Estimation of the generalized Bessel – Struve transform in a certain space of generalized functions
We investigate the so-called Bessel – Struve transform on certain class of generalized functions called Boehmians. By using different convolution products, we generate the Boehmian spaces, where the extended transform is well defined. We also show that the Bessel – Struve transform of a Boehmian is...
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Institute of Mathematics, NAS of Ukraine
2017
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507623161331712 |
|---|---|
| author | Al-Omari, S. K. Q. Аль-Омарі, С. К. К. |
| author_facet | Al-Omari, S. K. Q. Аль-Омарі, С. К. К. |
| author_sort | Al-Omari, S. K. Q. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:26:20Z |
| description | We investigate the so-called Bessel – Struve transform on certain class of generalized functions called Boehmians. By using
different convolution products, we generate the Boehmian spaces, where the extended transform is well defined. We also
show that the Bessel – Struve transform of a Boehmian is an isomorphism which is continuous with respect to a certain
type of convergence. |
| first_indexed | 2026-03-24T02:12:15Z |
| format | Article |
| fulltext |
UDC 517.9
S. K. Q. Al-Omari (Al-Balqa Appl. Univ., Amman, Jordan)
ESTIMATION OF THE GENERALIZED BESSEL – STRUVE TRANSFORM
IN A CERTAIN SPACE OF GENERALIZED FUNCTIONS
ОЦIНКА УЗАГАЛЬНЕНОГО ПЕРЕТВОРЕННЯ БЕССЕЛЯ – СТРУВЕ
В ДЕЯКОМУ ПРОСТОРI УЗАГАЛЬНЕНИХ ФУНКЦIЙ
We investigate the so-called Bessel – Struve transform on certain class of generalized functions called Boehmians. By using
different convolution products, we generate the Boehmian spaces, where the extended transform is well defined. We also
show that the Bessel – Struve transform of a Boehmian is an isomorphism which is continuous with respect to a certain
type of convergence.
Вивчається так зване перетворення Бесселя – Струве на деякому класi узагальнених функцiй, що називаються бьо-
мiанами. З використанням рiзних добуткiв типу згорток згенеровано простори Бьомiана, в яких розширене пере-
творення добре визначене. Також показано, що перетворення Бесселя – Струве для бьомiана є iзоморфiзмом, який
є неперервним вiдносно деякого виду збiжностi.
1. Introduction. While special types of what would later be known as Bessel functions were
studied by Euler, Lagrange, and the Bernoullis, the Bessel functions were first used by F. W. Bessel
to describe three body motion, with the Bessel functions appearing in the series expansion on
planetary perturbation and series solution to a second order differential equation that arise in many
diverse situations. On the other hand, Struve functions occur in many places in physics and applied
mathematics, e. g., in optics as the normalized line spread function, in fluid dynamics, and quite
prominently in acoustics for impedance calculations as well.
The normalized Bessel and Struve functions of index \alpha are, respectively, given by Watson [3] as
\jmath \alpha (z)2
\alpha \Gamma (\alpha + 1) z - \alpha \jmath \alpha (z) = \Gamma (\alpha + 1)
\infty \sum
n=0
( - 1)n
\Bigl( z
2
\Bigr) 2n
n!\Gamma (n+ \alpha + 1)
and
k\alpha (z) = 2\alpha \Gamma (\alpha + 1) z - \alpha H\alpha (z) = \Gamma (\alpha + 1)
\infty \sum
n=0
( - 1)n
\Bigl( z
2
\Bigr) 2n+1
\Gamma
\biggl(
n+
3
2
\biggr)
\Gamma
\biggl(
n+ \alpha +
3
2
\biggr) .
A kind of Fourier transforms named as Bessel – Struve transform was considered by S. Hamem et al.
as [2]
\bfitf \alpha
\beta ,s (f(x)) (\lambda ) =
\infty \int
- \infty
f(x)\sigma \alpha ( - i\lambda x) d\mu \alpha (x),
where \alpha > - 1
2
and \sigma \alpha is the Bessel – Struve kernel given by the equation
c\bigcirc S. K. Q. AL-OMARI, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9 1155
1156 S. K. Q. AL-OMARI
\sigma \alpha (x) = \jmath \alpha (ix) - ik\alpha (ix).
