World Academy of Science, Engineering and Technology International Journal of Mathematical and Computational Sciences Vol:2, No:3, 2008

Laplace Transformation on Ordered Linear Space of Generalized Functions K. V. Geetha and N. R. Mangalambal

Abstract—Aim. We have introduced the notion of order to multi- Let La,b denote the linear space of all complex valued normed spaces and countable union spaces and their duals. The smooth functions defined on R. Let topology of bounded convergence is assigned to the dual spaces. The aim of this paper is to develop the theory of ordered topological linear L (t)=eat, 0 ≤ t<∞ a,b spaces La,b, L (w, z), the dual spaces of ordered multinormed spaces = ebt, −∞

International Science Index, Mathematical and Computational Sciences Vol:2, No:3, 2008 waset.org/Publication/13650 L L( ) In this section we define the spaces a,b, w, z and apply fundamental system for G. A strict G-cone for the the class G the above ideas to these spaces and to their duals. of all τ-bounded sets in V (τ) is called a strict b-cone. Definition 4: The positive cone C of La,b when La,b is Manuscript received March 1, 2008. restricted to real-valued functions is the set of all non-negative K. V. Geetha is with the Department of Mathematics, St. Joseph’s L + L College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 0480- functions in a,b. Then C iC is the positive cone in a,b 2825358, Res. 0480-2824613, cell. 9447994135, fax. 0480-2830954, email. which is also denoted as C. [email protected]) Now we prove that the cone of L is not normal but is a N. R. Mangalambal is with the Department of Mathematics, St. Joseph’s a,b College, Irinjalakuda, Kerala, India. Pin.680 121 (Phone: Off. 0480-2825358, strict b-cone. Res. 0480-2709858, cell. 9495246832, fax. 0480-2830954, email. thot- Theorem 1: The cone C of La,b is not normal. [email protected]) Proof: Let La,b be restricted to real-valued functions. Let m be a fixed positive integer and (φi) be a sequence of

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L ∩ = sup{ ( ): ∈ } functions in a,b,Km C such that λi φi t t Km Proof: Every positive linear functional is continuous for ( ) L L converges to 0 but φi does not converge to 0 for a,b,Km . the Schwartz topology τa,b on a,b. Define L+ = (L R) − (L R) a,b C a,b, C a,b, ψi = λi,t∈ Km+1 where C(La,b, R) is the linear subspace consisting of all non- =0,t∈ K +1. m negative order bounded linear functionals of L(La,b, R), the L Let ξ be the regularization of ψ defined by linear space of all order bounded, linear functionals on a,b. i i L+ ⊆L  It follows that a,b a,b.    L ( )= ( − ) ( ) On the other hand, the space a,b equipped with the ξi t θα t t ψi t dt .  R (L L ) topology of bounded convergence β a,b, a,b and ordered ∈L ∀ C C L Then ξi a,b,Km+2 , i. by the dual cone of the cone in a,b is a reflexive space Also, 0 ≤ φi ≤ ξi, ∀i and (ξi) converges to 0 for τa,b.In ordered by a closed normal cone. Hence if D is a directed Proposition 1.3, Chapter 1, [2], it has been proved that the set (≤) of generalized functions that is either majorized in L (L L ) sup positive cone in an ordered linear space E(τ) is normal if and a,b or contains a β a,b, a,b -bounded section, then D L F( ) only if for any two nets {xβ : β ∈ I} and {yβ : β ∈ I} exists in a,b and the filter D of sections of D converges to sup (L L ) L+ = L in E(τ) if 0 ≤ xβ ≤ yβ for all β ∈ I and if {yβ : β ∈ I} D for β a,b, a,b . Hence a,b a,b and we conclude L (L L ) converges to 0 for τ then {xβ : β ∈ I} converges to 0 for τ.So that a,b with respect to the topology β a,b, a,b is both we conclude that C is not normal for the Schwartz topology. and topologically complete. (See Proposition It follows that C + iC is not normal for La,b, we conclude 1.8, Chapter 4, [2]). ≤ ≤ ∈ that the cone of La,b is not normal. Theorem 4: Let a c, d b. The restriction of any f L L L L L Theorem 2: The cone C is a strict b-cone in La,b. a,b to c,d is c,d when a,b, c,d are assigned the topology Proof: Assume that La,b is restricted to real-valued of bounded convergence. functions. Let B be the saturated class of all bounded cir- Proof: When a ≤ c, d ≤ b, Lc,d ⊆La,b. The topology L L cled subsets of La,b for τa,b. Then La,b = ∪B∈BBc where τc,d of c,d is stronger than the topology induced on c,d by L ∈L Bc = {(B ∩ C) − (B ∩ C):B ∈ B} is a fundamental system a,b. By Zemanian [3], the restriction of any f a,b to L L L L for B and C is a strict b-cone since B is the class of all τ- c,d is in c,d when c,d and a,b are assigned the topology bounded sets in La,b. of pointwise convergence. It follows that the above result is L L Let B be a bounded circled subset of La,b for τa,b. Then all true when c,d and a,b are assigned the topology of bounded

