On a class of generalized Meijer Laplace transforms of Fox function type kernels and their extension to a class of Boehmians

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1 Georgian Math. J. 218; 25(1): 1 8 Research Article Shrideh Khalaf Qasem Al-Omari* On a class of generalized Meijer Laplace transforms of Fox function type kernels their extension to a class of Boehmians DOI: /gmj Received October 16, 214; accepted March 11, 215 Abstract: In this paper, we investigate a Meijer Laplace transform enfolding Fox s H-functions on a class of Boehmians. The extended Meijer Laplace transform of a Boehmian is determined executed to preserve certain properties of the classical transform. The inverse problem related theorems are also discussed in some details. Keywords: Mellin Barnes-type integral, Meijer Laplace transformation, Fréchet space, Fox s H-function, Boehmians MSC 21: 46F12 1 Introduction H-functions are an extreme generalization of generalized hypergeometric functions beyond Meijer functions, have recently found applications in a large variety of problems connected with reaction, diffusion, reaction diffusion, engineering, communications, fractional differential integral equations many areas of theoretical physics statistical distribution theory as well. The usefulness importance of H-functions have been realized in recent years due to their occurrence as kernels of certain integral transforms. The conventional generalized Meijer Laplace transformation of a suitably restricted function φ(t) having Fox s H-function as kernel is defined by ([12, 25) μ g l (φ)(ω) = m,m+1 [ωt (η m + a m, A m ) φ(t) dt. (1.1) (η m, B m ), (p, β m+1 ) H m+1, Here H m,n p,q [ω is a compact notation of Fox s H-function adopted for (see [16, 2, 23, 26) Hp,q m,n (ω) = Hp,q m,n (a j, α j ) j=1,2,...,p [ω, (b j, β j ) j=1,2,...,q has an exemplification in terms of the Barnes-type integral ([15) Hp,q m,n (ω) = 1 2πi ȷ m,n p,q (ς)ω ς dς, L where L is a suitable path in the complex plane, ω ς = exp{ς(log ω + i arg ω)} ȷ m,n p,q (ς) = a(ς)b(ς) c(ς)d(ς), *Corresponding author: Shrideh Khalaf Qasem Al-Omari: Department of Applied Sciences, Faculty of Engineering Technology, Al-Balqa Applied University, Amman 11134, Jordan, s.k.q.alomari@fet.edu.jo Download Date 8/26/18 5:4 AM

2 2 S. K. Q. Al-Omari, On a class of generalized Meijer Laplace transforms where m a(ς) := Γ(b j β j ς), b(ς) := Γ(1 a j + α j ς), c(ς) := 1 q Γ(1 b j β j ς), d(ς) := m+1 n 1 p n+1 Γ(a j + α j ς), with m, p, q N, a j, b j C, α j, β j R +, n N := N {} satisfying < n < p < m < q, where C, R + N denote, respectively, the sets of complex numbers, positive real numbers positive integers. Let c = min 1 j m Re{η j /B j, p/b m+1 }, where η j are complex numbers whereas B j are positive real numbers for every choice of j (j = 1, 2,..., m). Let a be a fixed real number such that a c + 1 b >. We denote by H a,b (I) the space of all smooth functions φ(t) defined on I := (, ) such that (see [24) γ a,b,k (φ) = sup e bt t 1 a+k D k t φ(t) <, <t< where k =, 1, 2,..., D t = dk. dt k The topology of H a,b (I) is generated by the set of seminorms {γ a,b,k }. H a,b(i) is then a complete countably multinormed space, hence defines a Fréchet space. The generalized Meijer Laplace transform of a distribution f in the dual space H a,b (I) of H a,b(i) is defined by F(ω) = μ g l (f )(ω) = f(t), Hm+1, m,m+1 [ωt (η m + a m, A m ) (η m, B m ), (P, B m+1 ), where ω {ω : Re ω > }. The complex inversion formula for (1.1) is shown to be valid for the space of generalized functions H a,b (I) where the numbers a b are restricted in some way. An amendment H bd (I) of H a,b(i) is introduced as follows. Denote by H bd (I) the space of smooth functions φ of bounded support defined on I such that φ(t) = on (, 1 where k =, 1, 2,.... φ = sup e bt t 1 a k D k t φ(t) <, (1.2) <t< Theorem 1.1. Let φ H bd (I). Then we have μg l (φ)(ω) Hbd (I). Proof. Let ω be a fixed complex number such that Re ω > ; then we have e bt t 1 a k D k t μg l (φ)(ω) e bt t 1 a k D k t (φ(t)hm+1, m,m+1 [ωt (η m + a m, A m ) ) dt, (η m, B m ), (P, B m+1 ) which, by the Leibniz rule, can be put into the form e bt t 1 a k D k t μg l (φ)(ω) e bt t 1 a k k n=1 D k n t φ(t)d n t Hm+1, Using the the following property of Fox s H-functions (see [21): we put (1.3) into the form m,m+1 [ωt (η m + a m, A m ) dt. (1.3) (η m, B m ), (P, B m+1 ) D r xhp,q m,n [(ax + b) h (a p, A p ) (b q, B q ) = a r (ax + b) r Hm+1,n p+1,q+1 [(ax + (, h), (a p, A p ) b)h (b q, B q ), (γ, h), e bt t 1 a k D k t μg l (φ)(ω) k D k n t n=1 φ(t) t k n e bt a H m+1,1 m+1,m+2 [ωt (, 1), (η m + a m, A m ) (η m, B m ), (P, B m+1 ), (n, 1) dt. Download Date 8/26/18 5:4 AM