The Bessel – Struve kernel is the solution of the initial value problem \ell \alpha u(x) = \lambda 2u(x), where
u (0) = 1 and u\prime (0) =
\lambda \Gamma (\alpha + 1)
\surd
\pi \Gamma
\biggl(
\alpha +
3
2
\biggr) . It further satisfies the integral representation
\sigma \alpha (\lambda x) =
\lambda \Gamma (\alpha + 1)
\surd
\pi \Gamma
\biggl(
\alpha +
1
2
\biggr) 1\int
0
\bigl(
1 - t2
\bigr) \alpha - 1
2 e\lambda xtdt,
where x \in \BbbR and \lambda \in \BbbC .
Moreover, the Bessel – Struve transform is related to the Weyl integral transform [2]
\bfitw \alpha (f) (y) =
2\Gamma (\alpha + 1)
\surd
\pi \Gamma
\biggl(
\alpha +
1
2
\biggr) 1\int
| y|
\bigl(
x2 - y2
\bigr) \alpha - 1
2 xf (\mathrm{s}\mathrm{g}\mathrm{n} (y)x) dx
and it satisfies
\bfitf \alpha
\beta ,s (f) = \scrF f \circ \bfitw \alpha (f) , (1)
where f \in \bfitl 1\alpha (\BbbR ) and \scrF f is the Fourier transform of f,
\scrF (f(x)) (y) =
\infty \int
- \infty
f(x)e - ixy dx.
The Mellin-type convolution product of first kind was given in terms of the integral equation [10]
f \times g(y) =
\infty \int
0
f
\bigl(
yx - 1
\bigr)
x - 1g(x) dx. (2)
The space \bfitl p\alpha (\BbbR ) consists of those real valued measurable functions f defined on \BbbR such that
\| f\| p\alpha :=
\left\{
\left( \int
\BbbR
| f(x)| p d\mu \alpha (x)
\right) 1/p
<\infty , 1 \leq p <\infty ,
\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR
| f(x)| <\infty , p = \infty ,
where
d\mu \alpha (x) = A(x)dx and A(x) = | x|
2\alpha +1
.
By \bfitkappa (\BbbR ) we denote the space of test functions of bounded supports over \BbbR . Then, \bfitkappa (\BbbR ) is, indeed, a
dense subspace of \bfitl p(\BbbR ) for every choice of p. Here \bfitl 1(0,\infty ) denotes the Lebesgue space of complex
valued integrable functions defined on (0,\infty ) and \bfitl p\alpha (0,\infty ) denotes the restriction of \bfitl p\alpha (\BbbR ) to the
open interval (0,\infty ).
The following definition is very beneficial to our next investigation.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
ESTIMATION OF THE GENERALIZED BESSEL – STRUVE TRANSFORM IN A CERTAIN SPACE . . . 1157
Definition 1. Let \alpha > - 1
2
and \ttA (t) = | t| 2\alpha +1 and f, g in \bfitl 1(0,\infty ). Then we define the product
\otimes between f and g by the integral
f \otimes g(y) =
\infty \int
0
f (yt) g(t)d\mu (t), (3)
where d\mu (t) = \ttA (t)dt.
An assistance of (2) and (3) leads to the following proposition.
Proposition 1. Let f, g and h be integrable functions in \bfitl 1(0,\infty ) and y > 0. Then we have
f \otimes (g \times h)(y) = (f \otimes g)\otimes h(y).
Proof. Let the hypothesis of the theorem satisfy for f, g and h in \bfitl 1(0,\infty ). Then, appealing to
(2) and (3), we get
f \otimes (g \times h)(y) =
\infty \int
0
f (yt)
\infty \int
0
x - 1g
\bigl(
tx - 1
\bigr)
h(x)dxd\mu (t).
By Fubini’s theorem, we obtain
f \otimes (g \times h)(y) =
\infty \int
0
h(x)x - 1
\infty \int
0
g
\bigl(
tx - 1
\bigr)
f (yt) d\mu (t)dx.
Setting variables reveals
f \otimes (g \times h)(y) =
\infty \int
0
h(x)
\infty \int
0
f(yxz)g(z)d\mu (z)d\mu (x).
Proposition 1 is proved.
By the benefit of Proposition 2.1 of [2], it follows that \bfitw \alpha is a bounded operator from \bfitl 1\alpha (\BbbR ) into
\bfitl 1(\BbbR ). Hence, we have the following remark.
Remark 1. Let f \in \bfitl 1\alpha (\BbbR ). Then we have \bfitf \alpha
\beta ,s (f) \in \bfitl 1\alpha (\BbbR ).
Proof of this remark follows from equation (1) and the injectivity of \scrF . We therefore omit the
details.
