An extension of certain integral transform to a space

Journal of the Association of Arab Universities for Basic and Applied Sciences (2014) xxx, xxx–xxx

University of Bahrain

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ORIGINAL ARTICLE

An extension of certain integral transform to a space of Boehmians S.K.Q. Al-Omari

*

Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan Received 26 August 2013; revised 15 January 2014; accepted 11 February 2014

KEYWORDS 2

L transform; Function space; Generalized function; Boehmian

Abstract This paper investigates the L2 transform on a certain space of generalized functions. Two spaces of Boehmians have been constructed. The transform L2 is extended and some of its properties are also obtained. ª 2014 Production and hosting by Elsevier B.V. on behalf of University of Bahrain.

pffiffiffi y :

1. Introduction

Lð fðxÞÞðyÞ ¼ 2L2 ð fðx2 ÞÞ

Integral transforms are widely used to solve various problems in calculus, mechanics, mathematical physics, and some problems appear in computational mathematics as well. In the sequence of these integral transforms, the Laplace type integral transform, so-called L2 transform, is defined for a squared and an exponential function fðtÞ by David et al. (2007), as Z 1 2 2 L2 ð fðxÞÞðyÞ ¼ xex y fðxÞdx: ð1Þ

Let f and g be Lebesgue integrable functions; then the operation  between f and g is defined by

0 2

The L transform is related to the classical Laplace transform by means of the following relationships : 1  pffiffiffi L2 ð fðxÞÞðyÞ ¼ L f x ðy2 Þ 2 and * Tel.: +962 772357977. E-mail address: [email protected]. Peer review under responsibility of University of Bahrain.

Production and hosting by Elsevier

ð2Þ

ð f  gÞðtÞ ¼

Z

t

xf

ð3Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðt2  x2 Þ gðxÞdx:

0

The operation  is commutative, associative and satisfies the equation L2 ð f  gÞðyÞ ¼ L2 ð fÞðyÞL2 ðgÞðyÞ. Some facts about the transform L2 are given as follows:     2 (1) L2 sint2t ðyÞ ¼ 12 arctan y12 . 2 2

(2) L2 ðHðt  aÞÞðyÞ ¼ 2y12 ey a ; H being the heaviside unit function. cðnþ1Þ (3) L2 ðt2 ÞðyÞ ¼ 2y2nþ2 ; c being the gamma function. More properties, applications and the inversion formula of L2 transform are given by Yu¨rekli (1999a,b). 2. Abstract construction of Boehmians The minimal structure necessary for the construction of Boehmians consists of the following elements :

1815-3852 ª 2014 Production and hosting by Elsevier B.V. on behalf of University of Bahrain. http://dx.doi.org/10.1016/j.jaubas.2014.02.003 Please cite this article in press as: Al-Omari, S.K.Q. An extension of certain integral transform to a space of Boehmians. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.02.003

2

S.K.Q. Al-Omari (i) A set I; (ii) A commutative semigroup ðR; Þ; (iii) An operation  : I  R ! I such that for each x 2 I and t1 ; t2 ; 2 R,

x  ðt1  t2 Þ ¼ ðx  t1 Þ  t2 ; (vi) A collection D  RN satisfying : (a) If x; y 2 I; ðtn Þ 2 D; x  tn ¼ y  tn for all n, then x ¼ y; (b) If ðtn Þ; ðrn Þ 2 D, then ðtn  rn Þ 2 D. D is the set of all delta sequences. Consider A ¼ fðxn ; tn Þ : xn 2 I; ðtn Þ 2 D; xn  tm ¼ xm  tn ; 8m; n 2 Ng: If ðxn ; tn Þ; ðyn ; rn Þ 2 A; xn  rm ¼ ym  tn ; 8m; n 2 N, then we say ðxn ; tn Þ  ðyn ; rn Þ. The relation  is an equivalence relation in A. The space of equivalence classes in A is denoted by bðI; R; DÞ. Elements of bðI; R; DÞ are called Boehmians. Between I and bðI; R; DÞ there is a canonical embedding expressed as x!

