Bol. Soc. Paran. Mat. (3s.) v. ???? (??) : 1–11. c SPM –ISSN-2175-1188 on line ISSN-0037-8712 in press SPM: www.spm.uem.br/bspm doi:10.5269/bspm.43828

Fractional Hartley Transform on G-Boehmian Space

Rajakumar Roopkumar and Chinnaraman Ganesan

abstract: Using a special type of fractional convolution, a G-Boehmian space Bα containing integrable functions on R is constructed. The fractional Hartley transform (frht) is defined as a linear, continuous injection from Bα into the space of all continuous functions on R. This extension simultaneously generalizes the fractional Hartley transform on L1(R) as well as Hartley transform on an integrable Boehmian space.

Key Words: Fractional Hartley transform, Fractional convolution, Boehmians.

Contents

1 Introduction 1

2 Preliminaries 2

3 Fractional Convolutions and Fractional Hartley Transform 3

4 Fractional Hartley Transform on a G-Boehmian space 7

1. Introduction Hartley introduced a Fourier-like transform in 1942, which is called Hartley transform (see [7,11]). Like the fractional Fourier transform (frft)[21], many integral transforms have been generalized to the corresponding fractional integral transforms. In particular, fractional Fourier cosine transform (frfct), fractional Fourier sine transform (frfst) and fractional Hartley transform (frht) were defined and used extensively in signal processing [4,24]. In [14], Mikusi´nski, J. and Mikusi´nski, P., introduced Boehmian space, which in general, consists of convolution quotients of sequences of functions. In [15], an abstract Boehmian space B is constructed by using a complex topological vector space G, S G, ⋆ : G S G and a collection ∆ of sequences satisfying certain axioms. As many of these Boehmian⊂ spaces× conta→in the respective domains of various classical integral transforms, the research on Boehmian space includes extension of integral transforms to larger domains. For example, we refer [1,2,3,6,29,5,9,10,12,22,23,25,26,27]. Meanwhile, various versions of Boehmian spaces are introduced with new assumptions or slightly weaker assumptions than that are used in the general construction of a Boehmian space given in [15], by many authors [8,13,17,18,19]. Most recently, the G-Boehmian space is introduced in [9] as a generalization of the Boehmian space and the Hartley transform is extended to a suitable G-Boehmian space. In the present article, we introduce a special type of fractional convolution to construct a G-Boehmian space Bα containing the space of integrable functions on R. The fractional Hartley transform (frht) is extended consistently as a linear, continuous injection from Bα in to the space C(R), of all complex-valued continuous functions on reals. This paper is organized as follows. In Section 2, we recall fractional Hartley transform, the general construction of a G-Boehmian space and some of their properties. In Section 3, we shall prove all the preliminary results required for the construction of the G-Boehmian space Bα. In Section 4, we provide the extended frht on this G-Boehmian space and investigate its properties.

2010 Mathematics Subject Classification: 42A38, 44A15, 44A35. Submitted July 23, 2018. Published November 07, 2018

BSP Typeset by Mstyle. 1 c Soc. Paran. de Mat.

2 R. Roopkumar and C. Ganeasn

2. Preliminaries We now recall from [24], the definition of frht of f L 1(R) and some of its properties. frht of an arbitrary integrable function f was defined by ∈

2 2 2 iaα(x +u ) [Hα(f)](u)= cα f(x)e Cas (bαxu)dx, u R, (2.1) rπ Z ∀ ∈ R where cot α 1 eiα/2 Cas ( ) = cos( ) + sin( ),aα = ,bα = , and cα = . · · · 2 sin α √i sin α The frft and frht are obtained from one another through the following identities: 1+ i 1 i H (f) = F (f)+ − F ( f) α 2 α 2 α − H (f)+ H ( f) H (f) H ( f) F (f) = α α − i α − α − , α 2 − 2 where Fα(f) is the fractional Fourier transform of f, which is defined by