2. Generated spaces of Boehmians. Boehmians were used for all objects defined by an algebraic
construction similar to that of field of quotients and in some cases, it just gives the field of quotients.
The advent of Boehmians has recently brought drastic changes in the concept of applied functional
analysis. The idea of construction of Boehmians was initiated by the concept of Mikusinski regular
operators.
The minimal structure necessary for the construction of Boehmians consists of the following
axioms:
A(i) A nonempty set a.
A(ii) A commutative semigroup (b, \bullet ).
A(iii) An operation \star : a\times b \rightarrow a such that for each x \in a and s1, s2,\in b,
x \star (s1 \bullet s2) = (x \star s1) \star s2.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1158 S. K. Q. AL-OMARI
A(iv) A collection \Delta \subset b\BbbN such that:
(a) If x, y \in a, (sn) \in \Delta , x \bullet sn = y \bullet sn for all n, then x = y.
(b) If (sn) , (tn) \in \Delta , then (sn \bullet tn) \in \Delta .
Elements of \Delta are called delta sequences. Consider
Q =
\bigl\{
(xn, sn) : xn \in a, (sn) \in \Delta , xn \star sm = xm \star sn \forall m,n \in \BbbN
\bigr\}
.
If (xn, sn) , (yn, tn) \in Q, xn \star tm = ym \star sn \forall m,n \in \BbbN , then we say (xn, sn) \sim (yn, tn) . The
relation \sim is an equivalence relation in Q. The space of equivalence clases in Q is denoted by \bfitb .
Elements of \bfitb are called Boehmians.
Between a and \bfitb there is a canonical embedding expressed as x\rightarrow x \star sn
sn
. The operation \star can
be extended to \bfitb \times a by
xn
sn
\star t =
xn \star t
sn
. The relationship between the notion of convergence and
the product \star is given as:
(i) If fn \rightarrow f as n \rightarrow \infty in a and, \phi \in b is any fixed element, then fn \star \phi \rightarrow f \star \phi in
a (as n\rightarrow \infty ) .
(ii) If fn \rightarrow f as n\rightarrow \infty in a and (\delta n) \in \Delta , then fn \star \delta n \rightarrow f in a (as n\rightarrow \infty ).
The operation \star is extended to \bfitb \times b as follows: If
\biggl[
(fn)
(sn)
\biggr]
\in \bfitb and \phi \in b, then
\biggl[
(fn)
(sn)
\biggr]
\star \phi =
=
\biggl[
(fn) \star \phi
sn
\biggr]
.
Convergence in \bfitb is defined as follows :
A sequence (hn) in \bfitb is said to be \delta convergent to h in \bfitb , hn
\delta \rightarrow h, if there is a sequence
(sn) \in \Delta such that (hn \star sn) , (h \star sn) \in a \forall k, n \in \BbbN , and (hn \star sk) \rightarrow (h \star sk) as n \rightarrow \infty , in a,
for every k \in \BbbN .
A sequence (hn) in \bfitb is said to be \Delta convergent to h in \bfitb , hn
\Delta \rightarrow h, if there is a sequence
(sn) \in \Delta such that (hn - h) \star sn \in a \forall n \in \BbbN , and (hn - h) \star sn \rightarrow 0 as n\rightarrow \infty in a.
Several integral transforms were extended to various spaces of Boehmians by many authors such
as: Al-Omari and Kilicman [9, 15, 20], Al-Omari [13], Mikusinski and Zayed [16], Karunakaran and
Roopkumar [17], Karunakaran and Vembu [18], Roopkumar [19], Nemzer [21], Al-Omari, Loonker,
Banerji and Kalla [11] and many others to mention but a few. However, readers are assumed to be
acquainted with the abstract construction of Boehmian spaces. If it were otherwise we refer to [4 – 9,
11, 13] and [15 – 21]. We need the following lemma to be established.
Lemma 1. Let f \in \bfitl 1\alpha (0,\infty ) and \psi \in \bfitkappa (0,\infty ). Then we have
\bfitf \alpha
\beta ,s
\bigl(
f \times \psi (x);\lambda
\bigr)
=
\bigl(
\bfitf \alpha
\beta ,sf \otimes \psi (x)
\bigr)
(\lambda ).
Proof. Under the hypothesis of the theorem we write
\bfitf \alpha
\beta ,s
\bigl(
f \times \psi (x);\lambda
\bigr)
=
\infty \int
0
\infty \int
0
f
\bigl(
xt - 1
\bigr)
t - 1\psi (t)dt\sigma \alpha ( - i\lambda x) d\mu \alpha (x).