functions in B have their support in some compact set Km0 convergence by Theorem 2.15, [1]. L and there exists a constant M>0 such that |φ(t)|≤M, Theorem 5: If a

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Restricting D to the real-valued functions an order relation is Theorem 9: If T : U → V is linear and continuous its defined on D by identifying the positive cone to be the set adjoint T  : V  → U  is also linear and continuous where of all non-negative functions on D. The cone C + iC is the U, V are ordered multinormed spaces or ordered countable positive cone of D. union spaces. Definition 7: Let E be the linear space of all complex- Proof: For φ, ψ ∈ C(U) and α, β any two complex n valued smooth functions defined on R and (Km) a sequence numbers, we have Rn ⊆ ⊆ of compact subsets of such that K1 K2 ... and such    n (T f)(αφ + βψ)=α(T f)(φ)+β(T f)(ψ) that each compact subset of R is contained in some Km. Let γ (φ) = sup |Dkφ(t)| φ ∈E k =0, 1, 2,...  Km,k t∈Km , , . so that T f is a linear functional on U. {γ }∞ is a multinorm on E and E is equipped with the Km,k k=0 Let (φα)α∈J be a net in C(U) which converges to 0. topology generated by it. Since T : U → V is continuous, T (φ ) → 0 as α →∞. Note D E α For details on , , refer [1], [3]. (T f)(φ )=f(T (φ )) → 0 as α →∞since f ∈ U . D⊆L ⊆E D L D α α Result 1: a,b and is not dense in a,b but Thus T f is a continuous linear functional on U. i.e. T  is L( ) D E is dense in w, z for every w, z. is dense in also. It a mapping on V  to U . L E follows that a,b is dense in (refer [3]). Also, T (αf + βg)(φ)=(αT (f)+βT(g))(φ), f,g ∈ L( ) Equipped with the Schwartz topology w, z is complete C(V ), φ ∈ C(U), α, β any two complex numbers. So, =01 2 since it is a countable union space. For k , , ,... T  : V  → U  k −st is linear. t e ∈L(w, z) if and only if w0 φ>0 T (φ) > 0 T II. LINEAR MAPS ON ORDERED MULTINORMED SPACES, Proof: Let .For , since is f(T (φ)) > 0 ⇒ (T f)(φ) > 0 φ>0 ORDERED COUNTABLE UNION SPACES AND THEIR strictly positive. for . f>0 ⇒ T f>0 T  ADJOINTS Thus , is strictly positive. It follows that T  is positive and order bounded. In this section we study the properties of linear maps defined on ordered multinormed spaces, ordered countable union spaces and the adjoints of these maps defined on their A linear partial differential operator and its adjoint on gen- duals when the dual spaces are assigned the topology of eralized functions. n n bounded convergence. Also, we apply these results to some Let I be an open set in R or C and let t =(t1,...tn) ∈ I. L L( ) linear maps on a,b, w, z and their adjoints. Let θi(t), i =0, 1,...,mbe complex valued smooth functions Definition 8: Let U, V be ordered multinormed spaces or on I. Consider the linear partial differential operator ordered countable union spaces with positive cones C(U), =(−1)|k| k1 k2 km C(V ) respectively. A linear map T θ0D θ1D ...θm−1D θm

