THE GENERAL CHAIN TRANSFORM AND SELF-RECIPROCAL FUNCTIONS

A theory of a generalized form of the chain transforms of order n is developed, and various properties of these are established including the Parseval relation. Most known cases of the standard theory are derived as special cases. Also a theory of self-reciprocal functions is given, based on these general chain transforms; and relations among various classes of self-reciprocal functions are established.


INTRODUCTION.
The standard and the simplest definition of a function g to be the integral transforms of f with respect to k is that g(x) f(t)k(xt)dt where k is called the kernal, [1,VIII].If further f(x) I] g(t)k(xt)dt, (1.2) then we say that f and g are pair of k-transforms.Under less stringent conditions the equations (I.i) and (1.2) can be replaced by p(t)dt t-l f(t)kl (Xt)t I f(t)dt I t-lg(t)kl(Xt)d/ I x where k l(x) k(t)dt The above reciprocity formulae define the pair f and g, genera|ly known as Watson transforms.lany generalizations of this reciprocal relation have been given, and (I .3) (1.4) one of the directions was pointed out by Fox,[2], who introduced the idea of chain transforms.He showed that for chains of the form g2(x) I gl(U)kl(UX)du gn (x) gn-1 (U)kn-1 (ux)du gl (x) gn(U)k n (ux)du 0 there exists a theory similar to the standard theory for n 2, i.e., the Watson transformations.Also, Duggal, [3], considered the pair of equations.
! v()f(x)dx k()ff(z) (.5) z 1 u 0 using convolution of the functions f and and characterizing the relatonship between f and 9.If V 5 v are taken to be the Heavsde step functions, then (1.5) and (i.6) reduce to the equations sm[lar to the equations (1.1) and (1.2) above, thus extend- ng the Watson's definition.
In ths paper e shall combine th-above mentioned to extensions of the usuai atson integral transformations.e shall gve a generalized form of the chain trans- forms of order n and develop properties praliel to hose of the standard theory.
Nost of the ell-kno results are deduced as special cases.Further a theory of self-reciprocal functions s developed based on these generai chain transforms; and again many standard results are deduced as special cases.In the end various exapies re gven to llustrate the general nature of the main theory developed and its special cases.
If f and g L2(O, ) and have Mellin transforms F(s) and G(s) respectively, then ' ' a ' -s ' -d Definition 2.2.A function is said to be self-reciprocal with respect to the kernel k if f is its own k-transform, satisfying the identity f(x) f(t)k(xt)dt, 0 or its equivalent.We then say that f Rk (2.4)Although most of the results hold for functions involved under less stringent condi- tions, but we shall mainly work in the Hilbert space L 2, for convenience and elegance of the theory of integral transforms in this setting.
We consider the system of n integral equations ki(ux)/i+l (x)dx vi(ux)fi(x)dx where i 1,2 n and f(x) f().For simplicity we shall assume that no two are equal, although it is not essential.We shall refer to the functions k. and v.
as kernels.Now, we give one of our major results.Theorem 3.1.
PROOF.Due to the hypothesis () and the fact that fi g(0,) henever fi (0,), the ntegrals in (3.1) exist and are n fact absoluteIy convergent.
No due to the Parsevai theorem for MellOn transforms of 2-functtons, applied to boh sides of (3.1), we have Ki(s)Fi+1(s) (s)Fi(s a.e.+ fro).0r from (3.6), we have u F (x) =I P(s)F (s) u -s the integration inside the integral sign is justified because of uniform convergence due to hypotheses (i) and (iii).Now since P(s) is bounded, therefore P(s) DEFINITION 3.1.The sequence of functions {fi is said to be the general chain transform of order n, with respect to the kernels k i and vi, i 1,2,...,n, whenever the system (3.1)along with (3.3) is satisfied It is now an easy matter, to prove a stronger result than that of Theorem i, which we give below without proof.
THEOREM 3.2.Let the conditions of (i) and (ii) of THEOREM i hold.Then necessary and sufficient condition that (3.3) holds is that n v.(s) P(s) I-i 7 =1 .gi ()Next we shall give some special cases of the result of THEOREM i.
First let n I.
where Pl (X) i (3.8)Thus if k 1,v and fl are known, we have an inversion formula to retrlve the unknown function 2" If we further set Then THEOREM 2, reduces to a known result [3], that is: COROLLARY 3. If (i) ki,f i and f3 E L2(0,), i 1,2, and then f3 (x)dx x-lpl (ux)fl (x)dx 1 1%+i P(s) xl-S ds where P (x) J1/2_iil-Z Suppose, now, Pl(X) is differentiable, then from (3.4), we have p(s)x-Sds (l-x), and hence, its Mellin transform, P(s) 1 the above result, then, defines the chain transforms introduced by Fox.Note that all the above mentioned results are specializations of our Theorem i, showing the general nature of that result.Next we shall deduce a Perseval type relation for the general chain transforms as defined by Definition 3. Let sequences {fi and {gi} be chain trans- forms of order n with respect to the kernels k. and v.. Now consider where + () + () By virtue of THEOREM i and the consequence (3.3), we have x-lf:+1(ux)1(x)dx x-11(x)dx t-lp (uxt)?1(t)d t t-l (t)dt x -lpl (t)l 0 t (t)gn+l (ut)dt Hence x-l+* (uz)g (x)d t-l ()gn+l (ut)dt (3.9) establishing, formally, the Parseval relation.Further on differentiating both sides, formally, we have fn+1 (ux)1 (x) 71 (t)L+1 (ut)dt ettng n 1, reduces the above to the usual arseval Theorem, [1, It is now an east matter to develop an inversion theor for the general chain transforms.For instance Sf we assume that P(s)P(I ) i, then from (3.3), due to the standard inversion formula, we have 1 (X), x-lpl (zeX)fn+l (x) Thus the unknown function I is retrieved.On the other hand if we assume that 1 d EO, 0 -x P(e E0, being the Lagunerre-Polya class [4,VII], then due to the known inversion tech- niques, [5]   1 [fn+l ( Suppose {i is a sequence of general chain transforms of order n, with respe-t to the kernels k z. and v..z If we then set n+l(X) l(X)' in Theorem I, we have From the conclusion (4.3), t follows that 1 s self-reciprocal with respect to t,.
kernel Pl(z)' in the sense of definition 2.2.Sbol[cally we say that fl R