By Fubini’s theorem, this can be written as
\bfitf \alpha
\beta ,s
\bigl(
f \times \psi (x);\lambda
\bigr)
=
\infty \int
0
\psi (t)
\infty \int
0
f
\bigl(
xt - 1
\bigr)
\sigma \alpha ( - i\lambda x) d\mu \alpha (x)dt.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
ESTIMATION OF THE GENERALIZED BESSEL – STRUVE TRANSFORM IN A CERTAIN SPACE . . . 1159
On setting variables we get
\bfitf \alpha
\beta ,s
\bigl(
f \times \psi (x);\lambda
\bigr)
=
\infty \int
0
\bigl(
\bfitf \alpha
\beta ,sf(z);\lambda (t)
\bigr)
\psi (t)d\mu (t).
Hence, equation (3) reveals
\bfitf \alpha
\beta ,s
\bigl(
f \times \psi (x);\lambda
\bigr)
=
\bigl(
\bfitf \alpha
\beta ,sf \otimes \psi
\bigr)
(\lambda ).
Lemma 1 is proved.
Spaces we are generating here are the space \beta 1
\bigl(
\bfitl 1\alpha ,\bfitkappa ,\times ),\times ,\Delta
\bigr)
and the space \beta 2(\bfitl
1, (\bfitkappa ,\times ),
\otimes ,\Delta ). \Delta wherever it appears is the set of delta sequences (\delta n) from \bfitkappa (0,\infty ), where
\infty \int
0
\delta n(x)dx = 1, (4)
\infty \int
0
\bigm| \bigm| \delta n(x)\bigm| \bigm| dx < m, m is a positive real number, (5)
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} \delta n \subseteq [ - \varepsilon , \varepsilon ], \varepsilon \rightarrow 0 as n\rightarrow \infty . (6)
It is of importance that we recall here some properties of the product \times which we list as [12, 10]:
f \times g = g \times f, (7)
f \times (g + h) = f \times g + f \times h, (8)
f \times (g \times h) = (f \times g)\times h, (9)
(\alpha f)\times g = \alpha (f \times g) = f \times (\alpha g) , \alpha \in \BbbC . (10)
We merely generate the space \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
as the space \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
can be ge-
nerated similarly.
Theorem 1. Let f \in \bfitl 1\alpha (0,\infty ) and \psi \in \bfitkappa (0,\infty ), \alpha > - 1
2
. Then we have f \times \psi \in \bfitl 1\alpha (0,\infty ).
Proof. Let f \in \bfitl 1\alpha (0,\infty ) and \psi \in \bfitkappa (0,\infty ) be given. Let K = [a, b] , 0 < a < b, be a compact
subset of (0,\infty ) such that \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}\psi \subseteq K. Then, for \alpha > - 1
2
, we have
\infty \int
0
| f \times \psi (y)| d\mu (y) =
\infty \int
0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \int
0
f
\bigl(
yt - 1
\bigr)
t - 1\psi (t)dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (y) \leq
\leq
b\int
a
| \psi (t)| t - 1
\infty \int
0
\bigm| \bigm| f \bigl( yt - 1
\bigr) \bigm| \bigm| d\mu (y)dt.
By setting the variables z = yt - 1 we get
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1160 S. K. Q. AL-OMARI
\infty \int
0
| f \times \psi (y)| d\mu (y) \leq
b\int
a
| \psi (t)| t2\alpha dt
\infty \int
0
| f(z)| d\mu (z).
That can be interpreted as \bigm| \bigm| f \times \psi (y)
\bigm| \bigm| 1
\alpha
\leq m\ast \| f\| 1\alpha , (11)
where m\ast =
\int b
a
| \psi (t)| t2\alpha dt.
Theorem 1 is proved.
Theorem 2. Let f \in \bfitl 1\alpha (0,\infty ) and \psi 1, \psi 2 \in \bfitkappa (0,\infty ), \alpha > - 1
2
. Then we have
(i) f \times (\psi 1 + \psi 2) = f \times \psi 1 + f \times \psi 2,
(ii) f \times (\psi 1 \times \psi 2) = (f \times \psi 1)\times (\psi 2) ,
(iii) (\alpha f)\times \psi 1 = \alpha (f \times \psi 1) = f \times (\alpha \psi 1) , \alpha \in \BbbC .
Proof of identities (i) and (iii) follows from simple integral calculus. Identitity (ii) directly follows
from (9). This establishes the theorem.
