ABELIAN THEOREMS FOR WHITTAKER TRANSFORMS

Initial and final value Abellan theorems for the Whittaker transform of functions and of distributions are obtained. The Abelian theorems are obtained as the complex variable of the transform approaches 0 or in absolute value inside a wedge region in the right half plane.

[2, sections 8.6 and 8.7], Akhaury [3-4], Moharir and Saxena [5], and Tiwari and Ko [6] in the generality of the transform variable being in a wedge and in the generality of the parameters.
For s > the Whittaker function Wk, m(s) (Erdlyi et  Let p be a positive real parameter.For K > 0 being a fixed real number put {s Sl+iS2 e ' s I > 0 and Is21 < KSl); PK is a wedge in the right half plane PK >.Using (2.1) with the gamma function taken to the left of the equality and estimate analysis on the exponentials and powers as in Carmichael and Hayashi [8, Lemma 2.2, p. 70] and Carmichael and Hayashi [8, proofs of Theorems 3.1 and 3.2], we have the following important estimate for this paper" (2ml+l)/4 F (ml-kl+(i/2) IWk,m(pSt)I <_ (I+K2) exp(1Im21/2)ir(m_k+(i/2))l Wkl,ml(PSlt) SEPK t>0.
3. THE WHITTAKER TRANSFORM FOR FUNCTIONS.
Let k,m, and r be complex parameters, and let p and q be real parameters.
be a complex variable.The function (2,2) Let s ,m (s) (st)r-(1/2) p,q,r 0 exp(-qst/2) Wk,m(pSt) f(t) dt (3.1) is the Whittaker transform of the function f(t) where __Wk,m(pSt) is the Whittaker function.The general whittaker transform defined in (3.1) was first considered by Srivastava [9] for certain values of the parameters and variable which depend upon the order of growth of f(t).
In this section we prove initial and final value Abelian theorems for the 'whittaker transform defined in (3.1).For this purpose we now place conditions on the parameters and variable noted above; these restrictions will hold throughout the remainder of this section.The complex parameters k,m, and r satisfy Re(m-k+(1/2)) > 0 and Re(r) > Re(m) > 0 with Re(r) > (1/2).The real parameters p and q are positive.The complex variable s is in >, that is Re(s) > 0.
We now state and prove an initial value Abelian theorem.
THEOREM 3.1.Let q ql+iq2 be complex with Re(q) ql > (-1).Let f(t), 0 < t < , be a complex valued function such that there is a real number c > 0 for which (e -ct f(t)) is absolutely integrable over 0 <_ t < and such that (f(t)/t) is bounded on 0 < t < y for all y > 0. Let the Whittaker transform F k'm (s) of (t) p,q,r exist for s e @>.If there is a complex number m for which Rim f(t 3.2 (: t/0+ tq then for each fixed K > 0 im sq+l ,m (s) s e PK A(q,k,m,p,q",) (16), p. 216] we have tq (st)r-(1/2) exp(-qst/2) Wk,m(pSt) dt s A(q,k,m,p,q,r).
In this theorem we desire to prove (S.S) as li + (R), PK" As sl (R), PK, then necessarily s I Re(s) / .Thus we now assume without loss of generality that s I Re(s) > (2c/q) in the remainder of this proof for the fixed c > 0 and q > 0; and for such s I Re(s) we know that exp(-(qSl-2C)tl2) <_ l, t >_ O.
As an example for which Theorem 3.1 is applicable, let f(t) t/(l+t2), =l, c=l, and u=l.Another example is obtained if we take f(t) t2/(l+t2), O l+i, c=l, and c:--O.
We now obtain a final value Abelian theorem for the Whittaker transform of functions.
THEOREM 3.2.Let l + i2 with l > (-1).Let f(t), 0 <_ t < =, be a complex valued function such that there is a real number c > 0 for which (e -ct f(t)) is abso- lutely integrable over 0 <_t < and such that (f(t)/t0) is bounded on y <_ t < for all y > O. Let the Whittaker transform F k'm (s) of f(t) exist for s e >.If there is a p,q,r complex number (3.13) PROOF.Using (3.) and arguing as in (3.5) we have Is ml Fk'mp,q,r(S)-A(k,m,p,q,r) _< I I + 12 (3.1h) where I I and 12 are defined in (3.6) for y >0 arbitrary.Let K >_ 0 be arbitrary but fixed and s PK" Using the boundedness hypothesis on (f(t)/t), (2.2), and analysis as in (3.7) and (3.8) we have F (ml-kl+( 1/2 (3.15) and the right side of (3.15) is independent of s e PK" We now consider I I in (3.6).
For the real number c > 0 in the hypothesis we argue as in (3.9) and obtain where we have put p into the right side and hence have also put p there.The desired conclusion (3.13) in this theorem is to be obtained as Isl / 0, s e PK; as Isl 0, s e PK' then necessarily s I Re(s) 0 +; we thus may assume without loss of generality here that 0 < s I < (l/p) for the fixed parameter p > 0. As t / 0+, 0 < PSlt < t / 0+ for all s I < (l/p); by the growth condition at Wkl ,m 1 (PS It) Wkl ,m I (Pslt) <_ M (PSlt)(i/2)-ml, 0 < PSlt < t < T < y, s I < (l/p); (3.17 here M is independent of s I < (l/p) and of t < T and of PSl t < t < T and T is indepen- dent of s I < (l/p).Returning to (3.16) and using (3.17) and the fact that exp(-qslt/2) <_ l, t > 0, we have T Wkl,ml(pslt) e -ct (f(t) th) Idt I' In (3.18), as in (3.11), we have -ct (f(t)-t)i dt < ; (e -ct f(t)) is absolutely integrable over 0 !t < by assumption and (e -ct t o is absolutely integrable over 0 !t < since c > 0 and Re() > (-1).Now rl-(i/) Wkl (PSlt) ,ml(PSlt) is a continuous function of t on the closed bounded interval T !t !y for each fixed s l, 0 < s I < (l/p); hence this product attains its maximum on T !t y at a point tsl depending on s I < (l/p).Continuing (3.18) we then have The estimate (3.19) holds for all s I, 0 < s I < (i/p).The constants c,K, and M are independent of s I, 0 < s I < (i/p), as are all of the parameters p, k k I + ik2, m m I + im2, r r I + ir 2 and q ql + iq2 and the yet to be chosen constant y > 0.
Again as sl 0, s a PK' then s I Re(s) 0+.Recall that T <_ tsl <_y for 0 < s I < (I/p) in (3.19) and T is independent of Sl, 0 < s I < (i/p), but dependent on y > 0. (Recall (3.1).)As s I O+ then 0 < Psltsl <_PSlY 0+.Thus as sl 0, s PK' the growth condition Tiwari and Ko [6, line +, p. 351] and Whittaker and Watson [ii, Chapter 16] applied to Wkl,ml(PSltsl yields constants M' and T' which are independent of s I such that Wki,ml(Psitsi) Wki,mi(PSitsi Examples of the applicability of Theorem 3.2 similar to those after the proof of Theorem 3.1 for Theorem 3.1 can be constructed by the reader.

