ABELIAN THEOREMS FOR TRANSFORMABLE BOEHMIANS

A class of generalized functions called transformable Boehmians contains a proper subspace that can be identified with the class of Laplace transformable distributions. In this note, we establish some Abelian theorems for transformable Boehmians.


INTRODUCTION AND PRELIMINARIES
The Laplace transform has recently been extended to a class of generalized functions called transformable Boehmians [1].The object of this note is to present some Abelian theorems of the initial type for transformable Boehmians.Such theorems relate the behavior of a transformable Boehmian at zero to the behavior of its transform at irtfinity.
Our notation is the same as that used in [1].Let C be a subset of the real line tL The space of all continuous complex-valued functions on will be denoted by C(C).Throughout this note it will be assumed that if Q=(a,b) then a<0 and b>0.The space of all functions feC(R) such that f(t)=0 for t<0 will be denoted by C+(R).The support of a continuous function f, denoted by supp f, is the complement of the largest open set on which f is zero.
A pair of sequences (fn,6n) is called a quotient of sequences if fneC+(R) for n=1,2 {6n} is a delta sequence, and fk*6m fm*6k for all k and m.Two quotients of sequences (fn,6n) and (gn,q0n) are said to be equivalent if fk*q0m=gm*6k for all k and m.A straightforward calculation shows that this is an equivalence relation.The equivalence classes are called Boehmians.The space of all Boehmians will be denoted by f3 and a fn typical element of will be written as x 6 n By defining addition, multiplication, and scalar multiplication, on f3, i.e. fn gn (fn* q0 n + gn* gn) fn gn fn*gn and o.
where o is a 6 n 6n qn 6n n n n n complex number, 3 becomes a convolution algebra.

n
Since the Boehmian nn corresponds to the Dirac delta distribution, we denote it by 6. (n Moreover, the n th derivative of 6 is given by the formula Dn6 6 (n) where {6 n} 6 k is any infinitely differentiable delta sequence.In general, the n th derivative of xef3 is given by Dnx=x* 6 (n).
By the translation operator on C+(R), we mean the operator "r d' d real, such that fn (' d0(t) f(t-d).The translation operator can be extended to an element x nn f3 by "rdfn "dX= 6 n DEFINITION 1.1.Let Ct be an open subset of R. A Boehmian x is said to be equal to a continuous function f on C, denoted by x=f on O, if there exists a' delta sequence {6 n such that X*6nC(R for all n and X*6n-.funiformly on compact subsets of Q as n-, oo.
Two Boehmians x and y are said to be equal on an open set C, denoted by x=y on O, if x-y=0 on O.
The support of x3, written supp x, is the complement of the largest open set on which x is zero.For example, given any delta sequence {6 n} and E>0, 6n(t)-.0uniformly for tl>E as n-* oo.Thus, supp Dn6 {0} for n=0,1,2 fn For each x nn 3, it is not difficult to show that for each n supp fn G supp x + supp n For other results concerning Boehmians see [1], [2], and [3]. (1.1)

