The Laplace transform on time scales revisited - ECS - Baylor ...

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1304 J.M. Davis et al. / J. Math. Anal. Appl. 332 (2007) 1291–13073. <strong>The</strong> Dirac delta functionalLet f,g : T → R be given functions with f(x)having unit area. To define the delta functional,we construct the following functional. Let Cc ∞(T) denote the C∞ (T) functions with compactsupport. For g σ ∈ Cc ∞(T) and for all ɛ>0, define the functional F : C∞ c (T, R) × T → R byF ( g σ ,a ) { ∫ ∞0δa H =(x)gρ (σ (x)) x, if a is right scattered,∫ ∞ 1lim ɛ→0 0 ɛf ( )xɛg(σ(x))x, if a is right dense.<strong>The</strong> time scale Dirac delta functional is then given by〈δa (x), g σ 〉 = F ( g σ ,a ) .We need to see how the delta functional acts on functions from Cc ∞ (T). For this purpose, letg : T → R be given such that g σ ∈ Cc ∞ (T). Ifa is right dense, consider{ 1ɛ, if a x a + ɛ,f(x)=0, else,with the understanding that any sequence of ɛ’s we choose will be under the restriction thata + ɛ ∈ T for each ɛ in the sequence. <strong>The</strong>n for h(x) = g σ (x), we have (for any antiderivativeH(x) of h(x)),F ( g σ ,a ) ∫= lim f(x)h(x)xɛ→0∞If a is right scattered, thenF ( g σ ,a ) =0∫ a+ɛah(x) x= limɛ→0 ɛH(a+ ɛ) − H(a)= limɛ→0 ɛ= H (a) = h(a) = g ( σ(a) ) = g(a).∫ ∞0δ H a (x)gρ( σ(x) ) x = g ρ( σ(a) ) = g(a).Thus, in functional terms, the Dirac delta functional acts as〈g σ ,δ a〉= g(a),independently of the time scale involved. Also, if g(t) = e ⊖z (t, 0), then 〈δ a ,g σ 〉=e ⊖z (a, 0), sothat for a = 0, the Dirac delta functional δ 0 (t) has <strong>Laplace</strong> <strong>transform</strong> of 1, thereby producingan identity element for the convolution. However, the present definition of the delay operatoronly holds for functions. It is necessary to extend this definition for the Dirac delta functional.To maintain consistency with the delta function’s action on g(t) = e⊖z σ (t, 0), it follows that forany t ∈ T, theshift of δ a (τ) is given by δ a (t, σ (τ)) := δ t (σ (τ)). With this definition, our claimholds:
J.M. Davis et al. / J. Math. Anal. Appl. 332 (2007) 1291–1307 1305(δ 0 ∗ g)(t 0 ) =∫ t 00= g(t 0 , 0)= g(t 0 )==∫ t 00∫ t 00δ 0 (τ)g ( t,σ(τ) ) τg(τ)δ t0(σ(τ))τg(τ)δ 0(t0 ,σ(τ) ) τ= (g ∗ δ 0 )(t 0 ).In other words, when we perform the convolution 〈g,δ σ 〉, we must give meaning to this symboland do so by defining δ σ (t) to be the Hilger delta when t is right scattered and the Diracdelta if t is right dense. While this ad hoc approach does not address convolution with an arbitrary(shifted) distribution on the right, this will suffice (at least for now) since our eye is onsolving generalizations of canonical partial dynamic equations which will involve the Dirac deltadistribution.We now turn to uniqueness of the inverse of the Dirac delta. We would like an analogue ofthe diagram given in the proof of <strong>The</strong>orem 1.4 where we move from the space of continuousfunctions to its dual space (or at least restricted to the dual space of those functions that areinfinitely differentiable with compact support). As frequently happens when we move from aspace to its dual space, the diagram also becomes the dual of the original one. Indeed, the diagramin terms of the dual space is shown in Fig. 4.<strong>The</strong> mappings γ and γ −1 in the dual diagram act on the <strong>transform</strong> spaces just as in Fig. 3.It is worth comparing Figs. 3 and 4 since X ⊂ X ∗ . So how do we reconcile the two diagrams?We first must clarify what is meant by the identity map between the two dual spaces. <strong>The</strong> Diracdelta is defined in terms of its action on any function: in both dual spaces, 〈δ a ,g σ 〉=g(a).To make this agree with the map for the function spaces, note that for f ∈ C prd-e2 (T, R) andg ∈ C p-eo (R, R), the action of f(t) on h σ (t) = e⊖z σ (t, 0) should be the same as the action ofg(t) on ˜h(t) = e −zt . For example, the function on T that acts on h σ 1(t) with a result ofz 2 +1 isthe function f(t)= sin 1 (t, 0), while the function on R that acts on ˜h(t) with the same result isg(t) = sin(t). As another example, the function on T that acts on h σ (t) with a result of h(τ)zisthe time scale unit step function, while on R, the function that results in ˜h(τ)zis the continuousstep function. Thus, in terms of the dual spaces, the map on the left hand side of the diagram is(C ∞ c (R, R))∗ L R (C ∞ cL −1RT f (R)γ −1I I γ L T (T, R))∗ T f (T)L −1T= I ◦ L−1 R ◦ γ −1Fig. 4. Commutative diagram in the dual spaces.
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