∞ 2 2 cα iaα(x +u 2ux sec α) Fα(f)(u)= f(x)e − dx, u R. √2π Z ∀ ∈ −∞ If f L 1(R) and H (f) L 1(R) then H [H (f)] = f. ∈ α ∈ α α Using these identities along with the properties of frft, we have Hα(f) C0(R) and Hα(f) ∞ cα 1 ∈ k k ≤ | | f . Thus the frht is a continuous injective mapping from L (R) into C (R). √2π k k1 0 We shall devote the next part of this section to the general construction of a G-Boehmian space B given in [9]. According to [9], a G-Boehmian space is a quotient space defined as follows: Let B = B(Γ, S, ⋆, ∆), where Γ is a topological vector space over C, S Γ, ⋆ :Γ S Γ satisfies the following conditions: ⊆ × → A : (g + g ) ⋆s = g ⋆s + g ⋆s, g ,g Γ and s S. 1 1 2 1 2 ∀ 1 2 ∈ ∀ ∈ A : (cg) ⋆s = c(g⋆s), c C, g Γ and s S. 2 ∀ ∈ ∀ ∈ ∀ ∈ A : g⋆ (s⋆t) = (g⋆s) ⋆t = (g⋆t) ⋆s, g Γ and s,t S. 3 ∀ ∈ ∀ ∈ A : If g g as n in Γ and s S, then g ⋆s g⋆s as n in Γ. 4 n → → ∞ ∈ n → → ∞ and let ∆ be a collection of sequences from S such that (∆ ) If(s ), (t ) ∆, then (s ⋆t ) ∆. 1 n n ∈ n n ∈ (∆ ) If g Γ and (s ) ∆, then g⋆s g as n in Γ. 2 ∈ n ∈ n → → ∞ Let A = ((gn), (sn))/gn Γ, (sn) ∆ and gn ⋆sm = gm ⋆sn, m,n N . An equivalence relation on A is defined{ as follows:∈ ∈ ∀ ∈ } ∼ ((g ), (s )) ((h ), (t )) if g ⋆t = h ⋆s , m,n N n n ∼ n n n m m n ∀ ∈ gn B Denoting the equivalence class s containing ((gn), (sn)), we define the G-Boehmian space as h n i the set of all equivalence classes gn induced by the equivalence relation on A. It is clear that B is a sn ∼ vector space with respect to theh additioni and scalar multiplication defined as follows. g h g ⋆t + h ⋆s g cg n + n = n n n n , c n = n . sn   tn   sn ⋆tn  sn   sn 

B g⋆sn Every member g Γ can be uniquely identified as a member of by s , where (sn) ∆ is arbitrary ∈ h n i ∈ and the operation ⋆ is also extended to B S by gn ⋆t = gn⋆t . There are two notions of convergence × φn φn on B namely δ-convergence and ∆-convergence whichh i are definedh i as follows. FrHT on G-Boehmian Space 3

Definition 2.1. [9, δ-convergence] We say that X δ X as m in B, if there exist g ,g Γ, m → → ∞ m,n n ∈ gm,n gn m,n N and (sn) ∆ such that Xm = , X = and for each n N, gm,n gn as m in ∈ ∈ sn sn ∈ → → ∞ Γ. h i h i Definition 2.2. [9, ∆-convergence] We say that X ∆ X as m in B, if there exist g Γ and m → → ∞ n ∈ gm⋆sn (sn) ∆ such that Xm X = s and gm 0 as m in Γ. ∈ − h n i → → ∞ The major difference between a Boehmian space and a G-Boehmian space is that ⋆ should be com- mutative on S for a Boehmian space, which is not required for a G-Boehmian space. 3. Fractional Convolutions and Fractional Hartley Transform

In this section, we introduce a new type of fractional convolution #α, using which, we establish all the results required for constructing the fractional Hartley transformable G-Boehmian space Bα. Definition 3.1. For f,g L 1(R), we define the convolution # as follows: ∈ α 2 cα 2iaαy 2iaαxy 2iaαxy (f#αg)(x)= g(y)e [f(x + y)e + f(x y)e− ]dy, x R. √2π Z − ∀ ∈ R

1 We first point out that #α is not commutative on L (R). Indeed, we can give a pair of functions similar to that given in [10, Example 3.7].