Theorem 3. Let fn \rightarrow f \in \bfitl 1\alpha (0,\infty ) as n\rightarrow \infty and \psi \in \bfitkappa (0,\infty ), \alpha > - 1
2
. Then we have
fn \times \psi \rightarrow f \times \psi
as n\rightarrow \infty in \bfitl 1\alpha (0,\infty ).
Proof of this theorem follows from simple integration. We, therefore, omit the details.
Theorem 4. Let f \in \bfitl 1\alpha (0,\infty ) and (\delta n) \in \Delta , \alpha > - 1
2
. Then we get
f \times \delta n \rightarrow f
as n\rightarrow \infty in \bfitl 1\alpha (0,\infty ).
Proof. Let f \in \bfitl 1\alpha (0,\infty ) and (\delta n) \in \Delta be given. Since the space \bfitkappa (0,\infty ) is dense in \bfitl 1\alpha (0,\infty )
we find \psi \in \bfitkappa (0,\infty ) such that
\| f - \psi \| 1\alpha < \varepsilon (12)
for \varepsilon > 0.
Also, by (11) and the fact that (\delta n) \in \bfitkappa (0,\infty ), we obtain\bigm\| \bigm\| (f - \psi )\times \delta n
\bigm\| \bigm\| 1
\alpha
\leq m\ast \| f - \psi \| 1\alpha
for some real number m\ast .
Hence, inserting (12) into above equation we get\bigm\| \bigm\| (f - \psi )\times \delta n
\bigm\| \bigm\| 1
\alpha
\leq \varepsilon m\ast . (13)
Thus, we have obtained
\bigm\| \bigm\| \psi \times \delta n - \psi
\bigm\| \bigm\| 1
\alpha
=
\infty \int
0
| (\psi \times \delta n - \psi ) (y)| d\mu (y) =
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
ESTIMATION OF THE GENERALIZED BESSEL – STRUVE TRANSFORM IN A CERTAIN SPACE . . . 1161
=
\infty \int
0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\infty \int
0
\psi
\bigl(
yt - 1
\bigr)
t - 1\delta n(t)dt - \psi (y)
\infty \int
0
\delta n(t)dt
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| d\mu (y) \leq
\leq
\infty \int
0
\infty \int
0
\bigm| \bigm| \psi \bigl(
yt - 1
\bigr)
t - 1 - \psi (y)
\bigm| \bigm| | \delta n(t)| dtd\mu (y). (14)
Now, let gy(t) = \psi
\bigl(
yt - 1
\bigr)
t - 1, then gy(t) is uniformly continuous function in \bfitkappa (0,\infty ). Therefore,
we find \delta > 0 such that \bigm| \bigm| gy(t) - g(1)
\bigm| \bigm| < \varepsilon whenever | y - 1| < \delta .
Thus, inventing (4) in (14) gives
\bigm\| \bigm\| \psi \times \delta n - \psi
\bigm\| \bigm\| 1
\alpha
\leq
\infty \int
0
\infty \int
0
\bigm| \bigm| gy(t) - gy(1)
\bigm| \bigm| | \delta n(t)| dt d\mu (y) \leq \varepsilon
d\int
c
d\mu (y), (15)
where [a, b] is an interval containing the support of gy.
Therefore, (15) implies \bigm\| \bigm\| \psi \times \delta n - \psi
\bigm\| \bigm\| 1
\alpha
\leq A\varepsilon , (16)
where A =
\int d
c
d\mu (y).
On account of (13), (16) and (12), we reach to\bigm\| \bigm\| f \times \delta n - f
\bigm\| \bigm\| 1
\alpha
\leq
\bigm\| \bigm\| (f - \psi )\times \delta n
\bigm\| \bigm\| 1
\alpha
+
\bigm\| \bigm\| \psi \times \delta n - \psi
\bigm\| \bigm\| 1
\alpha
t \| f - \psi \| 1\alpha \leq \varepsilon m\ast +A\varepsilon + \varepsilon .
Hence, above equation gives \bigm\| \bigm\| f \times \delta n - f
\bigm\| \bigm\| 1
\alpha
\leq B\varepsilon ,
where B = m\ast +A+ 1.
Theorem 4 is proved.
The space \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
has therefore been generated.
