THE n-DIMENSIONAL DISTRIBUTIONAL MELLIN TRANSFORMATION

The n-dimensional distributional Mellin transformation is developed using the testing function space Mc,d and its dual M′c,d The standard theorems on analyticity, uniqueness and continuity are proved. A necessary and sufficient condition for a function to be an n-dimensional Mellin transformation is proved by the help of a boundedness property for distribution in M′c,d. Some operational transform formulas are also introduced.

For the sake of brevity, we shall use the following notations.R n and C n are respectively real and complex n-dimensional euclidean spaces.The symbols z and s stand for elements of C n representing the n-triples (Zl,Z 2 ,Zn) and (Sl,S2,...,sn) C n Rn, t Rn, Rn, Rn and s + i A respectively.We take x function on a subset of R n shall be denoted by h(x) h(Xl,X 2 ,Xn).By [x] we mean the product Xl,X2, .,xThus, [  Xn and e {e 1,...,e }.The notation x _< y and x < y mean respectively x -< y and x < y ( 1,2,...,n).The letters k and m shall denote non-negative integers in Rn, i.e., ku and m are non-negative integers.
Ox I

OXn kn
By a smooth function we mean a function that possesses partial derivatives of all orders at all points of its domain.
2. THE TESTING FUNCTION SPACE M cd" Let R denote the open domain 0 < x < oo.We define Nc,d(X) as the product (2.1) Qk denotes constants which depend upon the choices of k and f.
Any smooth function whose support is contained in R, is in M c,d"  The inverse mapping is given by PROOF.The proof of this theorem is easy and is therefore omitted.where _ is the tube of definition of the n-dimensional distributional Laplace transformation (see [3]).
In fact, the R.H.S. of (4.1) has a meaning because the application of h  h(e-p) is n-dimensional Laplace transformable.In such a case, Mh(x) H(x) Lh(e-p) for ever s 5.
Using Theorems 2.1 and 3.1, we can have the following theorems: THEOREM 4.2. (The Analiticity Theorem).If Mh H(s) for s h' then H(s) is analytic on and H s The proof is analogous to that given in [3].
THEOREM 4.3. (The Uniqueness Theorem).If Mh H(s) for s h and Mg G(s) for s g, if n g is non-void, and if H(s) G(s) for s n g, then h=g in the same sense of equality in M'c,d where c,d e h and c < d.The proof is ana- logous to that given in [3].(5.1) 6.A NECESSARY AND SUFFICIENT CONDITION FOR M(s) TO BE AN n-DIMENSIONAL MELLIN TRANSFORM.
A necessary and sufficient condition for a function M(s) to be the n-dimen- sional Mellin transform of a distribution h is that there be a tube c < Re s < d (c < d) on which M(s) is analytic and bounded when IM(s) < where P(Isl) is a polynomial in sl. (6.1) It can be easily proved by using the boundedness property of Section 5 and (Bochner [6], Theorem 60, p. 242 and 4, p. 244).Let us suppose that Mh(p) H(s) for s e h and p R Cn.We can easily have the following operational transform formulas (Using Theorem 4): (ii) M D kp h(p) s k H(s), s e h' ,...,sn and [e exp(-Sltl -'''-snnt d(X) Ix-d if e < x < In fact M is the linear space of all smooth functions f(x) defined on R with c,d values in C I which satisfy the following set of inequalities For each non-negative integer k, lc,d(X) [xk+l] Dk f(x) Qkx 0 < x < oo.
bers of M are Ix s-l] for c -< Re s d and [(log x) k xs-l] for c < Re s < d. c,d v represents a seminorm defi6ed by (f) max sup lc d(X [xk-l] Dk f(x) I .
the collection f} is a multinorm, being a separating collection of seminorms.Thus we can assign to M the topology generated by {} each fixed k, the functions c,d(X)[xk+l] Dkx fv(x.. converges uniformly on n R$ as v oo.Hence, Mc,d is sequentially complete.THEOREM 2.1.The mapping f(x) [e -p] f(e-p) g(p) (2.3)Other mem- is an isomorphism from M into L wlere L denotes the testing function space c,d c,d c,d defined by Sinha [3].

3 . 4 .
THE DUAL SPACE M' c,d" M'is the dual space of M Multiplication by a complex number, equality, c,d c,d" and addition are defined in the usual way.In fact, M' is a linear space over c,d CI.By <h, f> we mean a number that h[4], 1.6) of a distribution h is contained in a compact subset of R, ' c,d with c < d.Also, every member of M' is a distribution on R+. c,d Let us define a (weak) topology for M' by using the following separating c,d set of seminorms.For every f Mc,d, we define a seminorm f(h) on M'c,d by f(h) I<h,f>[, (h M',d).cIn fact, a sequence {h }o MIn view of Theorem i, we can relate to each h(x) log x), f(x)> <(p), g(p)>.(3.2) Using (3.1) and (3.2), we can easily have the following theorem: THEOREM 3.1.The mapping h(x) h(e-p) defined by (3.1), is an isomorphism THE n-DIMENSIONAL DISTRIBUTIONAL MELLIN TRANSFORMATION M. DEFINITION.We define the n-dimensional distributional Mellin transformation as the function H(s) on 2h into C I by (Mh)(s) H(s) <h(x), [xS-l] > for s e $h' (4.1) sp] and f(x) [x-1] g(-log x) [xs-l] and using Theorem 2.1, we can-have the following theorem:

THEOREM 4 . 1 .
The distribution h(x) is n-dimensional Mellin transformable if some c,d e R (c < d) and if Mhv H(s), then Lh H(s) exists for at least c < Re s < d and {H (s) converges pointwise in the tube of definition =i c < Re s < d to H(s).PROOF.Since [xs] is in M for each s satisfying c Re s d, the theorem c,d follows from the definition of convergence in M' and the fact that M' ,d, there exists a non-negative integer r R and a positive constant c R# such that, for all in M c,d' l<h,>l < c ().
7. SOME OPERATIONAL TRANSFORM FORMULAS FOR THE n-DImeNSIONAL DISTRIBUTIONAL MELL IN TRANS FOPMAT ION.