x  sn as n ! 1: sn

The operation  can be extended to bðI; R; DÞ  I by xn xn  t t¼ : tn tn The sum of two Boehmians and multiplication by a scalar can be defined in a natural way       ðxn Þ ðgn Þ ðxn  wn þ gn  tn Þ þ ¼ ðtn Þ ðwn Þ ðtn  wn Þ and     ðxn Þ axn ¼ ; a 2 C: a ðtn Þ tn The operation  and the differentiation are defined by       ðxn Þ ðgn Þ ðxn  gn Þ  ¼ ðtn Þ ðwn Þ ðtn  wn Þ and    a  ðxn Þ ðD xn Þ ¼ : Da ðtn Þ ðtn Þ h i nÞ In particular, if ðx 2 bðI; R; DÞ and d 2 R is any fixed eleðtn Þ ment, then the product , defined by     ðxn Þ ðxn  dÞ d¼ ; ðtn Þ ðtn Þ is in bðI; R; DÞ. Many a time Iðalso considered as a quasi-normed spaceÞ is also equipped with a notion of convergence. The intrinsic relationship between the notion of convergence and the product  is given by: ðiÞ If fn ! f as n ! 1 in I and / 2 R is any fixed element, then fn  / ! f  / as n ! 1 in I. ðiiÞ If fn ! f as n ! 1 in I and ðdn Þ 2 D, then fn  dn ! f as n ! 1 in I.

In bðI; R; DÞ, two types of convergence are: (1) A sequence ðhn Þ in bðI; R; DÞ is said to be d-convergent to d

h in bðI; R; DÞ, denoted by hn ! h as n ! 1, if there exists a delta sequence ðtn Þ such that ðhn tn Þ; ðhtn Þ 2 I; 8k; n 2 N, and ðhn tk Þ ! ðhtk Þ as n ! 1, in I, for every k 2 N. (2) A sequence ðhn Þ in bðI; R; DÞ is said to be D- convergent D to h in bðI; R; DÞ, denoted by hn ! h as n ! 1, if there exists a ðtn Þ 2 D such that ðhn  hÞ  tn 2 I; 8n 2 N, and ðhn  hÞ  tn ! 0 as n ! 1 in I. The following theorem is equivalent to the statement of d- convergence : d

Preposition 1. hn ! hðn ! 1Þ in bðI; R;hDÞiif and only h i if there

is fn;k ; fk 2 I and tk 2 D such that hn ¼

fn;k tk

; h¼

fk tk

and for

each k 2 N; fn;k ! fk as n ! 1 in I. For further discussion of Boehmian spaces and their construction; see Ganesan (2010), Karunakaran and Ganesan (2009), Al-Omari (2012, 2013a,b,c,d), Al-Omari and Kilicman (2011, 2012a,b, 2013a,b), Boehme (1973), Bhuvaneswari and Karunakaran (2010), Ganesan (2010), Karunakaran and Angeline (2011), Karunakaran and Devi (2010), Mikusinski (1983, 1987, 1995), Nemzer (2006, 2007, 2008, 2009, 2010) and Roopkumar (2009). 3. The Boehmian space bðp; ðd; Þ; ; Þ p denotes the space of rapidly decreasing functions defined on Rþ ðRþ ¼ ð0; 1ÞÞ. That is, /ðxÞ 2 p if /ðxÞ is a complex-valued and infinitely smooth function defined on Rþ and is such that, as jtj ! 1; / and its partial derivatives decrease to zero faster than every power of jtj1 . In more details, /ðtÞ 2 p iff it is infinitely smooth and is such that jtm /ðkÞ ðtÞj 6 Cm;k ; t 2 Rþ ;

ð4Þ

m and k run through all non-negative integers; see Pathak (1997). d denotes the Schwartz space of test functions of bounded support defined on Rþ .  denotes the Mellin-type convolution product offirst kind defined by Zemanian (1987), as Z 1   y 1 t gðtÞdt: ð f  gÞðyÞ ¼ f ð5Þ t 0 To construct the first Boehmian space bðp; ðd; Þ; ; DÞ, we need to establish the following necessary theorems. Theorem 2. Let / 2 p and u 2 d; then we have /  u 2 p. Proof. Let K be a compact subset in Rþ containing the support of u. Then, for all k 2 N and m 2 N, we, by (4) and (5), get that Z  y  m k m k t1 uðtÞ dt y D y / y Dy ð/  uÞðyÞ 6 t K Z Z y 1 jym j Dky / jt1 uðtÞjdt < MCm;k : ¼ jt uðtÞjdt 6 Cm;k t K K