L 1 R 2 L 1 R Lemma 3.2. If f,g ( ), then f#αg 1 cα π f 1 g 1 and hence f#αg ( ). ∈ k k ≤ | |q k k k k ∈ Proof. By using Fubini’s theorem, we obtain

2 cα 2ia y 2ia xy 2ia xy | | α α α f#αg 1 = √ g(y)e [f(x + y)e + f(x y)e− ]dy dx k k 2π R R − R R 2 cα 2ia y 2ia xy 2ia xy | | α α α √ g(y)e [f(x + y)e + f(x y)e− ] dy dx ≤ 2π R R | − | R R cα 2ia xy 2ia xy | | α α √ g(y) f(x + y)e + f(x y)e− dx dy ≤ 2π R | | R | − | R R cα | | √ g(y) f(x + y) dx + f(x y) dx dy ≤ 2π R | | R | | R | − |  R R R cα | | √ g(y) f(z) dz + f(z) dz dy ≤ 2π R | | R | | R | |  R R R 2 cα π f 1 g 1 ≤ | |q k k k k and hence f# g L 1(R).  α ∈ Lemma 3.3. If f,g,h L 1(R), then (f# g)# h = f# (g# h). ∈ α α α α 2 Proof. For a fixed α R, let k = cα . For x R, we obtain that ∈ α 2π ∈ 2 cα 2iaαy 2iaαxy 2iaαxy [f#α(g#αh)](x) = √ (g#αh)(y)e [f(x + y)e + f(x y)e− ]dy 2π R − R 2 2 2iaαz 2iaαyz 2iaαyz 2iaαy = kα h(z)e [g(y + z)e + g(y z)e− ]dz e R R −  R R 2iaαxy 2iaαxy [f(x + y)e + f(x y)e− ]dy 2 − 2 2iaαz 2iaα(y +xy+yz) = kα h(z)e f(x + y)g(y + z)e dy R R R R 2 2iaα(y +xy yz) + f(x + y)g(y z)e − dy R − R 2 2iaα(y xy+yz) + f(x y)g(y + z)e − dy R − R 2 2iaα(y xy yz) +f(x y)g(y z)e − − dy dz (by Fubini’s theorem) R − −  R 4 R. Roopkumar and C. Ganeasn

2 2 2iaαz 2iaα[(u z) +x(u z)+(u z)z] = kα h(z)e [f(x + u z)g(u)e − − − du R R − R R 2 2iaα[(u+z) +x(u+z) (u+z)z] + f(x + u + z)g(u)e − du R R 2 2iaα[(u z) x(u z)+(u z)z] + f(x + z u)g(u)e − − − − du R − R 2 2iaα[(z+u) x(z+u) (u+z)z] + f(x z u)g(u)e − − du dz − − } R 2 2 2iaαz 2iaα[u uz+xu xz] = kα h(z)e [f(x + u z)g(u)e − − du R R − R R 2 + f(x + z + u)g(u)e2iaα[u +uz+xu+xz]du R R 2 2iaα[u uz xu+xz] + f(x + z u)g(u)e − − du R − R 2 2iaα[u +uz xu xz] + f(x z u)g(u)e − − du dz R − − } R 2 2 = cα h(z)e2iaαz cα g(u)e2iaαu f(x + z + u)e2iaα(x+z)udu √2π  √2π R R 2 cα 2iaαu 2iaα(x+z)u 2iaαxz + √ g(u)e f(x + z u)e− du e 2π R −  R 2 cα 2iaαu 2iaα(x z)u + √ g(u)e f(x z + u)e − du  2π R − R 2 cα 2iaαu 2iaα(x z)u 2iaαxz + √ g(u)e f(x z u)e− − du e− dz 2π R − −   R 2 cα 2iaαz 2iaαxz 2iaαxz = √ h(z)e (f#αg)(x + z)e (f#αg)(x z)e− dz 2π R − − R   = [(f#αg)#αh](x). Since x R is arbitrary, the proof follows.  ∈ Lemma 3.4. If f,g,h L 1(R), then (f# g)# h = (f# h)# g. ∈ α α α α Proof. From the proof of Lemma 3.3, we have