The sum of two Boehmians in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
and multiplication by a scalar can be defined
as \biggl[
(fn)
(\delta n)
\biggr]
+
\biggl[
(gn)
(\psi n)
\biggr]
=
\biggl[
(fn)\times \psi n + (gn)\times (\delta n)
(\delta n)\times (\psi n)
\biggr]
and \alpha
\biggl[
(fn)
(\delta n)
\biggr]
=
\biggl[
\alpha (fn)
(\delta n)
\biggr]
,
where \alpha \in \BbbC , \BbbC being the space of complex numbers.
The operation \times and the differentiation are defined by\biggl[
(fn)
(\delta n)
\biggr]
\times
\biggl[
(gn)
(\psi n)
\biggr]
=
\biggl[
(fn)\times (gn)
(\delta n)\times (\psi n)
\biggr]
and \scrD \alpha
\biggl[
(fn)
(\delta n)
\biggr]
=
\biggl[
\scrD \alpha (fn)
(\delta n)
\biggr]
.
A sequence of Boehmians (\beta n)in\beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
is said to be \delta convergent to a Boehmian \beta
in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
, denoted by \beta n
\delta \rightarrow \beta , if there exists a delta sequence (\delta n) such that
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1162 S. K. Q. AL-OMARI
(\beta n \times \delta k), (\beta \times \delta k) \in \bfitl 1\alpha \forall k, n \in \BbbN ,
and
(\beta n \times \delta k) \rightarrow (\beta \times \delta k) as n\rightarrow \infty , in \bfitl 1\alpha , for every k \in \BbbN .
The equivalent statement for \delta convergence:
\beta n
\delta \rightarrow \beta (n\rightarrow \infty ) in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
if and only if there is (\varphi n,k), (\varphi k) \in \bfitl 1\alpha and (\delta k) \in \Delta
such that \beta n =
\biggl[
(\varphi n,k)
(\delta k)
\biggr]
, \beta =
\biggl[
(\varphi k)
(\delta k)
\biggr]
and for each k \in \BbbN , \varphi n,k \rightarrow \varphi k as n\rightarrow \infty in \bfitl 1\alpha .
A sequence of Boehmians (\beta n) in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
is said to be \Delta convergent to a Boehmian
\beta in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
, denoted by \beta n
\Delta \rightarrow \beta , if there exists a (\delta n) \in \Delta such that (\beta n - \beta )\times \delta n \in
\in \bfitl 1\alpha \forall n \in \BbbN , and (\beta n - \beta )\times \delta n \rightarrow 0 as n\rightarrow \infty in \bfitl 1\alpha .
Similarly, the following theorems generate the Boehmian space \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
.
Theorem 5. Let f \in \bfitl 1(0,\infty ) and \psi \in \bfitkappa (0,\infty ). Then we have f \otimes \psi \in \bfitl 1(0,\infty ).
Theorem 6. Let f \in \bfitl 1(0,\infty ) and \psi 1, \psi 2 \in \bfitkappa (0,\infty ). Then we obtain
(i) f \otimes (\psi 1 + \psi 2) = f \otimes \psi 1 + f \otimes \psi 2,
(ii) (\alpha f)\otimes \psi 1 = \alpha (f \otimes \psi 1) = f \otimes (\alpha \psi 1) , \alpha \in \BbbC .
Theorem 7. For f \in \bfitl 1(0,\infty ) and \psi 1, \psi 2 \in \bfitkappa (0,\infty ), we get f \otimes (\psi 1 \times \psi 2) = (f \otimes \psi 1)\otimes \psi 2.
Proof of Theorems 5 and 6 is, respectively, similar to that of Theorems 1 and 2. Proof of
Theorem 7 follows from Proposition 1.
Theorem 8. (i) Let fn \rightarrow f in \bfitl 1(0,\infty ) as n\rightarrow \infty and \psi \in \bfitkappa (0,\infty ). Then we have fn\otimes \psi \rightarrow
\rightarrow f \otimes \psi as n\rightarrow \infty .
(ii) Let fn \in \bfitl 1(0,\infty ) and (\delta n) \in \Delta . Then we have fn \otimes \delta n \rightarrow f as n\rightarrow \infty .
The proof of the Part (i) of the theorem follows from simple integration whereas proof of the
second part is analogous to that of Theorem 3. Hence, we prefer we delete the details.
The sum of two Boehmians in \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
and multiplication by a scalar can also be
defined as\biggl[
(fn)
(\delta n)
\biggr]
+
\biggl[
(gn)
(\varepsilon n)
\biggr]
=
\biggl[
(fn)\otimes \varepsilon n + (gn)\otimes (\delta n)
(\delta n)\times (\varepsilon n)
\biggr]
and \alpha
\biggl[
(fn)
(\delta n)
\biggr]
=
\biggl[
\alpha
(fn)
(\delta n)
\biggr]
=
\biggl[
\alpha (fn)
(\delta n)
\biggr]
,
\alpha \in \BbbC , space of complex numbers.