Please cite this article in press as: Al-Omari, S.K.Q. An extension of certain integral transform to a space of Boehmians. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.02.003

An extension of certain integral transform to a space of Boehmians Hence, by considering the supremum over all y 2 Rþ , we obtain that /  u 2 p. This completes the proof of the theorem. h Theorem 3. Let /1 ; /2 2 p; u1 ; u2 2 d; then we have

Z

3 1

jð/n  wn ÞðxÞjdx 6

Z

0

0

0

0

Finally, the inequality

Theorem 4. Let /n ! / in p as n ! 1 and u 2 d; then /n  u ! /  u as n ! 1. Proof of this theorem is straightforward from simple integration. Definition 5. Denoted by D the set of all sequences ðdn Þ 2 d that satisfying: R1

j/n ðx=yÞ/n ðyÞjy1 dydx:

0

suppð/n  wn Þ 6 ðsupp /n Þðsupp wn Þ:

Proof of (1) and (3) follows from the Reference (Pathak, 1997). Proof of (2) is straightforward from the properties of integration. Therefore, we prefer to omit the details. Hence the theorem is completely proved.

ðiÞ

1

Once again, change of variables, implies Z 1 Z 1 Z 1 jð/n . wn ÞðxÞjdx 6 j/n ðuÞjdu jwn ðyÞjdy 6 M1 M2 : 0

(1) /1  /2 ¼ /2  /1 . (2) ð/1 þ /2 Þ  u1 ¼ /1  u1 þ /2  u1 . (3) ð/1  u1 Þ  u2 ¼ /1  ðu1  u2 Þ.

1

Z

dn ¼ 1;

R01 ðiiÞ 0 jdn j < M; n 2 N; ðiiiÞ supp dn  ð0; eÞ; e ! 0 as n ! 1.

establishes our assertion. The space bðp; ðd; Þ; ; DÞ is therefore considered as a space of Boehmians. Definition 7. Let / 2 p and u 2 d. Between p and d define a product  as in the integral equation Z 1 ð/  uÞðyÞ ¼ t/ðytÞuðtÞdt: ð7Þ 0

We establish the second Boehmian space bðp; ðd; Þ; ; DÞ. Theorem 8. Let / 2 p and u 2 d; then we have /  u 2 p. Proof. By (7), we write Z 1 m k t/ðytÞuðtÞdt y Dy ð/  uÞðyÞ ¼ ym Dky Z 1 0 m k 6 y Dy /ðytÞ jtuðtÞjdt:

ð8Þ

Since / 2 p, we, from (8), find that Z m k jtuðtÞjdt < MCm;k ; y Dy ð/  uÞðyÞ 6 Cm;k

ð9Þ

0

Each ðdn Þ is called a delta sequence or an approximating identity which corresponds to the Dirac delta distribution. Theorem 6. Let / 2 p and ðdn Þ 2 D; then /  dn ! / as n ! 1. Proof. By Axiom ðiÞ of Definition 5, we write Z

as n ! 1. Hence, we have established that /  dn ! / as n ! 1. Hence the theorem has been proved.

h

Now, we assert that the product of delta sequences is a delta sequence. Detailed proof is as follows. Let /n ; wn 2 D. Then, we have Z 1 Z 1Z 1 ð/n  wn ÞðxÞdx ¼ /n ðx=yÞwn ðyÞy1 dydx ¼

1

0

¼

Z

0

1

0

Z Z

0

1

/n ðx=yÞdxwn ðyÞy1 dy 0 1

/n ðuÞduwn ðyÞdy ¼ 0

h

1

jð/  dn  /ÞðyÞj ! 0

Z

where K is a compact set in Rþ containing the support of u. Hence, (9) leads to the conclusion that /  u 2 p. This completes the proof of the theorem.

