NOTES FOR HARTLEY TRANSFORMS OF GENERALIZED FUNCTIONS. S.K.Q. Al-Omari. 1. Introduction

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1 italian journal of pure and applied mathematics n (21 30) 21 NOTES FOR HARTLEY TRANSFORMS OF GENERALIZED FUNCTIONS S.K.Q. Al-Omari Department of Applied Sciences Faculty of Engineering Technology Al-Balqa Applied University Amman Jordan Abstract. The classical Hartley transform, originally introduced by Hartley as a real transform with a number of properties being similar to the properties of Fourier transform. In this work, we extend the Hartley transform to certain space of distributions of compact support. Further, we establish that the Hartley transform and its inverse are one to one and onto mappings in the space of Boehmians. Moreover, continuity with respect to δ and convergence is discussed in some detail. Certain theorems are also proved. Keywords: Hartly transform; Boehmian space; convolution; smooth function; distribution of compact support Mathematics Subject Classification: Primary 54C40, 14E20; Secondary 46E25, 20C Introduction The Hartley transform is a spectral transform closely related to the Fourier transforms. It contain the same information that the Fourier transform does, and no advantage accrues in its use for complex signals. However, for real signal, the Hartley transform is real and this can offer computational advantages in signal processing applications that traditionally make use of Fourier transforms 15. Moreover, the Hartley transform can be analytically continued into the complex plane, and for real functions it is Hermitian symmetry on reflection in the real axis. The Hartley transform in the complex plane is an entire function of exponential type with zeros close to the real axis, a property shared with Fourier transform 15, p.p The Hartley transform H (υ), of f (x), is defined by (1) H (υ) = f (x) cas (2πxυ) dx, where R cas (2πυx) = cos (2πυx) + sin (2πυx).

2 22 s.k.q. al-omari Therefore, it follows that the inverse Hartley transform is given by (2) f (x) = H (υ) cas (2πxυ) dυ. R The scalling and linearity conditions of the Hartley transform have been described in 16. Theorems for the Hartley transform, analogous to those for the Fourier transform, can be easily derived from definitions. The more complicated, compared to the Fourier transform, is the convolution theorem which can sometimes amount to a disadvantage of the Hartley transform. For functions f and g, L 1 functions, the convolution theorem of the Hartley is defined by (3) H (f g) (υ) = 1 G ( Hg) (υ), 2 where (4) G (f g) (υ) = f (υ) g (υ) + f (υ) g ( υ) + f ( υ) g (υ) f ( υ) g ( υ). and is the usual convolution product. 2. Boehmian spaces The general construction of Boehmians is algebraic in nature. Boehmians were first constructed as a generalization of regular Mikusinski operators 5. The minimal structure necessary for the construction of Boehmians consists of the following elements: (i) A nonempty set X and a commutative semigroup (Y, ) ; (ii) An operation : X Y X such that for each x X and s 1, s 2, Y, x (s 1 s 2 ) = (x s 1 ) s 2 ; (iii) A collection Y N such that: a) If x, y X, (s n ), x s n = y s n for all n, then x = y; b) If (s n ), (t n ), then (s n t n ). Elements of are called delta sequences. Let Ϝ = {(x n, s n ) : x n X, s n, x n s m = x m s n, m, n N}. Consider (x n, s n ), (y n, t n ) Ϝ, x n t m = y m s n, m, n N, then we say (x n, s n ) (y n, t n ). The relation is an equivalence relation in Ϝ. The space of equivalence clases in Ϝ is denoted by β. Elements of β are called Boehmians. A typical element of β is written as x n sn. Between X and β there is a canonical embedding expressed as x x sn s n. The operation can be extended to β X by xn s n types of convergence: t = xn t s n. In β, two