The operation \otimes and the differentiation are respectively defined by\biggl[
(fn)
(\delta n)
\biggr]
\otimes
\biggl[
(gn)
(\varepsilon n)
\biggr]
=
\biggl[
(fn)\otimes (gn)
(\delta n)\times (\varepsilon n)
\biggr]
and \scrD k
\biggl[
(fn)
(\delta n)
\biggr]
=
\biggl[
\scrD k(fn)
(\delta n)
\biggr]
.
The notion of \delta and \Delta convergence in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
and \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
can be
defined in a natural way as above.
3. The Bessel – Struve transform of a Boehmian. Let \beta \in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
, \beta =
=
\bigl[
(fn)(\delta n)
\bigr]
, then, for every \alpha > - 1
2
, we define the Bessel – Struve transform of \beta as
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\delta n)
\biggr] \biggr)
=
\Biggl[ \bigl(
\bfitf \alpha
\beta ,sfn
\bigr)
(\delta n)
\Biggr]
. (17)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
ESTIMATION OF THE GENERALIZED BESSEL – STRUVE TRANSFORM IN A CERTAIN SPACE . . . 1163
The right-hand side of (17) belongs to \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
by the benefit of Remark 1. The above
definition is, indeed, well-defined. Let
\biggl[
(fn)
(\omega n)
\biggr]
=
\biggl[
(gn)
(\varepsilon n)
\biggr]
\in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
. Then, by the
notion of equivalence classes of \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
, we have
fn \times \varepsilon m = gm \times \omega n.
Employing (17) and the notion of equivalence classes of \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
yield
\bfitf \alpha
\beta ,sfn \otimes \varepsilon m = \bfitf \alpha
\beta ,sgm \otimes \omega n.
Hence, it follows that
\Bigl(
\bfitf \alpha
\beta ,sfn
\Bigr)
(\omega n)
\sim
\Bigl(
\bfitf \alpha
\beta ,sgn
\Bigr)
(\varepsilon n)
in \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
. Therefore, we get
\Biggl[ \bigl(
\bfitf \alpha
\beta ,sfn
\bigr)
(\omega n)
\Biggr]
=
\left[
\Bigl(
\bfitf \alpha
\beta ,sgn
\Bigr)
(\varepsilon n)
\right] .
This proves the claim.
Theorem 9. \u \bfitf \alpha
\beta ,s is an isomorphism from \beta 1
\bigl(
\bfitl 1\alpha ,\bfitkappa ,\times ),\times ,\Delta
\bigr)
into \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
.
Proof. Let us first establish that \u \bfitf \alpha
\beta ,s is injective. Given \u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr] \biggr)
= \u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(gn)
(\varepsilon n)
\biggr] \biggr)
. Then,
by Lemma 1 and notion of equivalent classes of \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
, it follows that
\bfitf \alpha
\beta ,sfn \otimes \varepsilon m = \bfitf \alpha
\beta ,sgm \otimes \omega n.
Therefore, Lemma 1 implies \bfitf \alpha
\beta ,s(fn \times \varepsilon m) = \bfitf \alpha
\beta ,s(gm \times \omega n). Employing \bfitf \alpha
\beta ,s gives
fn \times \varepsilon m = gm \times \omega n.
On the other hand, the notion of equivalent classes of \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
reveals that\biggl[
(fn)
(\omega n)
\biggr]
=
\biggl[
(gn)
(\varepsilon n)
\biggr]
.
Now, we establish that \u \bfitf \alpha
\beta ,s is a surjective mapping. Let
\Biggl[ \bigl(
\bfitf \alpha
\beta ,sfn
\bigr)
(\omega n)
\Biggr]
\in \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
be
arbitrary. Then we have
\bfitf \alpha
\beta ,sfn \otimes \omega m = \bfitf \alpha
\beta ,sfm \otimes \omega n
for every choice of m,n \in \BbbN . Hence fn, fm \in \bfitl 1\alpha (0,\infty ), for every m,n \in \BbbN , are satisfy
\bfitf \alpha
\beta ,s
\bigl(
fn \times \omega m
\bigr)
= \bfitf \alpha
\beta ,s
\bigl(
fm \times \omega n
\bigr)
.