2 2 [(f# g)# h](x) = cα h(z)e2iaαz cα g(u)e2iaαu f(x + z + u)e2iaα(x+z)udu α α √2π  √2π R R 2 cα 2iaαu 2iaα(x+z)u 2iaαxz + g(u)e f(x + z u)e− du e √2π −  R 2 cα 2iaαu 2iaα(x z)u + g(u)e f(x z + u)e − du  √2π − R 2 cα 2iaαu 2iaα(x z)u 2iaαxz + √ g(u)e f(x z u)e− − du e− dz 2π R − −   R 2 2 = cα g(u)e2iaαu cα h(z)e2iaαz f(x + u + z)e2iaα(x+u)zdz √2π  √2π R R 2 cα 2iaαz 2iaα(x+u)z 2iaαxz + √ h(z)e f(x + u z)e− dz e 2π R −  R 2 cα 2iaαz 2iaα(x u)z + √ h(z)e f(x u + z)e − dz  2π R − R 2 cα 2iaαz 2iaα(x u)z 2iaαxz + √ h(z)e f(x u z)e− − dz e− du 2π R − −   R (applying Fubini’s theorem) 2 cα 2iaαu 2iaαxu 2iaαxu = √ g(u)e [(f#αh)(x + u)e + (f#αh)(x u)e− ]du 2π R − R = [(f#αh)#αg](x). Since x R is arbitrary, the proof follows.  ∈ FrHT on G-Boehmian Space 5

1 1 Lemma 3.5. If fn f as n in L (R) and if g L (R), then fn#αg f#αg as n in L 1(R). → → ∞ ∈ → → ∞

Proof. From the proof of Lemma 3.2, we have the estimate

2 (f f)# g c f f g . (3.1) k n − α k1 ≤ | α|rπ k n − k1k k1

1 Since fn f as n in L (R), the right hand side of (3.1) tends to zero as n . Hence the lemma follows.→ → ∞ → ∞ 

2 1 iaα(y 2xy) Lemma 3.6. If f L (R) and if ξ(y)= f(x y)e ± f(x) dx, y R, then lim ξ(y)=0. ∈ R | ± − | ∀ ∈ y 0 R → 1 Proof. Since Cc(R) is dense in L (R), we find g Cc(R) such that f g 1 < ǫ. Using the continuity of the mapping y g from R in to L 1(R), (see∈ [28, Theorem 9.5]),k we− choosek 0 <δ<ǫ such that 7→ y g g <ǫ whenever s < δ, (3.2) k s − k1 | | where gy(x) = g(x y), x R. Let K be the compact support of g and C = sup x . Applying − ∀ ∈ x K | | 2 ∈ iaα(y 2xy) mean-value theorem, for the function y e ± on y <δ, for each fixed x K, we have 7→ | | ∈ 2 iaα(y 2xy) e ± 1 y 2y 2x 2 (δ + C) y . | − | ≤ | || ± |≤ | | For y <δ, we get that | | 2 iaα(y 2xy) ξ(y) = f(x y)e ± f(x) dx R | ± − | R 2 2 2 iaα(y 2xy) iaα(y 2xy) iaα(y 2xy) = f(x y)e ± g(x y)e ± + g(x y)e ± R | ± − ± ± R g(x y)+ g(x y) g(x)+ g(x) f(x) dx 2 − ± ± − − ia|α(y 2xy) f(x y) g(x y) dx + g(x y) e ± 1 dx ≤ R | ± − ± | R | ± | | − | R R + g(x y) g(x) dx + g(x) f(x) dx R | ± − | R | − | R R 2 iaα(y 2xy) = 2 f g 1 + g y g 1 + g(x y) e ± 1 dx k − k k ∓ − k | ± | | − | KR < 3ǫ + g y 2 (δ + C) dx k k∞ | | KR < 3ǫ + g 2 (δ + C) m(K)ǫ. k k∞ where m(K) is the Lebesgue measure of the compact set K. This completes the proof of the lemma. 

1 Definition 3.7. The collection of all sequences (δn) from L (R), satisfying the conditions

2 2 iaαt N ∆1 : cα π e δn(t) dt =1 n ; R ∀ ∈ q R ∆ : δ (t) dt M, n N, for some M > 0; 2 | n | ≤ ∀ ∈ R ∆ : supp δ 0 as n , where supp δ is the support of δ ; 3 n → → ∞ n n is denoted by ∆α.