T : U → V where ki denote non-negative integers and |k| = k1 +···+km. T denotes a sequence of operators: multiply by θm, differenti- is km ate according to D , multiply by θm−1 etc. Let U(I), V (I) International Science Index, Mathematical and Computational Sciences Vol:2, No:3, 2008 waset.org/Publication/13650 ( ( )) ⊆ ( ) ( ) ≥ 0 (i) positive if T C U C V , i.e., T φ whenever be testing function spaces on I. Let T  denote the adjoint of 0 ∈ φ> , φ U. T  (ii) strictly positive if T (φ) > 0 whenever φ>0. (T f)(φ)=f(T (φ)) = f(t)T (φ(t))dt. T U (iii) order bounded if maps each order bounded set in I into an order bounded set in V . By successive integration by parts this becomes Note. Every strictly positive linear map is positive and every  positive linear map is order bounded. km k1 φ(θmD ...D θ0f)dt. Definition 9: Let U, V be ordered multinormed spaces or I T : U → V  ordered countable union spaces and let be continu- Thus T may be identified as the operator ous and linear. T  : V  → U  defined by (T f)(φ)=f(T (φ)),  km k1 f ∈ V , φ ∈ U is the adjoint of T . θmD ...D θ0

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where D denotes the conventional differentiation. When T  acts upon f ∈ V (I), D denotes the generalized differentia- Theorem 14: Let λ be a fixed real number. For every a, ( ) → ( − ) (L ) tion. b, w, z, f t f t λ from C a,b to itself is linear, Theorem 11: The generalized differentiation is a continu- continuous, strictly positive and order bounded. Its inverse is (L ) (L( )) ous linear mapping of C a,b into itself and of C w, z continuous and order bounded. Hence the map is an order L L( ) L into itself and hence of a,b and w, z when the topology bounded automorphism on a,b. The map is an order bounded of bounded convergence is assigned to these dual spaces. automorphism on L(w, z). Proof: By the definition of seminorms γk, γk(−Dφ)= Proof: φ(t) → φ(t + λ) is linear, continuous and strictly γk+1(φ). By Lemma 1.10.1, [3], (−D) is a continuous linear positive on La,b and hence on L(w, z). The adjoint of this L ( ) → ( + ) (L ) mapping of a,b into itself. By Theorem 9 its adjoint operator map f t f t λ on C a,b is continuous by Theorem D which is a generalized differentiation is a continuous 9 and the map is strictly positive and hence order bounded by L (− ) linear mapping of a,b into itself. It follows that D is Theorem 2.4. It follows that the map is an order bounded L( ) L L( ) a continuous linear mapping of w, z into itself and that automorphism on a,b. From the definition of w, z it its adjoint operator D, the generalized differentiation is a follows that the map is an orderbounded automorphism on continuous linear mapping of L(w, z) into itself. L(w, z) Definition 10: Let S be a linear space of smooth functions defined as follows: θ(t) ∈ S if and only if it is smooth on III. THE TWO-SIDED LAPLACE TRANSFORMATION: (−∞, ∞) and for each non-negative integer k there exists an DEFINITION AND SOME BASIC PROPERTIES. integer Nk such that, In this section we introduce the two-sided Laplace Transfor- (1 + t2)−Nk Dk(θ(t)) mation and discuss some properties of the transformation and derive results which are used in solving differential equations. is bounded on (−∞, ∞). Definition 11: Let d(f) be a linear space of conventional Theorem 12: For θ ∈ S the mapping f → θf is a functions and f be a linear functional defined on d(f) which   continuous linear mapping of C (L (w, z)) into itself and satisfies the following properties L( ) hence of w, z into itself. (i) f is additive i.e. f(φ + ψ)=f(φ)+f(ψ), φ, ψ ∈ d(f) ∈ (L ) ∈ R Proof: Let φ C c,d for some c, d and let (ii) L ⊆ d(f) for at least one pair of real numbers a, b ∈ R ∈ a,b a, b such that a . isomorphism from a,b onto a−r,b−r. L( ) (L ) L From the definition of w, z , corresponding results follow f can be extended to C ci,di and to ci,di as follows: L( ) L( − − ) ∈ (L ) in the case of the map from w, z onto w r, z r . Let ψ C ci,di .