y t1  /ðyÞ jdn ðtÞjdt: jð/  dn  /ÞðyÞj 6 ð6Þ / t 0  The mapping wðyÞ ¼ / yt t1  /ðyÞ is uniformly continuous for each y 2 Rþ . Therefore, it follows that

0

K

Z

ðBy change of variablesÞ

1

/n ðuÞdu

0

let M1 and M2 be constants such that RNext, 1 jw n j 6 M2 ; then 0

Z

1

wn ðyÞdy ¼ 1

0

R1 0

Theorem 9. Let / 2 p and u; w 2 d; then /  ðu  wÞ ¼ ð/  uÞ  w. Proof. Let the hypothesis of the theorem satisfies for / 2 p and u; w 2 d: Then, using (5) and (7) yield Z 1 ð/  ðu  wÞÞðyÞ ¼ /ðytÞtðu  wÞðtÞdt 0 Z 1 Z 1  t 1 ¼ n wðnÞdndt /ðytÞt u n 0 0

  Z 1 Z 1 t dt wðnÞn1 dn: ð10Þ /ðytÞtu ¼ n 0 0 The change of variables t ¼ nz implies dt ¼ ndz. Hence, from (10), we obtain  Z 1 Z 1 ð/  ðu  wÞÞðyÞ ¼ /ðynzÞnzuðzÞndz wðnÞn1 dn 0 0  Z 1 Z 1 /ððynÞzÞzuðzÞdz nwðnÞdn ¼ Z0 1 0 ð/  uÞðynÞnwðnÞdn ¼ 0

¼ ðð/  uÞ  wÞðyÞ: j/n j 6 M1 and This completes the proof of the theorem.

h

Please cite this article in press as: Al-Omari, S.K.Q. An extension of certain integral transform to a space of Boehmians. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.02.003

4

S.K.Q. Al-Omari In view hof above, we define the extended L2 transform of a i

Theorem 10. Let /1 ; /2 2 p and u 2 d; then we have

Boehmian

ðiÞ ð/1 þ /2 Þ  u ¼ /1  u þ /2  u. ðiiÞ ða/Þ  u ¼ að/  uÞ; a 2 C. Proof of this theorem follows from the general properties of integration. Details are thus omitted. The theorem is therefore proved. Theorem 11. Let /n ! / in p as n ! 1; then /n  u ! /  u as n ! 1, for each u 2 d.

Theorem 12. Let / 2 p and ðdn Þ 2 D; then /  dn ! / as n ! 1. Proof. Utilizing Definition 5 and Axiom ðiÞ yield m k y Dy ðð/  dn ÞðyÞ  /ðyÞÞ Z 1 Z 1 Dky ðt/ðytÞÞdn ðtÞdt  ym Dky /ðyÞ dn ðtÞdt ¼ ym 0 0 m k 6 y Dy ðt/ðytÞ  /ðyÞÞ jdn ðtÞjdt

2

Z

1

2 2

ð/  uÞðnÞney n dn 0  Z 1 Z 1  n 1 2 2 ¼ t uðtÞdt ney n dn / t 0 0  Z 1 Z 1  n 2 2 ney n dn t1 uðtÞdt: / ¼ t 0 0

L ð/  uÞðyÞ ¼

The substitution n ¼ zt implies  Z 1 Z 1 2 2 2 L2 ð/  uÞðyÞ ¼ /ðzÞztey z t tdz t1 uðtÞdt 0

¼

0

Z

1

0

¼

Z

Z

1

/ðzÞzeðy

2 t2 Þz2

0

 dz tuðtÞdt

c2 : bðp; ðd; Þ; ; DÞ ! bðp; ðd; Þ; ; DÞ is wellTheorem 14. L defined. Proof. Let

h

i

ðun Þ ðrn Þ

¼

h

i

ðvn Þ ðwn Þ

in bðp; ðd; Þ; ; DÞ. Then, by the

write un  wm ¼ vm  rn ¼ vn  rm : Employing L2 on both sides of the above equation implies

L2 un L2 vn is equivalent to : rn wn ð11Þ

Therefore, we get  2   2  ðL un Þ ðL vn Þ ¼ : ðrn Þ ðwn Þ This completes the proof of the theorem.

h

The Boehmian space bðp; ðd; Þ; ; DÞ has been constructed.

