THE MEIJER TRANSFORMATION OF GENERALIZED FUNCTIONS

This paper extends the Meijer transformation, M μ , given by ( M μ f ) ( p ) = 2 p Γ ( 1 + μ ) ∫ 0 ∞ f ( t ) ( p t ) μ / 2 K μ ( 2 p t ) d t , 
where f belongs to an appropriate function space, μ  ϵ  ( − 1 , ∞ ) and K μ is the modified Bessel function of third kind of order μ , to certain generalized functions. A testing space is constructed so as to contain the Kernel, ( p t ) μ / 2 K μ ( 2 p t ) , of the transformation. Some properties of the kernel, function space and its dual are derived. The generalized Meijer transform, M ¯ μ f , is now defined on the dual space. This transform is shown to be analytic and an inversion theorem, in the distributional sense, is established.


I+D,
for the Bessel differential operator B t Dt where (-i,) and D d' Conlan and Koh [i] used the Meijer trans- formation M f given above.The operational formulas obtained mirrored those obtained by Koh  [2] by means of a Mikusinski-type calculus.The latter is algebraic in approach and allows for certain convolution operators not covered by the classical integral transform M f.To remedy this situation, the Meijer transformation has to be extended to generalized functions.This is the object of the present work.
The idea is to construct a testing function space ( which contains the kernel (pt)/2K(2 p).The generalized Meijer transformation M f is now defined on the dual space q' as follows: for feI' U,y 2p (Mf) (p) < f(t) (pt)/2K (2 p)> (i.i)where e(-,l).The definition in (I.i) coincides with the classical transformation whenever f is a regular distribution, that is, one that can be represented by a suit- able integrable function.We note that there are various extensions of classical transforms in which the operational calculus is geared for other Bessel-type operators among which are the K-transform [3] and the Hankel transform [3], [4], [5]   and [6].
In section two we shall describe the testing function space its dual and study some of their properties While , in section three we will give the definition of the generalized Meijer transform, show its analyticity in some region of the complex plane, and then derive an inversion theorem.
Throughout the sequel we shall make use of the following notations and facts and are stated for the sake of completeness.
We shall denote the interval (0,) by I, the Bessel differ- where the c's are suitable functions of u only.
The modified Bessel functions I (2 p) and K (2 p) of first and third kinds are defined by (see Watson [7]) ( is Euler's constant) and the finite sums are taken to be equal to zero whenever the upper limit is less than the lower limit.

2.
The Testing Function Space M and Its Dual.
In this section we shall define the testing functions space M its ,y' dual and study some of their properties.as those smooth functions on I that grow no faster than the exponential function e Yet for large t and behave like power functions near the origin.As will be seen in Lemma 2.1 below, the kernel of the Meijer transform is of this type.
The family {I k:0 1 2 is a countable multinorm y,k Indeed, 1 is a semi-norm for each k and 1 is a norm on y,k y,0 We assign to the topology generated by this family ,Y ,Y of multinorms.Thus is a countably normed space (a fund- amental space and members of the dual space M' are generalized e for functions).A sequence {#9} is Cauchy in all 9 and for every k 0, I, 2, y,k independently.We shall now show that the kernel, (pt)/2K (2 p), member of Lemma 2.1.For any fixed complex number p such that p 0, arg p < n and Re2/ > y, (pt) /2 K (2 p) e , Proof.Under the hypothesis on p, K (2 p) is an analytic function of t on the right half plane and hence a smooth function on I.
where, A' C' and E are constants depending upon only.
Applying L'Hdpital's rule to the second term in (2.3) and using the fact that < i, we imply that there is a real number T < 1 such that for Iptl < T, the right hand side of (2.3) is again bounded by a constant.Thus the left side of (2.2) is bounded for small t and < 1 for all k =0, i, 2, We now consider the finiteness of (2.1) for large t.By employing (1.7), we obtain, for -< arg p < and for Iptl > T, le/tl-pk(pt)/21< (2 p) < E' Iplk+-le(-Re2/)/{Iptl3/4-/2 where E' is a constant depending upon only.Since < i and y< Re 2, the right hand side of the last expression is bounded by a constant for Iptl > T.
Thus in either case we have shown that ((pt)/2K (2 p)) < y,k for all k 0, I, 2 Therefore, (pt)/2K (2 p)eM ,y" By employing the following fact Dt[(Pt)+/2K (2 p) -p(pt)-+/2-1/2K (2 p) and by arguments similar to the proof of Lemma 2.1, we have Lemma 2.2.For p 0, -< arg p < and Re 2/ > y, that Dp ,Y In the next few lemmas we shall investigate some properties of the space M and its dual.Fs -(t-s 0 s(int ins) 0. Since multiplication by a power of t or the application of D -i preserves uniform convergence on K, it follows from (2.5) that Ds$ converges on K. From (2.4), it follows that Dt converges uniformly on K as +=.Now the convergence of Therefore there exists a smooth function on I such that for each k 0, i, 2, and each tgI, Dtk(t) converges to Dtk(t) as +.We will be done if we can show that (t)M,y.
Since {#} is a Cauchy sequence, it follows that for each k and each > 0 there is N k such that if , > Nk, Iy, k (-) < e/2.Further, since e 7'r6tl-Bk_(-)l 0 as , we have for each teI and > N k that leYCrtl-B k ( -#) that there is a real number such that the restriction of f f to ,y is in ' if y > and is not in The real number of is called the abscissa of definition of f.
For any real numbers y and e with < e, we know that M,e is a subspace of , However, D' (I).

