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Stieltjes Transformation on Ordered Vector Space of Generalized Functions and Abelian and Tauberian Theorems

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Stieltjes Transformation on Ordered Vector Space of Generalized Functions and Abelian and Tauberian Theorems Int. J. Contemp. Math. Sciences, Vol. 3, 2008, no. 33, 1619 - 1628 Stieltjes Transformation on Ordered Vector Space of Generalized Functions and Abelian and Tauberian Theorems K. V. Geetha and N. R. Mangalambal Department of Mathematics, St. Joseph’s College Irinjalakuda, Kerala, India [email protected] [email protected] Abstract. Stieltjes transformation of the form ˆ ∞ f(t) f(x)= 0 (xm+tm)ρ dt, m, ρ > 0 has been studied extensively. We apply this form of the transform 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. The asymptotic be- haviour of the transform and its inverse are also studied. Mathematics Subject Classification: Primary 46F10, Secondary 46A04, 46A08 Keywords: Stieltjes transformation, tempered distributions, Strongasymp- totic behaviour 1. Introduction In an earlier paper [1] we had associated the notion of order to multinormed 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 fuction space Ma,b is defined and the topology of bounded convergence is assigned to its dual. Some properties m m of the cone in Ma,b and in (Ma,b) are studied in section 2. Stieltjes transformation and its properties have been studied by John J. K. [2]. In section 3 we apply some of these results to the new context when the topology of bounded convergence is assigned to the ordered vector space m (Ma,b) . It follows that the Stieltjes transform and its inverse are order bounded. 1620 K. V. Geetha and N. R. Mangalambal Order relation is defined on the space ζ of test functions of rapid descent and its dual ζ, the space of tempered distributions. The topology of bounded convergence is assigned to ζ. The notion of strongasymptotic behaviour is m introduced and the strongasymtotic behaviour of Sρ (f) with reference to the strongasymtotic behaviour of f and vice versa are studied in section 4. m 2. The testing function space Ma,b and its dual m Let Ma,b denote the linear space of all complex-valued smooth functions defined on (0, ∞). Let (Km) be a sequence of compact subsets of R+ such that K1 ⊆ K2 ⊆ ... and such that each compact subset of (0, ∞) is contained in one Kj, j =1, 2,.... On each Kj define m m(1−a+k) m a−b | 1−m d k | μa,b,Kj,k(φ) = sup t (1 + t ) (t ) φ(t) . t∈Kj dt a, b ∈ R, k =0, 1, 2,..., m ∈ (0, ∞). { m }∞ k k m μa,b,Kj,k k=0 is a multinorm on Ma,b,Kj where Ma,b,Kj is a subspace of Ma,b consisting of functions with support contained in Kj. The above multinorm m m m generates the topology τa,b,Kj on Ma,b,Kj . The inductive limit topology τa,b as m Kj varies over all compact sets K1,K2,... is assigned to Ma,b. With respect m to this topology Ma,b is complete. m On each Ma,b,Kj an equivalent multinorm is given by m m (1) μ¯a,b,Kj ,k(φ) = sup μa,b,Kj,k (φ) 0≤k≤k m m Definition 2.1. The positive cone C of Ma,b when Ma,b is restricted to real- m valued functions is the set of all non-negative functions in Ma,b. m Then C + iC is the positive cone in Ma,b which is also denoted as C. m Theorem 2.1. The cone C of Ma,b is not normal. m Proof. Suppose that Ma,b is restricted to real-valued functions. Let j be a fixed ∩ m +ve integer and let (φi) be a sequence of functions in C (Ma,b,Kj ) such that λi = sup{φi(t),t ∈ Kj} converges to 0 but (φi) does not converge to 0 for m Ma,b,Kj . λi,t∈ Kj+1 Define ψi = . 