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251 the Laplace-Stieltjes Transformation On Acta Math. Univ. Comenianae 251 Vol. LXXX, 2 (2011), pp. 251{257 THE LAPLACE-STIELTJES TRANSFORMATION ON ORDERED TOPOLOGICAL VECTOR SPACE OF GENERALIZED FUNCTIONS K. V. GEETHA and N. R. MANGALAMBAL Abstract. We have combined the Laplace transform and the Stieltjes transform of ^ R 1 f(t) the form f(x) = 0 (xm+tm)ρ dt, m; ρ > 0 and applied it to an ordered vector space of generalized functions to which the topology of bounded convergence is assigned. Some of the order properties of the transform and its inverse are studied. Also we solve an initial value problem and compare different solutions of the problem. 1. Introduction In an earlier paper [2] we associated the notion of order to multinormed spaces, countable union spaces and their duals. The topology of bounded convergence was assigned to the dual spaces. Some properties of these spaces were also studied. As illustrative examples the spaces D, E , La;b, L (w; z) and their duals were also studied. m In the present paper an ordered testing function space Ma;b;c is defined and the topology of bounded convergence is assigned to its dual. Some properties of the m m 0 cone in Ma;b;c and in (Ma;b;c) are studied in Section 2. The Laplace-Stieltjes transform and its properties were studied by Geetha K. V. and John J. K. [1]. We apply these results to the new context when the topology of m 0 bounded convergence is assigned to the ordered vector space (Ma;b;c) in Section 3. Some of the order properties of the transform are also studied. In Section 4, the inversion of the transform and the order properties of the inverse are studied. An operational transform formula which can be applied to boundary value prob- lems is established in Section 5. As the solutions are from an ordered vector space, comparison between different solutions is possible. Received December 12, 2010. 2010 Mathematics Subject Classification. Primary 46F10; Secondary 46A04, 46A08. Key words and phrases. Laplace-Stieltjes transformation; positive cone; order-bounded. 252 K. V. GEETHA and N. R. MANGALAMBAL m 2. The test function space Ma;b;c and its dual as ordered vector spaces m Let Ma;b;c denote the linear space of all complex-valued smooth functions φ(u; t) defined on (0; 1)×(0; 1). Let (Km) be a sequence of compact subsets of R+ ×R+ such that K1 ⊆ K2 ⊆ ::: and such that each compact subset of R+ × R+ is contained in one Kj, j = 1; 2;::: . Each Kj defines µ φ(u; t) = sup ecuuq+1Dq tm(1−a+b)(1 + tm)a−b(t1−mDl)φ(u; t) a;b;c;Kj ;q;l u t (u;t)2Kj where a; b; c 2 R, q; l = 0; 1; 2;::: , m 2 (0; 1) and @ @ D = D = : u @u t @t fµm g1 is a multinorm on M m , where M m is a subspace of a;b;c;Kj ;q;l q;l=0 a;b;c;Kj a;b;c;Kj m Ma;b;c consisting of functions with support in Kj. The above multinorm generates the topology τ m on M m . The inductive limit topology τ m as K varies a;b;c;Kj a;b;c;Kj a;b;c j m over all compact sets K1;K2;::: is assigned to Ma;b. With respect to this topology M m is complete. On M m an equivalent multinorm is given by a;b;c a;b;c;Kj m m µ (φ) = sup µ 0 0 (φ): a;b;c;Kj ;q;l a;b;c;Kj ;q ;l 0≤q0≤q 0≤l0≤l m By identifying a positive cone in Ma;b;c, an order relation is defined on it as follows. m m Definition 2.1. The positive cone of Ma;b;c when Ma;b;c is restricted to real- m valued functions is the set of all non-negative functions in Ma;b;c. When the field m of scalars is C, the positive cone in Ma;b;c is C + i C which is also denoted as C. m m We say φ ≤ in Ma;b;c if − φ 2 C in Ma;b;c. m Theorem 2.1. The cone C of Ma;b;c is not normal. m Proof. Suppose