TRANSFORMABLE BOEHMIANS
A Boehmian x is said to be transformable if there exists a delta sequence {6n and a normegative number cx such that x*6neC+(R for all n and x*6 n O(e t) as t-.oo for all n.The space of all transformable Boehmians will be denoted by 3 L.
If feC+(R) such that f(t) O(eCt) as t-.oo for some real number , then the Laplace transform of f is given by Throughout this note s will denote a complex variable, while or c will denote real variables.
Now, for x3 L where x* 6nC+(R) and x*6 n O(ect) as t-.oo for all n, the Laplace It can be shown [1] that the space of transformable distributions L+ [4] is a proper subspace of 3 L We state without proof the following theorem.THEOREM 2.1.For x,ye3 L, if X,(s) and LJ(s) are the Laplace transforms of x and y respectively, then:     The next theorem will be needed in the proof of an Abelian theorem (Theorem 3.2) in the next section.Also, since the Laplace transform of a Boehmian is an analytic function in some half-plane [1], Theorem 2.2 gives a necessary condition for an analytic function to be the Laplace transform of a Boehmian.THEOREM 2.2.Let xe3 L. For each k and E>0, fn PROOF.Letx =nn e 3L" We may assume that fn6C(R) for all n.For, if {q;n is an fn* q;n infinitely differentiable delta sequence that is, q; n6C (R) for all n), then x and 6n*q n fn* q; n6C (R) for all n.
Assume that supp 6 n c_ [0,an] for all n, where an-.0 as n-.oo.Now, by the Mean Value Theorem for Integrals, for each n there exists an (which depends on n and c) such that 0<<a n and L[6n](C f0e-Ct6n(t)dt e -C 06n(t)dt e -(3 > e-an (3 Hence, given an E>0, we may pick m such that 0<a m<E.Then, (2.1) 3) The proof is completed by observing that (2.3) is valid for all 8>0 and all nonnegative integers k.THEOREM 2.3.Suppose that F(s) is an analytic function in some half-plane Re s > and for some integer k and all >0 that ske-Sp(s) O(1) as s-, oo, Re s > c.Then, there exists x3 L such that L[x](s)=F(s), Re s > PROOF.Suppose that for some integer k and all E>0 ske-SF(s) O(1) as s-, oo, Re s > cx.
Re s > cx.For n=1,2,..., define fn(t) -.,yThen, for each n, fn is a continuous function such that: supp fn c [0,oo), fn(t) O(eY t) as too, and L[fn](S L[q0n](S)F(s), Re s > Now, L [fn*q0 ml L fn L [fro*q0 n] for all n and m.Thus, fn*q0 m fm *q0 n for all n and m.Let x f3 L. q0 n Hence L [x](s) F(s), Re s > Although the condition in the previous theorem is sufficient for an analytic function to be the Laplace transform of a Boehmian, as the next example demonstrates it is not necessary.(2ni! has Laplace transform REMARK 2.5.It is not difficult to show that the transform of a Boehmian is an analytic function in some half-plane [1].Hence, Theorem 2.2 provides a necessary condition for an analytic function to be the Laplace transform of some transformable Boehmian.Thus, the entire function g(s) e s is not the transform of a Boehmian.But, for each c>1 there is a transformable Boehmian xct such that Xc (s) is an entire function and for each E>0, Xo(s)=O(exp s (E +(1/c55), as Isl [1] (where this relation does not hold for any E<0).An interesting open problem is to characterize the class of transformable Boehmians by their Laplace transforms.

INITIAL VALUE THEOREMS
In classical analysis there are many different types of Abelian theorems (see [5] and [6]).
Abelian theorems of the final type relate the behavior of a function at infinity to the behavior of its transform at zero, while Abelian theorems of the initial type relate the behavior of a function at zero to the behavior of its transform at infinity.It is both interesting and important to extend such theorems to certain classes of generalized functions (see [4], [7], [8], and [9]).For example, Zemanian [4] has extended two Abelian theorems to transformable distributions.In [1] we presented an Abelian theorem of the final type for transformable Boehmians.In this section we will establish three Abelian theorems of the initial type (Theorems 3.2, 3.5, and 3.8). )is said to be in Y, if m does not change sign in [a,b] and [L[m](o)] -1 o(oneo(O)) as (3--, (for some integer n).DEFINITION 3.1.x,yef3 L. x--y as t-*0 + if there exist f,geC+[a,b], meY,, and an integer n such that x=Dnf and y=Dng on some neighborhood of [a,b], Lim f(t) t.,0 + m(t---1, and Lim g(t) t_,0 + m(t) 1. THEOREM 3.2.Suppose x,y13 L such that x--y as t-,0+.Then Lim ,(o) 1.
o4oo y(o) PROOF.x can be written in the form x Dnf + w, (3.1)where WL and supp w c [c,o) (c>0).Now, by a classical Abelian theorem (see [5]), Since y can also be written in the form of (3.1), to complete the proof it suffices to show  fn Now, w nn and by (1.1) supp fn c [c,oo) for all n.For each n, let gn e C(R) be defined gn by "cgn=fn and let z nn e f3 L. Thus w='o.z.Then, (for some y>0), By applying Theorem 2.2 to (3.3) we obtain (3.2), and hence the proof is complete.EXAMPLE 3.3.Let xe3 L such that x=6 on some neighborhood of [a,b].Since D2t=6, x-6 as t-*0 + and hence Lim X(o) 1.
REMARK 3.4.In Definition 3.1, the condition that the functions f, g, and m be continuous may be relaxed.We need only require that f,g,mLl(a,b).0+ cI [a,b] and 7 cx as t-* (cx complex and X real, X >-1), then o-*oo

F(X+I) =cx
Lim (where F(X+I) I o e -t t X dt).
(3.5) Now, for some positive constants M and a Is X-n+lL['dy](s) <M Re s k-n+le-aRe s, arg s <;<n/2.
For if fLl(a,b) and x=Dnf in some neighborhood of [a,b], then x=Dn+l(*f) in that neighborhood, where .is