3 S. K. Q. Al-Omari, On a class of generalized Meijer Laplace transforms 3 The asymptotic properties of H m,n p,q give e bt t 1 a k D k t μg l (φ)(ω) A k n=1 t k n D k n φ(t) dt, where A is a positive constant. Let [a, b, a > 1, b > 1, be a bounded set containing the support of φ(t). Then the hypothesis that φ(t) H bd (I) leads to the conclusion b e bt t 1 a k D k t μg l (φ)(ω) A kt k n A e bt t a 1 k+n dt < AA kt 2k t bt t a 1 dt (1.4) a a for some positive constant A. Hence, by considering the supremum, over all t ( < t < ), equation (1.4) yields φ < for every choice of φ H bd (I). Let H bd (I) denote the dual space of continuous linear forms on Hbd (I) is defined as Laplace transform of f H bd for every φ H bd (I). The right-h side of (1.5) is well defined by Theorem 1.1. Corollary 1.2. Let f H bd (I). Then we have μg l (f ) H bd (I). b (I). Then the distributional Meijer μ g l (f )(ω), φ(ω) = f(ω), μg l (φ)(ω) (1.5) Corollary 1.2 is interpreted to mean that the Meijer Laplace transform of f H bd (I) is a distribution in the same space. 2 Boehmian spaces, generalized distributions We assume that readers are familiar with the abstract construction of Boehmian spaces. For a more detailed account of the construction of Boehmians the transforms that are applied to Boehmians we refer to [1 11, 13, 14, 17 19, 22, 27 33, 35 In this section, we aim to establish certain spaces of Boehmians eligible for the extension of the transform μ g l. We denote by D(I) the stard notation of Schwartz space of test functions of compact support. The following constructive definition is very essential. Denote by the integral operator defined for F H bd (I) ψ D(I) as (F ψ)(t) = F(tβ)ψ(β) dβ. (2.1) Theorem 2.1. Let ψ D(I) F H bd (I). Then we have F ψ Hbd (I). Proof. Let F H bd (I). Then, using (1.2) integral (2.1), we have e bt t 1 a k D k t (F ψ)(t) ψ(β) e bt t 1 a k D k t F(βt) dβ. (2.2) The hypothesis ψ D(I) implies that ψ(β) =, where β (α, b ) for some positive real numbers α, b. Hence, taking the supremum over all positive real values of t, (2.2) shows that where A is a certain positive constant. (F ψ)(t) A ψ(β) dβ <, a b Download Date 8/26/18 5:4 AM