Proof. Let / 2 p and u 2 d; then we have

ð12Þ

That is,

Hence, from(11) and Definition 5ðiiÞ, we get m k y Dy ðð/  dn ÞðyÞ  /ðyÞÞ < MCm;k :

L2 ð/  uÞðyÞ ¼ ðL2 /  uÞðyÞ:

   2  c2 ð/n Þ ¼ ðL /n Þ 2 bðp; ðd; Þ; ; DÞ: L ðdn Þ ðdn Þ

L2 un  wm ¼ L2 vn  rm ; n; m:

Since / 2 p it follows m k y Dy ðt/ðytÞ  /ðyÞÞ < Cm;k ; m; k 2 N:

Theorem 13. Let / 2 p and u 2 d; then we have

in bðp; ðd; Þ; ; DÞ as

concept of quotient of sequences in bðp; ðd; Þ; ; DÞ, we

Proof of this theorem follows from Theorem 9.

Hence the theorem is therefore completely proved.

ð/n Þ ðdn Þ

h

c2 It is also interesting to know that the transform L is a linear mapping from bðp; ðd; Þ; ; DÞ into bðp; ðd; Þ; ; DÞ. h i h i ðvn Þ nÞ ; ðw Detailed proof can be given as follows : If ðu ðrn Þ nÞ 2 bðp; ðd; Þ; ; DÞ then

   

  c2 ðun Þ þ ðvn Þ c2 ðun  wn Þ þ ðvn  rn Þ ¼L L ðrn Þ ðwn Þ ðrn  wn Þ  2  L ðun  wn þ vn  rn Þ ¼ ðrn  wn Þ  2  L ðun  wn Þ þ L2 ðvn  rn Þ ¼ ðrn  wn Þ " #  L2 un  wn þ ðL2 vn  rn Þ ¼ ðrn  wn Þ  2   2  ðL un Þ ðL vn Þ þ : ¼ ðrn Þ ðwn Þ Hence,

        ðun Þ ðvn Þ c c c 2 2 ðun Þ 2 ðvn Þ þ ¼L þL : L ðrn Þ ðwn Þ ðrn Þ ðwn Þ Also, if a 2 C, the field of complex numbers, then we see that " # " 2 #   2 c2 ðun Þ ¼ a L un ¼ L ðaun Þ : aL ðrn Þ ðrn Þ ðrn Þ

1

L2 ð/ÞðytÞtuðtÞdt

0

¼ ðL2 /  uÞðyÞ: Hence the theorem is proved.

h

Hence,  

  c2 ðun Þ ¼ L c2 a ðun Þ : aL ðrn Þ ðrn Þ This completes the proof of the theorem.

Please cite this article in press as: Al-Omari, S.K.Q. An extension of certain integral transform to a space of Boehmians. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.02.003

An extension of certain integral transform to a space of Boehmians 5 h 2 i c2 : bðp; ðd; Þ; ; DÞ ! bðp; ðd; Þ; ; DÞand L c2 1 : Definition 15. Let ðLð/ fnÞÞ 2 bðp; ðd; Þ; ; DÞ; then we define Theorem 18. L n h 2 i c2 transform of ðL fn Þ as bðp; ðd; Þ; ; DÞ ! bðp; ðd; Þ; ; DÞ are continuous with respect to the inverse L ð/n Þ d and D-convergence.  2  " 2 1 2 #   c2 : bðp; ðd; Þ; ; DÞ ! ðL Þ ðL fn Þ ð fn Þ c 2 1 ðL fn Þ Proof. First of all, we show that L ¼ ; ð13Þ L ¼ ð/n Þ ð/n Þ ð/n Þ c2 1 : bðp; ðd; Þ; ; DÞ ! bðp; ðd; Þ; ; DÞ bðp; ðd; Þ; ; DÞ and L for each ð/n Þ 2 D.

are continuous with respect to d-convergence.

c2 : bðp; ðd; Þ; ; DÞ ! bðp; ðd; Þ; ; DÞ is an Theorem 16. L isomorphism. h i h i c2 ð fn Þ ¼ L c2 ðgn Þ . Using (12) and the concept Proof. Assume L ð/n Þ ðwn Þ