3 notes for hartley transforms of generalized functions 23 Type 1: A sequence (h n ) in β is said to be δ convergent to h in β, denoted δ by h n h, if there exists a delta sequence (s n ) such that (h n s n ), (h s n ) X, k, n N, and (h n s k ) (h s k ) as n,in X, for every k N. Type 2: A sequence (h n ) in β is said to be convergent to h in β, denoted by h n h, if there exists a (s n ) such that (h n h) s n X, n N, and (h n h) s n 0 as n in X. For more details, see 1-5, 7-9, 11, The Boehmian Space M (E, D,, ) By D (R), we denote the space of test functions of compact support and E (R) the space of all infinitely smooth functions on R equipped with the sequence of multinorms ξ k (f) = sup f (k) (x) for every f E (R), where K run through x K compact subsets of R. The kernel function of the Hartley transform, cas2πxυ, is certainly a member of E (R) for arbitrary but fixed υ. This justifies the extension of the Hartley transform to the context of distributions through the formula (υ) = f (x), cas2πυx for each f E (R), the strong dual of E (R) of distriutions of compact support 10, 19. Let be the family of sequences ( ) from D (R) such that (5) (x) dx = 1, for every n N. (6) (7) R R (x) dx M, for some positive M. supp (x) ( ε n, ε n ), ε n 0 as n. Members of are named as delta sequences. Lemma 3.1. Let f E (R) and φ D (R) then f φ E (R). Lemma 3.2. Given f 1, f 2 E (R) and φ D (R) then for every α C, (f 1 + f 2 ) φ = f 1 φ + f 2 φ and α (f 1 φ) = (αf 1 ) φ = f 1 (αφ). Lemma 3.3. Let f in E (R) and φ D (R) then φ f φ. Proofs of Lemmas are straightforward from the properties of the integral operator. Lemma 3.4. Let f in E (R) as n and φ D (R) then φ f as n. Proof. If f in E (R) then from Eq. (5) we get D k ( φ f) (x) Mξ k ( (x t) f (x)) 0 as n. Hence ξ k ( φ f) 0 as n. The lemma is completely proved.

4 24 s.k.q. al-omari The Boehmian Space M (E, D,, ) is constructed. The sum of two Boehmians and multiplication by a scalar in M (E, D,, ) can be defined in a natural f way n g + n f ψ n = n ψ n +g n f ψ n and α n = α, α C. The operation and g n g n ψ n D α the differentiation are defined by ψ n = and D α =. The relationship between the notion of convergence and the product is given by: 1 If f as n in E (R) and, φ D (R) is fixed, then φ f φ in E (R), as n ; 2 If f as n in E (R) and (δ n ), then δ n f in E (R), as n. The operation can be extended to M (E, D,, ) D in the sense that If /δ n M (E, D,, ) and φ D (R),then φ = φ δ n. Convergence in M (E, D,, ), is defined as follows: A sequence of (β n ) in M (E, D,, ) is said to be δ convergent to a Boehmian β in M (E, D,, ), de- δ noted by β n β, if there exists a delta sequence (δ n ) such that β n δ n, β δ n E (R), k, n N, and β n δ k β δ k as n, in E (R), for every k N. δ This can be interpreted to mean: β n β (n ) in M (E, D,, ) if and only fn,k f if there is,k, f k E (R) and (δ k ) such that β n =, β = k and for each k N,,k f k as n in E (R). It is more often convenient to use another kind of convergence: A sequence (β n ) in M (E, D,, ) is said to be convergent to a β in M (E, D,, ), denoted by β n β, if there exists a (δ n ) such that (β n β) δ n E (R), n N, and (β n β) δ n 0 as n in E (R). δ k δk 4. The Space M H ( E, D H, H, ) To extend the Hartley transform to Boehmians we describe another space of Boehmians as follows. First, it will be necessary to know that, if ( ) then (8) H (υ) 1 as n. and that (9) H ( υ) 1 as n. uniformly on compact subsets. Let a mapping between f and Hφ be defined by (10) (f Hφ) (υ) = 1 G (f Hφ) (υ). 2 G has the usual meaning in (4). Denote by D H (R), the set of all Hartley transform of functions from D (R). Define H = {H :, n N }. We prove the following Lemma 4.1. Let f E (R), Hφ D H (R) then (f Hφ) (υ) E (R).