4.
THE WHITTAKER TRANSFORM FOR DISTRIBUTIONS.
The proof is by Pathak [12, Lemma l, p. 6].Nothing different is intro- duced by having p > 0 in the exponential term and in Wk,m(pSt) here.

P
The set of seminorms {a} 01,2 which defines and generates the topology of V (0) is a mu_itinorm in the sense of 7.emanian [13 p. 8] as noted in a Pathak [12 p. 6] here Y0 is a norm.nus the hypotheses of Ze.n+/-an [13 Theorem 1.8-1 p. 18] are satisfied; by this result, given U V (0') there exist a positive constant and a nonnegative integer N which depend only on U such that < u, > < c :o,, , v()' va(o,). (.) We shall call the number N here the order of the generalized function U e Va(0,).
We obtain Abelian theorem.for the distributional Whitter transform; in so doing we use the results of section 3. Thus in the remainder of this section we assume that the complex parameters k and m satisfy Re(m-k + (1/2)) > 0. By comparing the forms (3.1) and (4.1) we see that the complex parameter r and the real parameter q of (3.1) and section 3 are now taken to be r m and q p in this section.We thus must make the restriction Re(m) > (i/2) in the remainder of this section because Re(r) was necessarily so restricted in section 3.
To prove our initial value theorem for wTk'm[u;s] we need the following lemma.P LEMMA 4.2.Let U e Va(0,) for a > U and let the support of U be in T it < , T > 0. Let s e PK with (p Re(s)) > max {l, p G U, 4a).There are constants B > 0 and Y0 > 0 which are independent of s such that m+kl+N-1/2 wTkm[u; s] <_B (i + s I) exp(-PSlY0/h), s I Re(s), (4.3) where N is the order of U.
PROOF.The proof of this lemma is in the spirit of that of Zemanian [2, Lemma i, p. 2h6].Choose a function (t) e C such that for any nonnegative integer m we have d(k(t) dt -<t <, where M is a constant which depends only on a; 0 < l(t) < I; k(t) i for T < t < ; and the support of k(t), denoted supp(k), is contained in Y0 !t < , 0 < Y0 < T.

