̧
r is given by
k
fwn g
fkwn g
=
; k 2 C:
f ng
f ng
8
S.K.Q. Al-Omari
Convergence of type
a Boehmian
: A sequence of Boehmians f
in ß̧
r; denoted by
(
n
?
n
n) ;
ng
in ß̧
r is said to be
convergent to
! ; if there exists a delta sequence f n g such that
( ?
n)
2 A;
¾ for every k; n 2 N;
and
(
n
?
k)
!( ?
k)
as n ! 1 in A;
¾ for every k 2 N:
Equivalent for convergence of type :
as (n ! 1) in ß̧
r if and only if there is wn;k ; wk 2 A¾ and f k g 2
n !
fwk g
fwn;k g
; =
and for each k 2 N;
f kg
f kg
such that
wn;k ! wk
as n ! 1 in A:
¾
Convergence of type : A sequence of Boehmians f n g in ß̧
r is said to be convergent to
a Boehmian in ß̧
r; denoted by n ! ; if there exists a f n g 2 Ŗ such that
(
)?
n
n
2 ß̧
r
for all n 2 N; and ( n
) ? n ! 0 as n ! 1 in ß̧
r:
Since the convolutions and the delta sequences used in ß̧
r and ßare di¤erent, it is not
possible to say one space is contained into the other.
5
Mellin Transform Associated with Quaternions
Definition 18 Let x =
f'n g
2ß
: Then we de…ne its extended Mellin transform by
f ng
f'n g
fwn g
=
2 ß̧
r
f ng
f ng
n o
where fwn g = f^
'n g and f n g = ^n :
z
To show this de…nition is well de…ned we assume
'n ]
m
f'n g
f'n g
=
in the sense of ß
: Then
f ng
f ng
= 'm ]
n:
Applying Mellin transform to both sides of the preceding equation yields
wn ?
n
= wm ?
m
where wn = '
^ n ; wm = 'm ;
n
=
n
and
n
= ^n :
fwn g
fwn g
and
are two equivalent classes in ß̧
r: This proves our assertion.
f ng
f ng
Theorem 19 The Mellin transform z : ß! ß̧
r is one to one.
fwn g
fwn g
Proof Assume that
=
in ß̧
r then by the concept of equivalence classes
f ng
f ng
we have
wn ? n = wm ? n :
Hence
9
On generalized space of quaternions and its Application ...
Let 'n ; 'n and n ; n be the preimages of wn ; wn and n ; n ; respectively: Then by using
\
\
De…nition 6.1 we get '
n ] n = 'n ] n for each n 2 N; and hence 'n ] n = 'n ] n which is
interpreted to mean
f'n g
f'n g
:
f ng
f ng
The theorem is therefore proved.
Theorem 20 The Mellin transform z : ß! ß̧
r is continuous with respect to convergence
of type .
Proof If xn 2 ßis such that xn ! 0 as n ! 1; then, by [14] ; we have
xn =
"
#
'n;i
f ig
; 'n;i ! 0 as n ! 1; where 'n;i 2 :
Employing the Mellin transform yields
wn;i ! 0 as n ! 1; wn;i = '
^ n;i :
Hence the theorem is established.
Theorem 21 The Mellin transform z : ß! ß̧
r is an onto linear mapping.
fwn g
Proof Linearity is obvious. Let
be in ß̧
r: Then wn ? m = wm ? n for every n 2 N:
f ng
Hence, we have
\
'\
n ] m = 'm ] n
for some f'n g ; f'm g 2
Hence
and f
ng ; f mg
2
:
f'n g
2ß
f ng
fwn g
:
f ng
Hence the theorem is completely proved.
is the preimage of
Theorem 22 The Mellin transform z : ß! ß̧
r is continuous with respect to convergence
of type .
Proof Let xn ! x 2 ß
: Then
such that
-convergence implies that there is 'n 2
(xn
x) ]
n
=
and f
ng
2
f'n g ] f n g
f ng
and 'n ! 0 as n ! 1:
Applying Mellin transform implies
(xn \
x) ?
n
=
fwn g ? f n g
f ng
n o
where f n g = ^n and fwn g = f^
'n g :
Hence from (30) ; x
^n ! x
^ as n ! 1:
This completes the proof of the theorem.
wn ! 0 as n ! 1
(17)
10
S.K.Q. Al-Omari
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