5 notes for hartley transforms of generalized functions 25 Proof. Let k N then for each f E (R) and Hφ D H E (R) we have Hφ ( υ) f (±υ) E (R). If K is a compact subset of R containing supp φ then by using Eq. (10) and Eq. (4) we get (11) sup D k υ (f Hφ) (υ) <. υ K Allowing K traverses the set of real numbers yields f Hφ E (R). This completes the proof of the lemma. Lemma 4.2. A mapping E D H E defined by (12) (f, Hφ) f Hφ satisfies the following (i) If Hφ, Hψ D H (R) then (Hφ Hψ) (υ) D H (R). (ii) If f, g E (R), Hφ D H (R) then ((f + g) Hφ) (υ) = (f Hφ) (υ) + (g Hψ) (υ). (iii) (Hφ Hψ) (υ) = (Hψ Hφ) (υ), Hφ, Hψ D H (R). (iv) If f E (R), Hφ, Hψ D H (R) then (f Hφ) Hψ = f (Hφ Hψ). Proof. (i) Let υ R then it is clear that (Hφ Hψ) (υ) = H (φ ψ) (υ). But Lemma 3.1 implies φ ψ D (R). Hence Hφ Hψ D H (R). (ii) is obvious. (iii) Since φ ψ D (R), φ ψ = ψ φ. Applying the Hartley transform yields H (ψ φ) (υ) = H (φ ψ) (υ). This implies Hφ Hψ = Hψ Hφ. (iv) can be easily established by routine calculation from Eq. (4) and Eq. (10). The lemma is completely proved. Lemma 4.3. Let f 1, f 2 E (R), H H, n, f 1 H = f 2 H, n, then f 1 = f 2 in E (R). Proof. From hypothesis f 1 H = f 2 H, n. Invoking Eq. (8) and Eq. (9) in Eq. (10), yields f 1 (υ) = f 2 (υ), υ R. Thus f = g. This completes the proof of the lemma. Lemma 4.4. (1) If f in E (R) as n and Hφ D H (R) then Hφ f Hφ as n. (2) If f in E (R) as n and H H then H f as n in E (R).

6 26 s.k.q. al-omari The proof of the above lemma is a result of Eq. (8) and Eq. (9). Lemma 4.5. Let (H ) 1, (Hξ n) 1 H then (H Hξ n ) 1 H. Proof. Let ( ), (ξ n ) then ( ξ n ). Thus, ( ξ n ) is the sequence such that (H Hξ n ) (υ) = H ( ξ n ) (υ) H. Hence the lemma is completely proved. Further, it can be observed that H Hξ n 1 as n, for every ( ), (ξ n ). Hence, H ( satisfies the necessary conditions for delta sequences. The Boehmian space M H E, D H, H, ), or M H, is constructed. Addition, scalar multiplication, differentiation, convolution and convergence can be defined in a natural way. For some detail, the sum and multiplication by a scalar ( on M H E, D H, H, ) is defined as n Hg H + n Hψ n = n Hψ n+hg n H H Hψ n and α n H = α n H, α C. The operation and the differentiation are defined by n Hg H n Hψ n = n Hg n H Hψ n and D α H D H = α H H. Convergence on ( M H E, D H, H, ) is defined by: ( A sequence of (Hβ n ) in M H E, D H, H, ) is said to be δ convergent to a δ Boehmian Hβ, denoted by Hβ n Hβ, if there exists a delta sequence (Hδ n ) such that Hβ n Hδ n, Hβ Hδ n E (R), k, n N, and Hβ n Hδ k Hβ Hδ k as δ n,in E (R), for every k N. This can be interpreted to mean: Hβ n Hβ (n ) if and only if there is H,k, k E (R) and (Hδ k ) H such that n,k Hβ n = Hδ k, Hβ = k Hδ k and for each k N, H,k k as n in E (R). It is more often convenient to use another kind of convergence: A sequence (Hβ n ) M H (E, D H, H, ) is said to be convergent to a Hβ M H (E, D H, H, ), denoted by Hβ n Hβ, if there exists a (Hδ n ) H such that (Hβ n Hβ) Hδ n E(R), n N, and (Hβ n Hβ) Hδ n 0 as n in E(R). Theorem 4.6. The mapping (13) ( E M H E, D H, H, ) f Hγ f n H ( is a continuous imbedding of E (R) into M H E, D H, H, ) with respect to δ convergence. f Hγ Proof. To show the mapping is one to one let n g Ht H = n Ht n. Then (f H ) Ht m = (g Ht m ) H. For large values of m and n, Ht m, H 1. The above equation is therefore reduced to f = g. To establish continuity of Eq. (13) with respect to δ convergence,