Lemma 3.8. If (δ ), (ψ ) ∆α then (δ # ψ ) ∆α. n n ∈ n α n ∈ 6 R. Roopkumar and C. Ganeasn

Proof. Let (δ ), (ψ ) ∆α. By using Fubini’s theorem, we get that n n ∈

2 2 2 iaαy cα iaαy 2iaαu 2iaαyu 2iaαyu e [δn#αψn](y) dy = √ e ψn(u)e [δn(y + u)e + δn(y u)e− ]dudy R 2π R R − R R R 2 2 cα 2iaαu iaαy 2iaαyu 2iaαyu = √ ψn(u)e e [δn(y + u)e + δn(y u)e− ]dydu 2π R R − R 2 R 2 2 cα iaαu iaα(y +u +2yu) = √ ψn(u)e e δn(y + u)dy 2π R R R 2 2 R iaα(y +u 2yu) + e − δn(y u)dydu R − R 2 2 cα iaαu iaα(y+u) = √ ψn(u)e e δn(y + u)dy 2π R R R 2 R iaα(y u) + e − δn(y u)dydu R − R 2 2 = 2cα ψ (u)eiaαu eiaαs δ (s)ds du √2π n n R R 1 2 2cα 2 − iaαu α = √ cα π ψn(u)e du, (by condition ∆2 of ∆ ) 2π  q  R 1 R 1 1 2cα 2 − 2 − 2 − = √ cα π cα π = cα π . 2π  q   q   q 

α Next, by Lemma 3.2 and property (∆2) of ∆ , we get that

2 2 δ # ψ c δ ψ < c P P , n N, k n α nk1 ≤ | α|rπ k nk1k nk1 | α|rπ 1 2 ∀ ∈ where P1 > 0 and P2 > 0 are such that

δn(t) (t) dt P1 and ψ (t) (t) dt P2, n N. Z | | ≤ Z | n | ≤ ∀ ∈ R R

N ǫ ǫ For a given ǫ> 0, we choose N such that supp δn, supp ψn ( 2 , 2 ) for all n N. Using the fact that ∈ ⊂ − ≥

supp (δ # ψ ) [supp δ + supp ψ ] [ supp δ supp ψ ], n α n ⊂ n n ∪ n − n

ǫ ǫ ǫ ǫ we get that supp (δn#αψn) ( 2 , 2 )+( 2 , 2 ) = ( ǫ,ǫ), for all n N. Hence it follows that (δn#αψn) ∆α. ⊂ − − − ≥ ∈

Theorem 3.9. Let f L 1(R) and let (ϕ ) ∆α, then f# ϕ f as n in L 1(R). ∈ n ∈ α n → → ∞

Proof. Let ǫ> 0 be given. By the property (∆ )of(ϕ ), there exists M > 0 with ϕ (t) dt M, n 2 n | n | ≤ ∀ ∈ R N. Using Lemma 3.6, choose δ > 0 such that

2 iaα(y 2xy) ǫ f(x y)e ± f(x) dx < (3.3) Z | ± − | 2 R

N whenever y < δ. By the property (∆3) of (ϕn), there exists N with supp ϕn [ δ,δ], n N. For x R,| we| have ∈ ⊂ − ∀ ≥ ∈ FrHT on G-Boehmian Space 7

2 cα 2iaαy 2iaαxy 2iaαxy (f#αϕn)(x) f(x) = √ ϕn(y)e [f(x + y)e + f(x y)e− ]dy f(x) − 2π R − − R 2 cα 2iaαy 2iaαxy 2iaαxy = √ ϕn(y)e [f(x + y)e + f(x y)e− ]dy 2π R − R 2 2 iaαy cα π f(x)e ϕn(y)dy − q R R 2 2 2 cα iaαy iaα(y +2xy) iaα(y 2xy) = √ ϕn(y)e [f(x + y)e + f(x y)e − ]dy 2π R − R 2 2 f(x)eiaαy ϕ (y)dy − n R δ 2 2 = cα ϕ (y)eiaαy [f(x + y)eiaα(y +2xy) f(x) √2π n δ − −R 2 iaα(y 2xy) +f(x y)e − f(x)]dy. − − This implies that for each n N, ≥