0,t∈ Kj+1 Let ξi be the regularization of ψi defined by ∞ ξi(t)= θα(t − t )ψi(t )dt 0 ∈ m Then ξ Ma,b,Kj+2 for all i. m Also, 0 ≤ φi ≤ ξi and (ξi) converges to 0 for τa,b. In Proposition 1.3, chapter 1, [3], it has been proved that the positive cone in an ordered topological vector space E(τ) is normal if and only if for any two nets {xβ : β ∈ I} and Stieltjes transformation 1621 {yβ : β ∈ I} in E(τ)if0≤ xβ ≤ yβ for all β ∈ I and if {yβ : β ∈ I} converges to 0 for τ then {xβ : β ∈ I} converges to 0 for τ. So we conclude that C is not normal for the Schwartz topology. It follows that C + iC is not normal for m Ma,b m Theorem 2.2. The cone C is a strict b-cone in Ma,b. m Proof. Let Ma,b be restricted to real-valued functions. Let B be the saturated m m class of all bounded circled subsets of Ma,b for τa,b. m ∪ Bc Bc { ∩ − ∩ ∈ B} Then Ma,b = B∈ where = (B C) (B C):B is a fundamental system for B and C is a strict b-cone, since B is a collection of all τ-bounded m sets in Ma,b. m m Let B be a bounded circled subset of Ma,b for τa,b. Then all functions in B have their support in some compact set Kj0 and there exists a constant M>0 such that |φ(t)|≤M, for all φ ∈ B, t ∈ Kj0 . Let M, t ∈ Kj +1 ψ(t)= 0 0,t∈ Kj0+1. Then the regularization ξ of ψ is defined by ∞ ξ(t)= θα(t − t )ψ(t )dt 0 and ξ has its support in Kj0+2. Also, B ⊂ (B + ξ) ∩{ξ} ⊂ (B + ξ) ∩ C − (B + ξ) ∩ C It follows that C is a strict b-cone. We conclude that C + iC is a strict b- cone. m m Order and topology on the dual of Ma,b. Let (Ma,b) denote the linear m space of all linear functionals defined on Ma,b. An order relation is defined on m m (Ma,b) by identifying the positive cone of (Ma,b) to be the dual cone C of C in m 0 m m Ma,b. The class of all B , the polars of B as B varies over all σ(Ma,b, (Ma,b) )- m m bounded subsets of Ma,b is a neighbourhood basis of 0 in (Ma,b) for the locally m m m convex topology β((Ma,b) ,Ma,b). When (Ma,b) is ordered by the dual con C m m and is equipped with the topology β((Ma,b) ,Ma,b) it follows that C is a normal m m m m cone since C is a strict b-cone in Ma,b for τa,b. Note that β((Ma,b) ,Ma,b)is the G-topology corresponding to the class G of all compete, bounded, convex, m m m circled subsets of (Ma,b) for the topology σ(Ma,b, (Ma,b) ). Anthony L. Perissini [3] has observed that a subset L(E,F), the vector 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). 1622 K. V. Geetha and N. R. Mangalambal m + Theorem 2.3. The order dual (Ma,b) is the same as the topological dual m m (Ma,b) when (Ma,b) is equipped with the topology of bounded convergence. Proof. Every positive linear functional is continuous for the Schwartz topology m m τa,b on Ma,b. m + m m (Ma,b) = C(Ma,b,R) − C(Ma,b,R) m where C(Ma,b,R) is the linear subspace consisting of all non-negative order m m bounded linear functionals on Ma,b of L(Ma,b,R), the linear space of all order m m + m bounded linear functionals on Ma,b. It follows that (Ma,b) ⊆ (Ma,b) . m On the other hand, the space (Ma,b) equipped with the topology of bounded m m convergence β((Ma,b) ,Ma,b) and ordered by the dual cone C of the cone C m in Ma,b is a reflexive space ordered by a closed a normal cone. Hence if D is m a directed (≤) set of elements that is either majorized in (Ma,b) or contains m m m a β((Ma,b) ,Ma,b)-bounded section, then sup D exists in (Ma,b) and the filter m m F(D) of sections of D converges to sup D for β((Ma,b) ,Ma,b). m + m Hence (Ma,b) =(Ma,b) . m m m We conclude that (Ma,b) with respect to the topology β((Ma,b) ,Ma,b)is both topologically complete and order complete 3. The Stieltjes transformation In this section we study some properties of the Stieltjes transformation. The inverse of the transformation is also discussed here. m For φ ∈ Mα,β the Stieltjes transform is defined as ∞ m ˆ φ(t) Sρ (φ)=φ(x)= m m ρ dt 0 (x + t ) for a fixed m>0, ρ ≥ 1.
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