7 notes for hartley transforms of generalized functions 27 let 0 as n then H 0 as n. Hence n. The theorem is completely proved. H H 0 as 5. Hartley transform of Boehmians Let M (E, D,, ) then, in view of analysis established in Section 4, we define the extended Hartley transform by fn n (14) κ =. H in M ( E, D H, H, ). Theorem 5.1. κ : M (E, D,, ) M H is well-defined. g n Proof. Let = then t m = g m. Employing the Hartley transform and using Eq. (10) we get H Ht m = Hg m H. Therefore n Hgn =. H Ht n t n The theorem is completely proved. Theorem 5.2. κ:m (E, D,, ) M H is one to one. g n Proof. Let κ = κ then H Ht m = Hg m H. Using the fact that the classical Hartley transform is one to one and upon employing Eq. (10) we get f t m = g m. Therefore n g = n t n. Hence the theorem. t n Theorem 5.3. κ:m (E, D,, ) M H is continuous with respect to δ convergence. fn,i γ i Proof. Let x n 0 in M (E, D,, ) as n, then using 7, x n = for some,i where,i 0 as n. Applying the Hartley transform yields H,i 0 as n. Thus κx n 0 as n. This completes the proof. Theorem 5.4. κ:m (E, D,, ) M H is linear. Proof of this theorem is straightforward. Definition 5.5. Let n H M H then we define the inverse generalized Hartley transform to be the mapping (15) κ 1 n fn = H in M (E, D,, ).

8 28 s.k.q. al-omari Theorem 5.6. κ 1 : M H M (E, D,, ) is a well-defined and linear mapping. The proof is analoguous to that of Theorems 5.1 and 5.4, and thus avoided. Theorem 5.7. κ 1 : M H M (E, D,, ) is one to one. Theorem 5.8. κ 1 : M H M (E, D,, ) is continuous with respect to δ convergence. The proof of Theorems 5.7 and 5.8 are analoguous to that of Theorem 5.2 and Theorem 5.3, respectively. Details are avoided. Theorem 5.9. The mapping κ:m (E, D,, ) M H is surjective. Proof. Let n H M H be arbitrary, then H Hγ m = m H for every m, n N. Using Eq. (10), H ( γ m ) = H (f m ), for every m, n N. Hence the Boehmian M (E, D,, ) satisfies the equation κ complete the proof of the lemma. = H H. This Theorem κ:m (E, D,, ) M H, κ 1 : M H M (E, D,, ) are continuous with respect to convergence. Proof. Let x n x in M (E, D,, ) as n. Then, there is E (R) f and ( ) such that (x n x) = n and 0 as n. Employing the Hartley transform implies κ ((x n x) ) = H H H(fn γn) H. Hence κ ((x n x) ) = H H 0 as n in M H. Therefore κx n κx as n. Next, let y n y in M H as n, then we find F k E (R) such F that (y n y) = n H H and F n 0 as n for some ( ) and F n = H. Next, applying Eq. (10), H κ 1 1 (F n H ) ((y n y) H ) =. Thus κ 1 f ((y n y) H ) = n 0 as n in E (R). Thus κ 1 ((y n y) H ) = (κ 1 y n κ 1 y) 0 as n. Hence, κ 1 y n κ 1 y as n in M (E, D,, ). This completes the proof of the theorem. References

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