f#αϕn f 1 = (f#αϕn)(x) f(x) dx k − k R | − | R δ 2 cα ia (y +2xy) | | ϕ (y) f(x + y)e α f(x) dx √2π n ≤ δ | | R | − | −R R 2 iaα(y 2xy) + f(x y)e − f(x) dxdy R | − − | R δ cα ǫ ǫ | | ϕ (y) + dy, by using equation (3.3) ≤ √2π | n |{ 2 2 } Rδ Mǫ c−α | | , by using the property (∆ ) of (ϕ ). ≤ √2π 3 n Since ǫ> 0 is arbitrary, it now follows that f# ϕ f as n in L 1(R).  α n → → ∞ Lemma 3.10. If f f as n in L 1(R) and (ϕ ) ∆α, then f # ϕ f as n in L 1(R). n → → ∞ n ∈ n α n → → ∞ Proof. For any n N, using Lemma 3.2, we have ∈ f # ϕ f = f # ϕ f# ϕ + f# ϕ f k n α n − k1 k n α n − α n α n − k1 (f f)# ϕ + f# ϕ f ≤ k n − α nk1 k α n − k1 2 c f f ϕ + f# ϕ f , ≤ | α|r π k n − k1 k nk1 k α n − k1 2 M c f f + f# ϕ f . ≤ | α|rπ k n − k1 k α n − k1

1 Since fn f as n in L (R) and by Theorem 3.9, the right hand side of the last inequality tends to zero as→n .→ Hence ∞ the lemma follows.  → ∞ 1 1 α Thus the G-Boehmian space Bα = B(L (R), L (R), #α, ∆ ) has been constructed. 4. Fractional Hartley Transform on a G-Boehmian space

In order to extend the frht to the Boehmian space Bα, we have to first obtain a suitable convolution 1 theorem for fractional Hartley transform. For this purpose, we introduce the function Cα on L (R), defined as

∞ 2 2 iaαx [Cα(f)](t)= cα f(x)e cos(bαxt) dx, t R. (4.1) rπ Z ∀ ∈ −∞ As the limits of the integration varies over the entire real line, this function Cα differs from the usual fractional Fourier cosine transform. 8 R. Roopkumar and C. Ganeasn

Theorem 4.1 (Convolution theorem). If f,g L 1(R), then H (f# g)= H (f) C (g). ∈ α α α · α

2 R cα Proof. Let t be arbitrary and kα = 2π be define as in the proof of Lemma 3.3. By using Fubini’s theorem, we obtain∈ that

2 2 H 2 iaα(x +u ) [ α(f#αg)](u) = cα π (f#αg)(x)e Cas (bαxu)dx q R R 2 2 2 iaα(x +u ) 2iaαy 2iaαxy = 2kα e Cas (bαxu) g(y)e [f(x + y)e R R R 2iaαxy R +f(x y)e− ]dydx 2 2 2 − iaα(y +u ) iaα(x+y) = 2kα g(y)e Cas (bαxu)[f(x + y)e R R R 2 R iaα(x y) +f(x y)e − ]dxdy 2 2 2 − iaα(y +u ) iaαz = 2kα g(y)e Cas (bα(z y)u)f(z)e dz R R − R R 2 iaαz + Cas (bα(z + y)u)f(z)e dzdy R R 2 2 iaα(y +u ) = 2kα g(y)e [ Cas (bα(z y)u) R R − R R 2 iaαz + Cas (bα(z + y)u)]f(z)e dzdy 2 2 iaα(y +u ) = 4kα g(y)e [cos(bαzu) cos(bαyu) R R R R 2 iaαz + sin(bαzu) cos(bαyu)]f(z)e dzdy 2 2 2 iaα(y +u ) iaαz = 4kα g(y)e Cas (bαzu) cos(bαyu)f(z)e dzdy R R R 2 R 2 2 iaαy iaα(z +u ) = 4kα g(y)e cos(bαyu) Cas (bαzu)f(z)e dzdy R R R R 2 2 H iaαy = cα π [ α(f)](u) g(y)e cos(bαyu)dy q R = [H (f)](u) [C (g)](u). α · α Thus we have H (f# g)= H (f) C (g).  α α α · α

Theorem 4.2. If f,g L 1(R), then C (f# g)= C (f) C (g). ∈ α α α · α

Proof. Proof of this theorem is much similar to that of the previous theorem and hence we prefer to omit the details. 