are independent of
We now prove an initial value Abelian theorem for the distributional Whittaker transform where the element U e V (0,) is assumed to have support in [0,).THEOREM h.l.Let U e V a (0,) for a > U such that over some right neighborhood (0, t 0) of zero U is a regular distribution corresponding to a complex valued function f(t) such that there is a real number c > 0 for which (e -ct f(t)) is absolutely inte- > -1, let (f(t)/tD) be bounded on 0 < t grable over 0 <_ t <_ t O For i+i2, i < y' for all y' < t O Let the Whittaker transform Fa'_m,m (, s), s e >, exist for the p function which is f(t) on 0 < t < y' and which is zero on y' <_ t < for all y' < t O If there is a complex number for which f(t) (.8) t0+ t PROOF.Since f(t) is Lebesgue integrable over 0 <_ t < then (e -ct f(t)) is absolutely integrable over 0 <_ t < for any fixed c > 0. The result thus follows immediately by Theorem We now proceed to obtain a final value Abelian theorem for the distributional Whittaker transform.First we need to make some comments concerning this transform.of a distribution of the form assumed in the final value theorem below.Let [t O > O. Assume over some interval U e V (0,), a > o U, with supp(U) ,), t O a y < t < with 0 < t o < y that U is a regular distribution corresponding to a complex valued function f(t).Then U can be decomposed as U U 1 + U 2 with supp(U I) _ [t0,T1 and supp(U 2) [T,), T > y.We then have WT k'm [U;s] WT k'm [Ul;S] + WT k'm [U2;s].

P P P (h.13)
From the definition (h.l) WT k'm [U;s] is defined for s e U in which Re(s) > O U P Because of the form of the U V (0,) assumed above and the assumptions on f(t) a in the final value Theorem h.2 below, where we use the above decomposition, we can take G U 0 here, and (h.13) will be well defined in Theorem h.2 for s I Re(s) > 0.
Because of this we may let sl0, s PK' in the final value theorem below as desired.
We now obtain a needed lemma for the final value result.for some nonnegative integer N, the order of U, where the g(t), e 0,i, ,N, are continuous functions with support in an arbitrary neighborhood [t O -g, T +g ], > 0, of [t O T].Since g> 0 is arbitrary we assume here that t O -g > O. Using distributional differentation and the calculation Slater [lh, (2..17   Using analysis like that in obtaining the estimate (4.7) in the proof of Lemma 4.2 we have (pst)m-(i/2)-(cx'/2) exp(-pst/2) Wk+(a/2),m_(c12) (pst)l <_ for constants % with a 0,i, ,N and s e PK" Again recall that as Isl 0, s e PK' then necessarily s I Re(s) 0 +.Thus by combining (4.15), (h.16), (4.17), and (4.18) we obtain (4.14) and the proof is complete.
We now obtain a final value Abelian theorem for the distributional Whittaker trans form.THEOREM 4.2.Let U e Va(0,), a > OU, with supp(U) _C [t0,) t O > 0.Over some interval y < t < , 0 < t O < y, let U be a regular distribution corresponding to a complex valued function f(t) for which there is a real number c > 0 such that e -ct f(t)) is absolutely integrable over y <_ t < .For l + i2' hl > (-1), let f(t)/t be bounded on y' <_ t < for all y' >_ y; and assume that F k'm (s), s e >, exists for the function which is f(t) on y' < t < and which is p,p,m zero on 0 <_ t <_y' for all y' >_ y.If there is a complex number for which with supp(U l) [t0,T] and supp(U 2) [t,), T > y, as in the paragraph above which contains equation (h.13); and by the discussion in that paragraph, (h.13) holds for s I Re(s) > 0. Let the function f(t) in the hypothesis, which is known on y < t < , be extended to O<t<by (t) {0 0 <t <T, f(t), T< t< Then (t) satisfies the hypotheses of Theorem 3.2.Now [U2;s] [(t);s] P P with this latter Whittaker transform equaling the function Whittaker transform of (t) defined in (3.1).Thus the conclusion (h.20) follows from (h.13), Lemma h.3, and Theorem 3.2.The proof is complete.
In future work we hope to extend the Abelian theorems of Joshi and Saxena [16] and Malgonde and Saxena [17] for the H-transform to the general setting that the complex variable of the transform approaches 0 or inside a wedge region in the right half plane and for more general parameters, To do so we will need to use the proper- ties of the H-function as in Srivastava et al [18].Analysis which is associated with that in this paper and with the corresponding H-transform problem is contained in Sinha [19] and Joshi and Saxena [20].The MeiJer transform is studied in Pathak [21]  which we note here because of the similarity of the MeiJer and Whittaker transforms. 5.

ACKNOWLEDGEMENTS.
In a letter to one of us (R.D.C.) Professor H. M. Srivastava has made some comments concerning certain analysis contained in this paper.We thank Professor Srivastava for this contribution to our paper.
One of us (R.D.C.) expresses his sincere appreciation to the Department of Mathematical Sciences of New Mexico State University for the opportunity of serving as Visiting Professor during 1984-1985 when the research on this paper began.
The research of Richard D. Carmichael in this paper is based upon work supported by the National Science Foundation under Grant No. DMS-8418h35.
Some of the research of R. S. Pathak in this paper is based upon work which was partially supported by a grant from the William C. Archie Fund for Faculty Excellence of Wake Forest University.This author expresses his appreciation for this support while visiting Wake Forest University. i.

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