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Define f(ψ)=f(λψ)+f((1 − λ)ψ). Result 4: For a fixed complex number α if Lf = F (s), (L ) L f is continuous and linear on C ci,di and hence on ci,di . −αt L(e f(t)) = F (s + α),s+ α ∈ Ωf . Using the same methods f may be extended to L(σ1,σ2). This extension of f is unique. Result 5: If Lf = F (s), for a fixed real number β Definition 13: Let f be a Laplace transformable general- L(f(t − β)) = e−sβF (s),s∈ Ω . ized function, σ1 = inf ∧f , σ2 = sup ∧f , f

Ω = {s : σ1 < Re s<σ2}. f IV. INVERSION AND UNIQUENESS The Laplace transformation F of f is defined as a conventional The results on inversion of the Laplace transform and Ω ( )=(L )( )= ( ) −st ∈ Ω function on f by F s f s f t ,e , s f . Uniqueness Theorems proved by Zemanian [3] hold good in ∈ Ω For any fixed s f the right hand side has a meaning as the ordered dual spaces when they are assigned the topology ∈L( ) −st ∈L( ) the application of f σ1,σ2 to e σ1,σ2 . of bounded convergence. In some situations the domain is Theorem 16: If f(t) is a positive locally integrable function ( ) restricted to the positive cone of the respective spaces. We such that f t is absolutely integrable on −∞ 0, β>0. Since Lf>0 for f>0, f ∈ W it follows −∞ that L is strictly positive and hence is order bounded. converges in C(La,b) to φ(τ) and hence in La,b as r →∞. Notation. Theorem 19: Let Lf = F (s) for σ1 <σ<σ2, r be a real d ∂ ∂ D ≡ ,D ≡ ,D ≡ . variable. Then for a fixed real number σ, σ1 <σ<σ2, dt t ∂t s ∂s  1 σ+ir k k st Result 2: L(D f(t)) = s F (s), s ∈ Ωf , k =1, 2,..., f(t) = lim F (s)e ds →∞ 2 Lf = F (s). r πi σ−ir By Theorem 11 the generalized differentiation is a con- where the convergence is with respect to the topology of   tinuous linear mapping of L (σ1,σ2) into itself. For f ∈ bounded convergence in D , the dual of D. (For details see  L (σ1,σ2) [1]). L = ( ) ∈ Ω L = ( ) Dkf(t),e−st = f(t), (−D )ke−st Theorem 20: If f F s for s f and h H s t ∈ Ω Ω ∩ Ω = ( )= ( ) − for s h and if f h φ and if F s H s for = f(t),ske st  s ∈ Ωf ∩ Ωh then f = h in the sense of equality in L (w, z) k = s F (s),s∈ Ωf ,k=1, 2,... where the interval w<σ