4 4 S. K. Q. Al-Omari, On a class of generalized Meijer Laplace transforms Now, we denote by the usual Mellin-type convolution product of first kind (see [34), namely, (ψ 1 ψ 2 )(t) = α 1 ψ 1 (tα 1 )ψ 2 (α) dα. (2.3) Then we find it worthwhile to describe its properties briefly as follows: (i) φ ψ = ψ φ, (ii) (φ ψ) ψ 1 = φ (ψ ψ 1 ), (iii) (φ ψ) ψ 1 = (φ ψ 1 ) φ, (iv) φ (ψ + ψ 1 ) = φ ψ + φ ψ 1, (v) (αφ) ψ 1 = α(φ ψ 1 ), where φ, ψ 1 ψ are integrable functions defined on I. We have the following theorem. Theorem 2.2. We have (F (ψ 1 ψ 2 ))(t) = ((F ψ 1 ) ψ 2 )(t) for every F H bd (I) ψ 1, ψ 2 D(I). Proof. Let F H bd (I) ψ 1, ψ 2 D(I) be given arbitrarily. Then the operator (2.1) reveals (F (ψ 1 ψ 2 ))(t) = F(tβ)(ψ 1 ψ 2 )(β) dβ. Hence, with the aid of (2.3) Fubini s theorem, we obtain That is, (F (ψ 1 ψ 2 ))(t) = ψ 2 (α)α 1 F(tβ)ψ 1 (βα 1 ) dβ dα = ψ 2 (α) F(tαγ)ψ 1 (γ) dγ dα. Hence, the theorem is proved. (F (ψ 1 ψ 2 ))(t) = (F ψ 1 )(tα)ψ 2 (α) dα. The proof of the next theorem follows from simple integration. We therefore prefer to omit the details. Theorem 2.3. Let F n, F, F 1, F 2 H bd (I), F n F as n ψ D(I). Then we have (i) ((F 1 + F 2 ) ψ)(t) = (F 1 ψ)(t) + (F 2 ψ)(t), (ii) ((α 1 F) ψ)(t) = α 1 (F ψ)(t), α 1 C, (iii) (F n ψ)(t) (F ψ)(t) as n. Let be the subset of sequences ( ) of D(I) such that the following hold: (C1) (β) dβ = 1 for all n N, (C2) (β) < M, where M is a positive constant, (C3) supp (β) (a n, b n ), with a n, b n as n. Then is a set of delta sequences that correspond to the delta distribution. Theorem 2.4. Let ( ) F H bd (I). Then we have F F as n in H bd (I). Proof. By the topology of H bd (I) (C1), we write e bt t 1 a k D k t (F )(t) F(t) e bt t 1 a k D k t (F(tβ) F(t)) (β) dβ. (2.4) Since F = F(tβ) F(t) H bd (I), from (2.4) it follows that e bt t 1 a k D k t (F )(t) F(t) F (β) dβ. Download Date 8/26/18 5:4 AM

5 S. K. Q. Al-Omari, On a class of generalized Meijer Laplace transforms 5 By (C2) (C3), we get e bt t 1 a k D k t ((F )(t) F(t)) F M(a n, b n ) (2.5) for some positive constant M. Taking the supremum with respect to both sides of (2.5) over all t ( < t < ), we obtain (F )(t) F(t) as n. Hence, the theorem is proved. We state without proving the following corollary. Corollary 2.5. Let F 1, F 2 H bd (I), () (F 1 )(t) = (F 2 )(t). Then we have F 1 (t) = F 2 (t) for all t. Theorem 2.6. Let ( ), (ϵ n ). Then we have ϵ n. The proof of this theorem is a straightforward consequence of the definition of the set the product. Hence, we avoid the details. The space (I), (D, ),, ) is therefore a Boehmian space. In this space, the sum multiplication by a scalar of two Boehmians can be defined as + [ g n ε n = + g n ε n α = [α f n = [ αf n, α C. The operation the differentiation are defined by Let space [ g n ε n = g n ε n D k = [ Dk f n (I), (D, ),, ) ω Hbd (I) by the formula (I), (D, ),, ) Hbd ω = ω (I). Then the operation can be extended to the A sequence of Boehmians (β n ) in (I), (D, ),, ) is said to be δ convergent to a Boehmian β in (I), (D, ),, ) ( is denoted by β n δ β) if there exists a delta sequence ( ) such that (β n δ k ), (β δ k ) H bd (I) for all k, n N (β n δ k ) (β δ k ) as n in H bd (I) for every k N. The equivalent statement for δ convergence reads as follows: β δ n β (n ) in (I), (D, ),, ) if only if there exist f n,k, f k H bd (I) (δ k) such that β n =,k δ k, β = [ f k δ k, for each k N, f n,k f k as n in H bd (I). A sequence of Boehmians (β n ) in (I), (D, ),, ) is said to be convergent to a Boehmian β in (I), (D, ),, ) ( is denoted by β n β) if there exists ( ) such that (β n β) H bd (I) for all n N, (β n β) as n in H bd (I). Considering the properties of enumerated above the proofs of the previous construction of the space (I), (D, ),, ), we easily establish the space B(Hbd (I), (D, ),, ). Download Date 8/26/18 5:4 AM

