THE STIELTJES TRANSFORM OF DISTRIBUTIONS

In the present work, two complex inversion formulas of Byrne and Love for generalized Stieltjes transformation are shown to be valid for a class of distributions. This is accomplished by transfering the complex inversion formulas on the testing function space of a class of distributions and then showing that the limiting process in the resulting formula converges in the topology of the testing function space.

U.N. TIWARI and J. N. PANDEY for all s in the "cut plane", that is all complex numbers except those which are negative real or zero, the StleltJes Transform in its general form is defined by: (,+)P The follorl inversion theorems for particular values of p and s are well known.
THEKEEM A (Widder).If f(t) belongs to L(O,R) for every positive R and is such that he integral o :+x converges for x > O, then F(s) exists for complex s in the cut plane and llm f (-f -i) -F(-+i) +)+f ( -) 2i 2 for any positive at which f(+) and f(-) both exist.THBOREM B (Smer).If p > 0, f(t) is locally integrable in [0,=], the improper Lebesgue integral.o (+t) converges (for a certain value of s in the cut plane and so for all), t > 0 and the limits f(t_+0) exist, then: t . [(,:+o)+(,:-o)] li,.j' d ' (z)P'lF'(z) where C is a contour in the cut plane from -x-i to -x+i.
2. THE TESTING FUNCTION SPACE, S(1) AND ITS DUAL.
An infinitely dlfferentlable complex valued function (x) defined over I (0,) belongs to the testing function spaces S(1) if, sup .l+x_ where is a fixed real number.Clearly, S(I) is a vector space with respect to the field of complex numbers.The zero element of the vector space S(1) is the function defined over I which is identically zero.The topology over S(1) is generated by the collection of semlnorms }= [24; p. 8].We say that a sequence | where belongs to converges in S(1) to (x) if for each fixed k, yk( I) tends to zero as tends to .The space S(I) is a locally convex Hausdorff tologlcal vector space.The space B(1) is a vector subspace of S(1) and the topology of D(I) is stronger than the topology induced on D(1) by S(I) and as such the restriction of any member of S(1) to D(1) is in D'(1), where S(1) and D'(I) denote the dual spaces of S(I) and D(I) respectively.We say that a sequence 1 where (x) belongs to S(I) is a Cauchy sequence in S(I) if yk( -) goes to zero for any non-negative integer k as and both tend to infinity independently of each other.It can be readily seen that S(1) is sequentlally complete.
3. THE D.I.STRIB,UIONAL S,TIELT,J,E,S .TNSFORMA.TION 1 For a complex s not negative or zero.belongs to S' where a < Rep.
(s+x)P Therefore, the distributional Stieltes transformation F(s) of an arbitrary element f S', a < Re p, is defined by = (s+x)P where s belongs to the complex plane cut along the negative real axis including the origin.
THEOREM 3.1.If m and k both assume non-negatlve integral values and is a compact set of the complex plane not meeting the negative real axis, then for fixed non-negative integers m and k, there exists a constant B satisfying uniformly for all s lying in the compact set of the complex plane not meeting the negative real axis on the origin.

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PROOF.Using the compactness of the set and the fact that S'; (s+x)e +k < Re p, the theorem is immediate.
THEOREM 3.2.For an arbitrary f S' and a < Re p let F(s) be defined by the equation (3.1).Then, for m I, p+m where (p)m p(p+l)(p+2) (p+m-1).
PROOF.If p is such that s p does not have a branch cut in the complex plane, the proof can be given in a way similar to that given in [3, Lemma 2a].If p is such that s p has a branch cut (along the negative real axis for the sake of deflnlteness), then choose the contour of integration as shown in the complex- plane cut along the negative real axis below.where L is the length of the contour C and B is the uniform botmd of c (p)k(l+t)a'tk (z+t) +k for all z lying on the closed contour Camd t > 0.