Lemma 4.3. If (ϕ ) ∆α then C (ϕ ) 1 as n uniformly on each compact subset of R. n ∈ α n → → ∞ Proof. Let ǫ > 0 and a compact subset K of R be given. Choose M > 0, M > 0 and N N such 1 2 ∈ that ∞ ϕ (t) dt M , n N, K [ M ,M ] and supp ϕ [ ǫ, ǫ] for all n N. For u K and | n | ≤ 1 ∀ ∈ ⊂ − 2 2 n ⊂ − ≥ ∈ n NR−∞, we have ≥

2 [Cα(ϕ )](u) 1 cα ϕ (s) cos(bαus) 1 ds | n − | ≤| |rπ Z | n | | − | R ǫ 2 = cα ϕ (s) cos(bαus) 1 ds, n N | |rπ Z | n | | − | ∀ ≥ ǫ − FrHT on G-Boehmian Space 9

ǫ

ϕ (s) bα us sinz ds, ≤ Z | n | | | | || | ǫ − (where 0

bα M2ǫ ϕ (s) ds ≤| | Z | n | ǫ − b M M ǫ. ≤| α| 2 1 This completes the proof. 

fn B Definition 4.4. For β = ϕ α, we define the extended fractional Hartley transform of β by h n i ∈ [H (β)](t) = lim [Hα(fn)](t), (t R). n →∞ ∈ R The above limit exists and is independent of the representative ((fn), (ϕn)) of β. Indeed, for t , choose k such that [C (ϕ )](t) = 0. Then, applying Theorem 4.1, we obtain that ∈ α k 6 H H H H [ α(fn#αϕk)](t) [ α(fk#αϕn)](t) [ α(fk)](t) C [ α(fn)](t)= C = C = C [ α(ϕn)](t). [ α(ϕk)](t) [ α(ϕk)](t) [ α(ϕk)](t)

[Hα(fk )](t) Therefore, using Lemma 4.3, we get [Hα(fn)](t) C , as n uniformly on each compact [ α(ϕk)](t) R → → ∞ N subset of . If ((fn), (ϕn)) ((gn), (ψn)), then fn#αψm = gm#αϕn for all m,n . Again using Theorem 4.1, we get ∼ ∈

[Hα(fk)](t) [Hα(gk)](t) lim [Hα(fn)](t)= = = lim [Hα(gn)](t). n C C n →∞ [ α(ϕk)](t) [ α(ψk)](t) →∞

L 1 R f#αϕn If f ( ) and β = ϕ , then ∈ h n i [H (β)](t) = lim [Hα(f#αϕ )](t) = [Hα(f)](t) lim [Cα(ϕ )](t) = [Hα(f)](t), n n n n →∞ →∞ C R as [ α(ϕn)](t) 1 as n uniformly on each compact subset of . This shows that the extended fractional Hartley→ transform→ ∞ is consistent with the frht on L 1(R). Theorem 4.5. If β B , then the extended fractional Hartley transform H (β) C(R). ∈ α ∈ Proof. As H (β) is the uniform limit of Hα(fn) on each compact subset of R and each Hα(fn) is a continuous function on R, H (β) is a continuous{ function} on R. 

Theorem 4.6. The extended fractional Hartley transform H : B C(R) is linear. α → 1 Proof. Proof is straightforward from the convolution theorem and the linearity of Hα on L (R). 

Theorem 4.7. The extended fractional Hartley transform H : B C(R) is one-to-one. α → fn B H R R N Proof. Let β = ϕ α be such that [ (β)](s)=0, s . For t , choose k with h n i ∈ ∀ ∈ ∈ ∈ [C (ϕ )](t) =0. Then α k 6 0 = [Cα(ϕ )](t) [H (β)](t) = [Cα(ϕ )](t) lim [Hα(fn)](t) = [Hα(fk)](t). k k n · · →∞ H C H C H N Using [ α(fm)](t)[ α(ϕk)](t) = [ α(fk)](t)[ α(ϕm)](t), it follows that α(fm)=0, m . Since frht is injective, we get f = 0 in L 1(R), for every m N. Thus β =0, proving the result.∀ ∈  m ∈

Theorem 4.8. The extended fractional Hartley transform H : Bα C(R) is continuous with respect to δ-convergence and ∆-convergence. → Proof. As the proof of this result is routine, we prefer not to give the details.  10 R. Roopkumar and C. Ganeasn

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R. Roopkumar, Department of Mathematics, Central University of Tamil Nadu, Thiruvarur-610005, India. E-mail address: [email protected] and

C. Ganesan, Department of Mathematics, V. H. N. Senthikumara Nadar College (Autonomous), Virudhunagar-626001, India. E-mail address: [email protected]