6 6 S. K. Q. Al-Omari, On a class of generalized Meijer Laplace transforms We define the sum multiplication by a scalar in (I), (D, ),, ) as follows: + [ g n ε n = ε n + g n ε n α = [α f n = [ αf n, α C. The operation the differentiation are defined in (I), (D, ),, ) as Let space [ g n ε n = g n ε n D k = [ Dk f n (I), (D, ),, ) ω Hbd (I) by (I), (D, ),, ) Hbd ω = ω (I), then the operation can be extended to the A sequence of Boehmians (β n ) in (I), (D, ),, ) is said to be δ convergent to a Boehmian β in (I), (D, ),, ) ( is denoted by β n δ β) if there exists a delta sequence ( ) such that (β n δ k ), (β δ k ) H bd (I) for all k, n N (β n δ k ) (β δ k ) as n in H bd (I) for every k N. The equivalent statement for δ convergence reads as follows: β δ n β (n ) in (I), (D, ),, ) if only if there exist f n,k, f k H bd (I) (δ k) such that β n =,k δ k, β = [ f k δ k, for each k N, f n,k f k as n in H bd (I). A sequence of Boehmians (β n ) in (I), (D, ),, ) is said to be convergent to a Boehmian β in (I), (D, ),, ) ( is denoted by β n β) if there exists ( ) such that (β n β) H bd (I) for all n N, (β n β) as n in H bd (I). 3 The extended μ g l transform for Boehmians To extend the transform μ g l to Boehmians, we establish the following useful theorem. Theorem 3.1. We have μ g l (F ψ)(t) = (μg l (F) ψ)(t) for every F Hbd (I) ψ D(I). Proof. Let F H bd (I) ψ D(I) be given arbitrarily. Then putting (2.3) in (1.1) gives μ g l (F ψ)(ω) = m,m+1 [ωt (η m + a m, A m ) (η m, B m ), (P, β m+1 ) α 1 F(tα 1 )ψ(α) dα dt. H m+1, Employing the change of variables tα 1 = β t = αβ dt = αdβ leads to μ g l (F ψ)(ω) = ψ(α)h m+1, Hence, by taking into account integral (1.1), we have Thus, the theorem is proved. m,m+1 [ωαβ (η m + a m, A m ) F(β) dβ. (η m, B m ), (P, β m+1 ) μ g l (F ψ)(ω) = μ g l (F)(ωα)ψ(α) dα = (μg l (F) ψ)(ω). Download Date 8/26/18 5:4 AM

7 S. K. Q. Al-Omari, On a class of generalized Meijer Laplace transforms 7 As a corollary of Theorem 3.1, we deduce the following definition. Definition 3.2. Let [ F n (I), (D, ),, ) by in the space (I), (D, ),, ). (I), (D, ),, ). Then we define the extension μ g l ([ F n ) = ([ μg l F n ) The right-h side of the above equation is well defined by Theorem 1.1. μg l of μg l to the space Theorem 3.3. (i) The operator μ g l is well defined linear. (ii) The operator μ g l is an isomorphism from B(Hbd (I), (D, ),, ) onto B(Hbd (I), (D, ),, ). (iii) The operator μ g l is continuous with respect to δ -convergence. (iv) The extended transform μ g l : B(Hbd (I), (D, ),, ) B(Hbd (I), (D, ),, ) is compatible with μ g l : Hbd (I) Hbd (I). Proof. Similar proofs of parts (i) (iii) are available in many cited papers of the author. To prove the last part of the theorem, let σ H bd (I), let [ σ be its representative in the space (I), (D, ),, ); then [ σ, where ( ) for all n N. It is clear that ( ) is independent of the representative for all n N. Therefore, by Theorem 3.1, we have μ g l ([ σ ) = μ g l δ ([ σ n which is the representative of μ g l σ in Hbd (I). ) = [ μg l (σ ) = [ μg l σ, 4 The inverse problem Definition 4.1. Let [ F n (I), (D, ),, ), F n = μ g l f n f n H bd (I). We define the inverse transform g l ) 1 of μ g l as for each ( ). g l ) 1 [ F n = [ (μg l ) 1 F n = Theorem 4.2. Let [ F n (I), (D, ),, ) ϕ D. Then we have g l ) 1 ([ F n ϕ) = ϕ. Proof. Assume that [ F n (I), (D, ),, ), F n = μ g l f n, f n H bd (I), let ϕ D be given arbitrarily. Then, by Definition 4.1, we have g l ) 1 ([ F n ϕ) = ( μ g l δ ) 1 ([ F n ϕ ) = [ (μg l ) 1 (F n ϕ) n Hence, by using Theorem 3.1 Definition 4.1, we obtain g l ) 1 ([ F n This completes the proof of the theorem. ϕ) = [ (μg l ) 1 F n ϕ = ϕ. Download Date 8/26/18 5:4 AM

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