COI,LE INVERSION OREMS
We are now ready to prove our first inversion theorem.
THEOREM 4.1.For a fixed < 1 and Re p > I, let f S'(1) and let F(s) be the StleltJes transform of f(t) as defined by (3.1).Ten, (y+ -) For fixed x and t, Since, in view of Theorem 3.2, F(s) is analytic in the cut plane, the left-hand side integral in (4.1) is meaningful.By using the technique of Riemann sums it can be shown that for > 0, [by Lenna 5,1, p. 333] cp -i   One can "easily check that as 0+, (-x+ -i)P -I 0 in S (I) for Re p > I > and for fixed %, and x.Therefore, letting 0, we get: In view of Lemma 3.5* [7, p. 12], it follows that: as % for fixed x and . (4.3) Therefore, letting k in (4.2), we obtain * The proof was provided by Professor E.R. Love. (4.4) Using a similar argument, we can show that f (x+t)P -2F (t+i)dt < f(y), p.--y-x+i > -x Combining Equations (4.4) and (4.5), we get 2i J (x+t)P Now using the technique of Riemann sums, we obtain (4.6) where the support of (x) D(1) Is contained in (a,b), b > a > 0. Usi e same techniques as followed in proving Theorem 2 of (3) one can show that b a (y -x) 2.
Thus, the proof of the lemma is complete.
LEMMA 4.3.Let Re p > I > .Assume that t, x, I and are all positive numbers and (t) D(1).Then, I as 0+ in the topology of S where the support of (t) E D(1) is contained in (a,b); b > a > 0.
It can be easily shewn that uniformly for all satisfying 0 < < I. So, (l+x)xmDxm(x,) 0 as x (R)   uniformly for all (0,i).Therefore, for > 0 there exists N > 0 such that (4.I) uniformly for all x >N, and 0 < < I. Now consider the case 0 < x <_ .We will first give the proof for m 0 and complete the proof for m I, 2, 3, by using the result for the case m= 0. ]: (x,,Q) ]:2 (x,) First, consider I l(x,).a (x-t)2+ In view of Lemma 4.2, the rlght-hand side converges to 0 uniformly for all x > 0 as 0+.
The case m 1, 2, 3, A careful computation along with integration by parts will show that Using the technique of induction, we obtain: uniformly for all x > 0 and each fixed m O, I, 2, since is arbitrary our claim is established.THEOREM 4.5.For a fixed < I < Re p, let f(t) S'(I) and let F(s) be the StleltJes transform of f(t) defined by (3.1).Then, llm <p-I 2i J (x+t)P'2[F(t'i)'F(t+i)]dt' {(x) >= <f,{> for all 6 D(1) and A > 0.
PROOF.The result follows quite easily in view of Theorems 4.1 and 4.4.
ACKNOWLEDGEMENTS.The authors are grateful to Professor E.R. Love, for his valuable suggestions, in particular for providing the proof of Lemma 4.3.The work of the second author was supported by a National Research Council Grant Number A52 98.

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: arcs of two concentric circles with centre at origin and C is the contour of integration as shown, The radii of C 1 and C 2 and paths L I and L 2 are so chosen that the point s is contained in the region bounded by the contour.d Let d inf l-sl and choose IAsl < NOW F(s+as)-F(s).< (t), p z-s-s z-s (z.s)2 j dz > where C is the contour shown in the diagram < f(t), 8As >

THEOREM 3 . 3 .
The function F (m)(x) for real x where F(s) is the StleltJes transform of f S', satisfies the following relation: -i)-(+t)]dt, (x) > < , >This completes the proof of the theorem.

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llT err X'm Pl/ A Re p Therefore we can find a constant B(p) independent of x and such that Z(x,)l <_ B (+x) b j.

THEOREM 4 . 4 .
Let Re p > I > and f(t) S'.If F(s) is the StieltJes transform of f(t) defined by (3.1) then for > 0 and each im <P-2i f(y-i)-F(y+i)] (y+t)P-2dy, (t) > 0 X PROOF.By using the same technique as used in proving Theorem 4.1, it can be shown that where the support of (t) is contained in (a,b), b > a > 0, < f(), a > (t)dt. (By using Riemann's sum technique) Letting 0+, the result follows in view ofLemma 4.3.