A CHARACTERIZATION OF THE GENERALIZED MEIJER TRANSFORM

The purpose of the note is to prove a representation theorem for the generalized Meijer transform defined in [2]. In particular, we shall state and prove necessary and sufficient conditions for a function F(p) to be the generalized Meijer transform of a generalized function.

The generalized Meijer transform has been defined and studied in [2] and [3] and is given by 2p (/l,f)(p) r(1-t) < f(t), (pt)t'/2Kt,(2x/')> (I.I) where p > -I,K t, is tl,e modified Bessel function of third kind and order p, p belongs to a region of the complex plane and f belongs to the dual M, of the space Hit,.defined by and A,,() ..p I .-.((t))l,t 0,,2,... 7 being any real number and B_ t, ttDtl-tD(D ) is the Bessel differential operator.The properties of the space Mt,, and its dual have been studied in [2].Furthermore, in [2] the transform (I.I) has bee, shown to be analytic and an inversion theore;n, in the distributional sense, has been established.We note here that if J'(t) is locally integrable on I (0, oo) and f(t)e-rvqt-l+t' is absolutely integrable on I, then we obtain the classical Meijer transform In [3], we applied the generalized Meijer transforln to a bundary value probletn with distributional conditions.To arrive at the solution, it was necessary to use a characterization of the Meijer transform.llt this note we shall state and prove necessary and sucient conditions for a function F(p) to be the generalized Meijer transform of a generalized function f in M,.This will be the content of Section 3 while Section 2 will be devoted to preliminary results and background material.

PRELIMINARIES.
For the sake of completeness, we shall collect in this section the background material that will be nded in proving the representation threm.
For any feM,.and pt! {pCIRe2v/ > 7 > 1, P 0,1argpl < } the following have been established in [2]: which is a basis for an operational calculus of the transform (ii) f is analytic in fl! and where p(0) ?s/4e secs 0/2, T is fixed rel number in /nd the limit is to be understood in the sense of convergence in D(I), the dul of the spce D(l) of ll smith Functions on I whose support is co.tined i compact subset K of I equipped with the semi-norms Finally, we remark that if j'(t) is locally integrabh on I and J'(t)e-'V/t -+' is absolutely integrabh on I, then (t) generates a regular member of M;,, vi As noted earlier for such functions the transform in (1.1) reduces to the clsical Meijer transform given in (1.2).
A result that will be nded in our proof of the representation theorem is THEOREM A. [Threm 4 [1]].If Ret, -, Re > 70 0 and F(p)is analytic and bounded according to I(p)l < MIPIwhere q < Ret= + 2, then for reM e > 7o and Re > c, F(p) where 1(0 r(] +/')-"/ 2x F(p)p-1-,/, I, 2)dp.ne=c 3. MAIN RESULT.In this section we shall give a necsary and sucient condition for a function F(p) to be the generized Meijer transform of a function f in M,.As we shl s lxter in the proof of the necessy part, the real number p must be rtricted to -] p < 1.
Before we state the result, we nd the following lemmx stated in our cmRext (s [4], p. 18).

/tevf=c
We will show next that e-Z-+g() is absolutely integrable on I nd conclude from (2.10) that generates a regular member g of M-We consider two ces (i) for Iptl 1, 0 < < , (2.3)impli that le-(p0-a+/l(2)l Me-lPl which is integrable on I for Rep > O.