That is,
\biggl[
(fn)
(\omega n)
\biggr]
\in \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
is such that
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1164 S. K. Q. AL-OMARI
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr] \biggr)
=
\Biggl[ \bigl(
\bfitf \alpha
\beta ,sfn
\bigr)
(\omega n)
\Biggr]
.
Theorem 9 is proved.
In addition to above, we derive the extension formula of \times to \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
as follows:
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr]
\times \phi
\biggr)
= \u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr] \biggr)
\otimes \phi .
Justification is as follows: by aid of (17) we write
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr]
\times \phi
\biggr)
=
\left[
\Bigl(
\bfitf \alpha
\beta ,s(fn \times \phi )
\Bigr)
(\omega n)
\right] .
Lemma 1 therefore gives
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr]
\times \phi
\biggr)
=
\left[
\Bigl(
\bfitf \alpha
\beta ,sfn \otimes \phi
\Bigr)
(\omega n)
\right] .
The definition of the product \times implies
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr]
\times \phi
\biggr)
=
\Biggl[ \bigl(
\bfitf \alpha
\beta ,sfn
\bigr)
(\omega n)
\Biggr]
\times \phi .
Once again, (17) yields
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr]
\times \phi
\biggr)
= \u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr] \biggr)
\otimes \phi .
Hence, we have reached to the conclusion that
\u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr]
\times \phi
\biggr)
= \u \bfitf \alpha
\beta ,s
\biggl( \biggl[
(fn)
(\omega n)
\biggr] \biggr)
\otimes \phi .
Theorem 10. \u \bfitf \alpha
\beta ,s : \beta 1
\bigl(
\bfitl 1\alpha , (\bfitkappa ,\times ),\times ,\Delta
\bigr)
\rightarrow \beta 2
\bigl(
\bfitl 1, (\bfitkappa ,\times ),\otimes ,\Delta
\bigr)
is continuous with respect to
\delta and \Delta — convergence.
Proof of this theorem follows from similar technique to that followed below in the citations.
References
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ESTIMATION OF THE GENERALIZED BESSEL – STRUVE TRANSFORM IN A CERTAIN SPACE . . . 1165
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Received 10.12.15
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
|
| id | umjimathkievua-article-1767 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:15Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/62/ad050f85dff53ed2967bb0d9e587b062.pdf |
| spelling | umjimathkievua-article-17672019-12-05T09:26:20Z Estimation of the generalized Bessel – Struve transform in a certain space of generalized functions Оцiнка узагальненого перетворення Бесселя–Струве в деякому просторi узагальнених функцiй Al-Omari, S. K. Q. Аль-Омарі, С. К. К. We investigate the so-called Bessel – Struve transform on certain class of generalized functions called Boehmians. By using different convolution products, we generate the Boehmian spaces, where the extended transform is well defined. We also show that the Bessel – Struve transform of a Boehmian is an isomorphism which is continuous with respect to a certain type of convergence. Вивчається так зване перетворення Бесселя – Струве на деякому класi узагальнених функцiй, що називаються бьо- мiанами. З використанням рiзних добуткiв типу згорток згенеровано простори Бьомiана, в яких розширене пере- творення добре визначене. Також показано, що перетворення Бесселя –Струве для бьомiана є iзоморфiзмом, який є неперервним вiдносно деякого виду збiжностi. Institute of Mathematics, NAS of Ukraine 2017-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1767 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 9 (2017); 1155-1165 Український математичний журнал; Том 69 № 9 (2017); 1155-1165 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1767/749 Copyright (c) 2017 Al-Omari S. K. Q. |
| spellingShingle | Al-Omari, S. K. Q. Аль-Омарі, С. К. К. Estimation of the generalized Bessel – Struve transform in a certain space of generalized functions |
| title | Estimation of the generalized Bessel – Struve transform in a certain space of
generalized functions |
| title_alt | Оцiнка узагальненого перетворення Бесселя–Струве
в деякому просторi узагальнених функцiй |
| title_full | Estimation of the generalized Bessel – Struve transform in a certain space of
generalized functions |
| title_fullStr | Estimation of the generalized Bessel – Struve transform in a certain space of
generalized functions |
| title_full_unstemmed | Estimation of the generalized Bessel – Struve transform in a certain space of
generalized functions |
| title_short | Estimation of the generalized Bessel – Struve transform in a certain space of
generalized functions |
| title_sort | estimation of the generalized bessel – struve transform in a certain space of
generalized functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1767 |
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