3.
The Generalized Meijer Transform.In this section we shall give the extension of the Meijer transformM f to generalized functions belonging to ' We shall also prove the analyticity ,Y of the transform and exhibit an inversion theorem which is a generalization of the classical inversion formula (see [i]) For fe' e(-,l) and pelf {PEIRe 2/ > y > o,r p 0 where Je(-l,).We shall next prove the analyticity of the transform M (f) However, we need the follow lemma.
Lemma 3.1.Let y be any real number and p be a fixed point in f.Then sup eY/ (pt) I-/2 K 0<t< .3) where C is a constant depending only on .
Proof.The idea of the proof is similar to that of Lemma 2.1.Since K (2 p/) is analytic except at t 0 and t , we need only to show the boundedness of eY/(pt)l-/2K (2/) as t 0 and as t m.Employing the series expansion of K(2 p/)__ (1.4) with 0, -i, -2, we obtain leY/E(pt where A and A' are constants depending upon only.For T < 1 such that Ptl < T, the right hand side of the last expression is bounded by a constant R independent of p and t.Similarly, if we employ (1.4) with 0, -I, -2, then, for Iptl < T, we have ey/E(pt where Q is independent of p and t. For t large we employ (1.7) to obtain le/(pt) I-/2K (2 p) _< L1l pt /2+ 3/4e (y-Re2) where L is again a constant independent of p and t.Let E max{R,Q,L}.Then sup 0<t< (pt)l-/2K (2 p/) < E (l+Iptl-/2+ 3/4 (y-Re2)/ U e < EU (l+IPl -u/2+3/4) (l+Itl /2+ 3/4 e (v,-Re2)/ and since Re2/ > y and pe(-,l), it follows that the right hand side of the above expression is less than Cp(l+Ipl -p/2+3/4) with C a constant depending upon p.This completes the proof.

P
We will now show that M f is analytic.Namely, Proof.For pelf,let C be the circle about p of radius r so chosen that i lies entirely in f and let gpl < r I < r for some r I (f Dpp(pt) 2Ku (2) ].
In order to prove the result, it suffices to show that Ap(t) 0 as Ap 0 in U,y.Since by Lemma 2.2 Dpp(pt)O/2K(2/)e,, it follows that (3.4) is well-defined.Further, since for k=0, i, 2 B k Dpp(pt)/2K (2 p) D pk+l(pt)/2K (2) v p v by equation (1.5) it follows that B k is an analytic function -U Ap of p on f.By Cauchy Integral formula, we have (-P) (-p-Ap) Since I-Pl r l-p-Apl > r-r I and eY/(t) I-V/2K (2 /6) is bounded by C (l+ I -/2 + 3/4 it follows that The rest of this section is to prove a generalization of the inversion theorem for the classical Meijer transform (3.2)   due to Conlan and Koh [1].Our result is restricted to generalized functions belonging to the space D(I) (Remark 2.2).In particular, we shall prove the following inversion theorem Theorem 3.3.Let (Mf) (p) be the Meijer transform of fe' for pefif.Then  The proof of Theorem 3.3 will be developed in a sequence of lemmas.We will state the lemmas and give them their proofs in the Appendix.Lemma
APPENDIX.In this appendix we shall give the proofs of Lemma 3.2 through Lemma 3.6.
Proof of Lemma 3. on the domain (x,Y)I--< x < , < r < (/ + --) }.So given any e > 0 we can make the first term in (A5) less than e/3 by choosing small enough, say o i" The second term of (AS) converges to zero as 81 .Thus J1 + J4 < e/3.
Therefore the This implies that given any e > 0, we can choose sufficiently small, say o 2 such that IJ21 < e/3.
By a similar argument, for e > 0 we can choose sufficiently small, say 3. such that J31 < e/3.
Proof of Lemma 3.6.Since (t) has a compact support and since the integrand is smooth, it follows that B k H8 () 1 B k (pr) Letbe any real number and e(-,l).Define ,,y {eC (I)lly,k(the weight function eY/tl-one may think of the elements of be a smooth function on I and s t fixed point of I. Define (D-I)(t) r | (T)dT.
uniformly on K as /.

(
1) if f(t) is a regular generalized function (see Remark 2.1) then we obtain the classical Meijer transform
zero as p 0 and this completes the proof.
real number in f.

(
The limit is to be understood in the sense of convergence in D' (I)).