k k Result 3: L(t f(t))=(−Ds) F (s), s ∈ Ωf , k = 1, 2, 3, V. O PERATIONAL CALCULUS AND SOLUTIONS OF ..., tk ∈ S for k =1, 2, 3, .... DIFFERENTIAL EQUATIONS Let S be a fixed complex number such that for σ1 < Re s< σ2, The following characterization for the Laplace transform has tf(t),e−st = f(t),te−st been modified to suit the present situation of ordered Laplace transformable generalized functions where we consider only L ( ( )) = − ( ) ∈ Ω i.e., f tf t DsF s ,s f . the elements of the positive cone we have defined. International Science Index, Mathematical and Computational Sciences Vol:2, No:3, 2008 waset.org/Publication/13650 For k =1, 2, 3,... Theorem 21: Let F (s) be a strictly positive function. The necessary and sufficient condition for F (s) to be the Laplace  k ( ) −st =  k−1 ( ) −st t f t ,e t f t ,te transform of a positive generalized function f and for the k−1 −st = −Dst f(t),te  corresponding strip of definition to be k−2 −st = −Dst f(t),te  Ωf = {s : σ1 < Re s<σ2} 2 k−2 −st =(−Ds) t f(t),e  is that F (S) be analytic on Ωf and for each closed substrip = ... {s : a ≤ Re s ≤ b} of Ωf where σ1

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The polynomial will in general depend on the choices of a REFERENCES and b. [1] Geetha K. V. and Mangalambal N. R. : On Dual Spaces of Ordered Corollary 1: Let F (s) be a strictly positive function and let Multinormed Spaces and Countable Union Spaces, Bull. KMA, No.2, Vol. 4, Dec 2007, pp. 63–74. Lf = F (s) for s ∈ Ωf . Choose three fixed real numbers a, ( ) [2] Perissini A. L.: Ordered Topological Vector Spaces, Harper and Row, σ and b such that a<σ

Then in the sense of equality in L(a, b)  1 σ+i∞ ( ) ( )= ( ) F s st F t Q Dt 2 ( )e ds, a < σ < b. πi σ−i∞ Q s  where Dt represents generalized differentiation in L (a, b) and the integral converges in the conventional sense to a continu- ous function that generates a regul;ar member of C(L(a, b)).

Operational Calculus Consider the linear differential equation

n n−1 Lu(t)=(anD + an−1D + ···+ a0)u(t)=g(t)

where the ai’s are constants, an =0and g(t) is a given Laplace transformable generalized function. Applying Laplace transform to both sides,

B(s) ∪ (s)=G(s)

n n−1 n−1 where B(s)=ans + an−1s + s + ···+ a0

∪(s)=Lu

( )=L ∈ Ω = { : } G s g, s g s σg1 < Re s<σg2

If B(s) has no zeros in Ωg then by Theorem 21 there exists ( ) G(s) a generalized function u t whose Laplace transform is B(s) Ω ( ) L( ) on g. u t is a unique member of σg1 ,σg2 and satisfies the given equation. If B(s) has a finite number of zeros in Ωg there exists a set of m adjoint substrips = σg1 σ0 < Re s<σ1,σ1 < Re s<σ2,... σm−1 < Re s< = G(s) σm σg2 on which B(s) is analytic and satisfies the growth condition of Corollary 1. For any given substrip say σi < Re s<σi+1 there exists a unique member u(t) of L(σi,σi+1) G(s) and whose Laplace transform is B(s) on σ1 < Re s<σi+1. For any other choice of the substrip there will be a different solution.

International Science Index, Mathematical and Computational Sciences Vol:2, No:3, 2008 waset.org/Publication/13650 If u(t), v(t) are two solutions of the above differential equation such that u(t) for all t lies to the left of the line say Re s = α and v(t) lies to the right for all t then u ≤ v by the order relation we have defined on the Laplace transformable generalized functions.

ACKNOWLEDGMENT Authors acknowledge gratefully the help and guidance of Dr. T. Thrivikraman, Dr. M. S. Chaudhari and Dr. J. K. John in the preparation of this paper.

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