Let bn ! b in bðp; ðd; Þ; ; DÞ as n ! 1; then we show that c2 b as n ! 1. By virtue of Preposition 1 we can find c2 b ! L L n fn;k and fk in p such that     fn;k fk and b ¼ bn ¼ /k /k

d

of quotients we get L2 fn  wm ¼ L2 gm  /n . Therefore, Theorem 13 implies L2 ð fn  wm Þ ¼ L2 ðgm  /n Þ: Properties of L2 implies fn  wm ¼ gm  /n . Therefore, from the concept of qoutients of equivalent classes of bðp; ðd; Þ; ; DÞ, we get     ð fn Þ ðgn Þ ¼ : ð/n Þ ðwn Þ h i c2 is surjective, let ðL2 fn Þ 2 bðp; ðd; Þ; ; DÞ. To establish that L ð/n Þ Then L2 fn  /m ¼ L2 fm  /n for every m; n 2 N. Once again, Theorem 13 implies L2 ð fn  /m Þ ¼ L2 ð fm  /n Þ. Hence h i ð fn Þ 2 bðp; ðd; Þ; ; DÞ is a Boehmian satisfying the equation ð/n Þ    2  ðL fn Þ c 2 ð fn Þ ¼ : L ð/n Þ ð/n Þ This completes the proof of the theorem. h h 2 i Theorem 17. Let ðLð/nfnÞÞ 2 bðp; ðd; Þ; ; DÞ and / 2 d; then we have

 2     c2 1 ðL fn Þ  / ¼ ð fn Þ  / and L ð/n Þ ð/n Þ

    2  c2 ð fn Þ  / ¼ ðL fn Þ  /: L ð/n Þ ð/n Þ Detailed proof of the first part is as follows : Applying (13) yields " ! " #! #  L2 fn L2 fn  / c2 1 c2 1 / ¼ L L ð/n Þ ð/n Þ "  # 1 ðL2 Þ ðL2 fn Þ  / ¼ : ð/n Þ

and fn;k ! fk as n ! 1 for every k 2 N. Hence, L2 fn;k ! L2 fk as n ! 1 in the space p. Thus,  2   2  L fn;k L fk ! /k /k as n ! 1 in bðp; ðd; Þ; ; DÞ. d To prove the second part, let gn ! g in bðp; ðd; Þ; ; DÞ as h 2 i L f n ! 1. Then, once again, by Preposition 1, gn ¼ /kn;k and h 2 i g ¼ L/ fk and L2 fn;k ! L2 fk as n ! 1. Hence fn;k ! fk in k h i h i f bðp; ðd; Þ; ; DÞ as n ! 1. Or, /n;k ! /fk as n ! 1. Using k

k

(13) we get  2   2  c2 1 L fn;k ! L c2 1 L fk as n ! 1: L /k /k c2 and L c2 1 with respect Now, we establish continuity of L to D-convergence. D

Let bn ! b in bðp; ðd; Þ; ; DÞ as n ! 1. Then, there exist h i fn 2 p and ð/n Þ 2 D such that ðbn  bÞ  /n ¼ ð fn/Þ/k and k

fn ! 0 as n ! 1. Employing (12) gives  2  L ðð fn Þ  /k Þ c 2 : L ððbn  bÞ  /n Þ ¼ /k Hence, we have  2  c2 ððb  bÞ  / Þ ¼ ðL fn Þ  /k ¼ L2 f ! 0 L n n n /k as n ! 1 in p. Therefore   c2 ððb  bÞ  / Þ ¼ L c2 b  L c2 b  / ! 0 as n ! 1: L n n n n D c c2 b ! Hence, L L2 b as n ! 1. n

D

Finally, let gn ! g in bðp; ðd; Þ; ; DÞ as n ! 1; then we h 2 i find L2 fk 2 p such that ðgn  gÞ  /n ¼ L f/k /k and L2 fk ! 0

By using (12), we obtain

 2       c2 1 ðL fn Þ  / ¼ ð fn Þ  / ¼ ð fn Þ  /: L ð/n Þ ð/n Þ ð/n Þ

as n ! 1for some ð/n Þ 2 D.

h i  h i c2 ð fn Þ  / ¼ ðL2 fn Þ  / is The proof of the second part L ð/n Þ ð/n Þ similar. This completes the proof of the theorem.