that Ma;b;c is restricted to real-valued functions. Let j be a fixed positive integer and let (φ ) be a sequence of functions in C \ (M m ) i a;b;c;Kj such that λ = supfφ (u; t):(u; t) 2 K g converges to 0 for M m . i i j a;b;c;Kj Define ( λi; (u; t) 2 Kj+1 i = 0; (u; t) 62 Kj+1: Let ξi be the regularization of i defined by 1 1 Z Z 0 0 0 0 0 0 ξi(u; t) = θα(u − u ; t − t ) i(u ; t )du dt : 0 0 Then ξ 2 M m , for all i. i a;b;c;Kj+2 In [3, Proposition 1.3, Chapter 1] it was proved that the positive cone in an ordered vector space E(τ) is normal if and only if for any two nets fxβ : β 2 Ig and fyβ : β 2 Ig in E(τ) if 0 ≤ xβ ≤ yβ for all β 2 I and if fyβ : β 2 Ig converges LAPLACE-STIELTJES TRANSFORMATION ON TOPOLOGICAL SPACE 253 to 0 for τ, then fxβ : β 2 Ig converges to 0 for τ. So we conclude that C is not normal for the Schwartz topology. It follows that C + i C is not normal for m Ma;b;c. m Theorem 2.2. The cone C is a strict b-cone in Ma;b;c. m Proof. Let Ma;b;c be restricted to real-valued functions. Let B be the saturated m m m class of all bounded circled subsets of Ma;b;c for τa;b;c. Then Ma;b;c = [B2BBc where Bc = f(B \ C) − (B \ C): B 2 Bg is a fundamental system for B and C m m is a strict b-cone since B is a collection of all τa;b;c-bounded sets in Ma;b;c. m m Let B be a bounded circled subset of Ma;b;c for τa;b;c. Then all functions in B have their support in a compact set Kj0 and there exists a constant M > 0 such that jφ(u; t)j ≤ M, for all φ 2 B,(u; t) 2 Kj0 . Let ( M; (u; t) 2 Kj0+1 (u; t) = 0; (u; t) 62 Kj0+1: The regularization ξ(u; t) of (u; t) is defined by 1 1 Z Z 0 0 0 0 0 0 ξ(u; t) = θα(u − u ; t − t ) (u ; t )du dt 0 0 and ξ has its support in Kj0+2. Also, B ⊆ (B + ξ) \ fξg ⊆ (B + ξ) \ C − (B + ξ) \ C: It follows that C is a strict b-cone. We conclude that C + i C is a strict b-cone. m Order and Topology on the dual of Ma;b;c. m 0 m Let (Ma;b;c) denote the linear space of all linear functionals defined on Ma;b;c. m 0 m 0 An order relation is defined on (Ma;b;c) by identifying the positive cone of (Ma;b;c) 0 m 0 to be the dual cone C of C in Ma;b;c. The class of all B , the polars of B as m m 0 m B varies over all σ(Ma;b;c; (Ma;b;c) )-bounded subsets of Ma;b;c is a neighbour- m 0 m 0 m hood basis of 0 in (Ma;b;c) for the locally convex topology β((Ma;b;c) ;Ma;b;c). m 0 0 When (Ma;b;c) is ordered by the dual cone C and is equipped with the topology m 0 m 0 β((Ma;b;c) ;Ma;b;c) it follows that C is a normal cone since C is a strict b-cone m m m 0 m in Ma;b;c for τa;b;c. Note that β((Ma;b;c) ;Ma;b;c) is the G -topology correspond- m 0 ing to the class C of all complete, bounded, convex, circled subsets of (Ma;b;c) m m 0 for the topology σ(Ma;b;c; (Ma;b;c) ). Anthony L. Perissini [3] observed that the subset L (E; F ), the linear space of all continuous linear mappings of E into F is bounded for the topology of pointwise convergence if and only if it is bounded for the G -topology corresponding to the class of all complete, bounded, convex, circled subsets of E(τ). m + Theorem 2.3. The order dual (Ma;b;c) is the same as the topological dual m 0 m 0 (Ma;b;c) when (Ma;b;c) is assigned the topology of bounded convergence. 254 K. V. GEETHA and N. R. MANGALAMBAL Proof. Every positive linear functional is continuous for the Schwartz topology m m τa;b;c on Ma;b;c. m + m m (Ma;b;c) = C(Ma;b;c; R) − C(Ma;b;c; R) m where C(Ma;b;c; R) is the linear subspace consisting of all non-negative order m m bounded linear functionals on Ma;b;c of L(Ma;b;c; R), the linear space of all or- m der bounded linear functionals on Ma;b;c.
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