P
The particular case when n 2 is of special interest.In this case, many useful properties of self-reciprocal functions are established, including a procedure for generating self-reciprocal functions with respect to a given kernel.Now let n 2.
Then, COROLLARY i.If (i) ki,v i and fi L2(0')' i 1,2, (ii) f k (um)2(x)o!rf V (um)l (x)ci  L() By the Parseval theorem for Mellin transform of L2-functions (4.8) and (4.9) give, re spec t ively, f2 (X)Clr x-lm(um where and The functions m and e are defined by the integrals which exist in the mean-square 1 sense, since both M(8) and L(8) are bounded on 8 + i, < < =, due to the M(8) L(8) both hypothesis (iii) of Corollary 1 above, and consequently iL and 1 8 1 1 L2( i, + i).Thus, combining the above results, we have, Corollary 2. If (i) fl and f2 L2(0')' then fl R where gl,ml and Pl are defined above.
One can show that f m" Then g(x) xql ql where "e Q(s) x ds Again the special case when n 2, is interesting, since it gives us a procedure for generating self-reciprocal functions.COROLRY 5.If (i) fi,vi and k i L (0,), i 1, .e. f (R q If we consider the functions L(8) and M(s), defined earlier, then as before, we obtain F2(8) M(s)F2(I s) and El(S) L(s)F2(I s), a.e.
The last two equations, then give, respectively, 0 f2(x)dx IIm(ux)f2(m)dzx-  f2(u) 0 m(ux)f2(x)dx fl(U) I e(um)f2 (ii) then fl(u) J q(uz)fl(x)c5:.0 The kernel ql (u) q(z)ce and the kernels (z) and m(m) are defined similarly.Note that the above results gives us a procedure for generating self-reciprocal functions.For example, let re(x) x J (x) then its Mellin transform is given by [7; 326 K being the usual Bessel function of third kind and its Mellin transform is 2-1/2 3 whence q(x) x 1/2 J (x).Thus, we have as a special case of Corollary 7.  where Fs,Fc and L denote the Fourier sine, Fourier cosine and the Laplace transforms Re, [7;61(7)], as predicted.
successive elimination and using (3.2), in the mean-square sense, since P( + iT) is bounded and exists and L2(0,).Also fl(x) U I[F l(l_s); xl, therefore due to the Parseval theorem applied to the right-hand side of (3.7), we obtain u fn+l (x)dxx-Ip (ux) (x)dx as desired.

COROLLARY 8 . 2 and
If f H and 1 7F(++1) (xt)f(t)dt, class H denotes the class of self-reciprocal functions with respect to theHankel transforms of order .To varify this result, let f(x) x v+ e -

F
where R and R denote the classes of self-reciprocal functions with respect to Fourier sine and Fourier cosine transforms, respectively.In the operational notation, we have that if s functions and then g(x) predicted, thus verifying the result of Corollary 4. Next we shall give a slightly different version of Theorem 4.1, so that the function fl' rather than the function I is involved in the conclusion.
g (x), as Now from the hypothesis ([[), by using the Parseval theore or Nellin transfos, the It is not difficult to prove the above result rigorously, the functions fl and f2