" # 1 ðL2 Þ ðL2 fk  /k Þ c 2 1 : L ððgn  gÞ  /n Þ ¼ /k

k

Now, using (13), we obtain

Please cite this article in press as: Al-Omari, S.K.Q. An extension of certain integral transform to a space of Boehmians. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.02.003

6

S.K.Q. Al-Omari

Theorem 13 implies   c2 1 ððg  gÞ  / Þ ¼ ð fn Þ  /k ¼ f ! 0 as n ! 1 in p: L n n n /k Thus   c2 1 ððg  gÞ  / Þ ¼ L c2 1 g  L c2 1 g  / ! 0 as n ! 1: L n n n n D c c2 1 g ! From this we find that L L2 1 g as n ! 1 in n bðp; ðd; Þ; ; DÞ. This completes the proof of the theorem. h

c2 transform is consistent with L2 . Theorem 19. The extended L Proof. For every f 2 p, let b be its representative in h i bðp; ðd; Þ; ; DÞ; then b ¼ fðuðunnÞÞ , where ðun Þ 2 D; 8n 2 N. Its clear that ðun Þ is independent from the representative, 8n 2 N. Therefore

  c2 ðbÞ ¼ L c2 f  ðun Þ L ðun Þ  2  L ð f  ðu n ÞÞ i:e: ¼ ðun Þ  2  L f  ðun Þ ¼ ðun Þ which is the representative of L2 f in p. Hence the proof is completed. h h i ðgn Þ 2 bðp; ðd; Þ; ; DÞ; then the necessary Theorem 20. Let ðw nÞ h i ðgn Þ c2 is that to be in the range of L and sufficient condition that ðw nÞ gn belongs to range of L2 for every n 2 N. h i c2 ; then of course g Proof. Let ððgwn ÞÞ be in the range of L n n

belongs to the range of L2 ; 8n 2 N. To establish the converse, let gn be in the range of L2 ; 8n 2 N. Then there is fn 2 p such that L2 fn ¼ gn ; n 2 N. h i ðgn Þ Since, ðw Þ 2 bðp; ðd; Þ; ; DÞ, n

gn  wm ¼ gm  wn ; 8m; n 2 N. Therefore, L2 ð fn  un Þ ¼ L2 ð fm  un Þ; 8m; n 2 N; where fn 2 p and un 2 D; 8n 2 N. The fact that L2 is injective, implies that fn  um ¼ fm  un ; m; n 2 N. Thus, ufn is qoutient of sequences in bðp; ðd; Þ; ; DÞ. Hence, n





ð fn Þ c2 2 bðp; ðd; Þ; ; DÞ and L ðun Þ



ð fn Þ ðun Þ



 ¼

 ðgn Þ : ðwn Þ

Hence the theorem is proved. h h i h i ð jn Þ 2 Theorem 21. Let b ¼ ððufnnÞÞ 2 bðp; ðd; Þ; ; DÞ and c ¼ ð/ nÞ bðp; ðd; Þ; ; DÞ; then c2 ðb  cÞ ¼ L c2 b  c: L

Proof. Assume the requirements of the theorem are satisfied for some b and c 2 bðp; ðd; Þ; ; DÞ. Then

  ð fn Þ  ðjn Þ c c 2 2 L ðb  cÞ ¼ L ðun Þ  ð/n Þ  2  L ðð fn Þ  ðjn ÞÞ ¼ ðun Þ  ð/n Þ  2  ðL fn Þ  ðjn Þ ¼ ðun Þ  ð/n Þ  2    ðL fn Þ ðj n Þ  : ¼ ðun Þ ð/n Þ Therefore, c2 ðb  cÞ ¼ L c2 ðbÞ  c: L This completes the proof of the theorem.

h

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Please cite this article in press as: Al-Omari, S.K.Q. An extension of certain integral transform to a space of Boehmians. Journal of the Association of Arab Universities for Basic and Applied Sciences (2014), http://dx.doi.org/10.1016/j.jaubas.2014.02.003