Hankel transforms in generalized Fock spaces

. A classical Fock space consists of functions of the form, where 0 e C and Cq e L p (Rq), q _> 1. We will replace the q, q >_ 1 with test functions having Hankel transforms. This space is a natural generalization of a classical Fock space as seen by expanding functionals having abstract Taylor Series. The particular coefficients of such series are multilinear functionals having distributions as their domain. Convergence requirements set forth are somewhat in the spirit of ultra differentiable functions and ultra distribution theory. The Hankel transform oftentimes implemented in Cauchy problems will be introduced into this setting. A theorem will be proven relating the convergence of the transform to the inductive limit parameter, s, which sweeps out a scale of generalized Fock spaces.

development in this setting can be found in reference (Bogolubov et al [6]).A state vector, belonging to this Fock space is described by an arbitrary sequence, I, {I,q}q 0, satisfying the condition, I, a__ y q'q < o.The Fock space is equipped with the natural scalar q=0 product given by the fdrmula, (,) a q (q,q), where each (,I,q,tI,q), q > 0 is the inner product given with the Hilbert space, :Bq.A principal problem with this development together with the test space, :Bg, is that the kernel of the Hankel transform is not a member of the test space, , and the Dirac delta is not a member of the space, LP(Iq), (Zemanian [4]).These problems are overcome when one defines the distributional Hankel transform, H, p_>-1/2.These will be briefly reviewed in section 3.
However the number of independent variables belonging to the q-dimensional orthant still remains to be finite.
Our present development will implement the procedures developed in Schmeelk [7] together with a general setting developed in Schmeelk and Takai [8].With these settings in place, we will then extend the Hankel transform into inductive and projective limit spaces (Zarinov [9]).These spaces will enjoy all of the classical Hankel transform results together with an approach to solve the infinite number of independent variables problem.
We will conclude our paper with a generalization of the Hankel transform for the Dirac delta functional, 6(k)(m+P), into our setting.The transform for 6(k)(m=+P) is developed in Aguirre and Trione [10] and is based on the notion of distributions applied to surfaces (Gelfand and Shilov [11]).The extension of a particular case of 6(k)(m2+P) will then enjoy the infinite number of independent variable setting.
2. SOME NOTIONS AND NOTATIONS.We begin with recalling some fundamental conditions placed on our sequences of positive constants and sequences of functions.The prerequisites on the sequences lead us in a natural way into the approach in [12] and then into our generalized Fock spaces.
We next suppose that a sequence, o (Mp('))peo, of continuous functions on q is given.We require the usual conditions, (e),(i) and (i) hold as in reference [13] as well as the inequalities, Mo(t)_< M(t)_< tR q.
The family of norms, ([[.p)peN0, defines a locally convex topology on %(Mp)which in view of condition (P) turns this space into a Fr6chet space.It also has several other mathematical properties.For a detailed account of spaces of type %(Mp)see references [11, 14].
Observe that the sequence of norms,{ I1" II-p}pe 0, satisfies x 0 _> x II--> for any x e %'(Mp).The sequence of positive numbers, r (mq)qdlo, and the sequence of continuous functions,-'0 (Mp('))pd0, will play an essential role in the definition of the generalized Fock space, F r''g, in Section 4. Throughout the paper the notation, N, will indicate the natural numbers and N o indicates the natural numbers and zero.
3. THE SPACES,Yz AND Y.
_N__.Y,...,Yq_ 0 0 The Nncion, Jo(y), (1 7 q), -, is he Bessel Nnction of the first kind given by the formula, Several properties regarding this definition of the Hankel transform on functions defined on R' can be found in reference [4] and the Rq, q > 2, case in reference [2].
For/ _> -1/2 the generalized Hankel transform, H defined on distributions, Fe , is taken to be the adjoint of the Hankel transform, H/ given by the equation, (HF,) a= (F,H), for every e / and F e .A survey of the many properties for this definition of the generalized Hankel transform can be found in references [4, 5, 14, 1]. 4. GENERALIZED FOCK SPACES, Fr''At.
Let the sequencer= (mq)qe and .A o (Mp( .))pelbe given with the properties given in Section 2. We then define aq: %'(Mp).x"L" x %'(Mp) -C q-copies to be a multilinear continuous functional, q e N, and by definition select a o C.    Recall the definition of x _p as given in expression (2.2).REMARK.Physicists prefer to represent the elements from our generalized Fock space, I 's'r'Ag, as column vectors, for instance, Since this is convenient also when working with the Hankel transform, we shall do likewise.
Let us first observe that in view of (4.3) and (4.4), the canonical inclusion r s'r'Ars',r'Ah, ( is continuous provided that s' > s > 1.So in view of reference [9], we can now give the following definition.
DEFINITION 4.1.A generalized Fock space, Fr'Mh, is the inductive limit of the spaces, Fs'r'A, i.e., F r''A ind Fs'r''A'.

q= 1Rq
In keeping with the spirit of such a constraint, we shall indicate that the elements from the inductive limit, I"r'Ab, are L r summable for any re(1,oc).For this result we cite a well known lemma.PROOF.See reference [8].
The state vectors, (I' Fr'ML, can also enjoy an alternate representation called its kernel representation.However for this to be true we must require that each member, Mp(.)e ML0, decrease sufficiently fast as infinity so that our test space, %(Mp), for example contain the rapid descent test functions [7].Assuming this to be true, we briefly review the kernel construction.
Since each aq, q >_ is a multilinear functional on %'(Mp) x...x%'(Mp) we can define q-copies Cq(to,. Cq where a o & o is a scalar and Cq, q _> 1 are each defined in expression (4.7).We use this alternate representation given in expression (4.9) when,addressing the dual space of Fr''Ah.\

GENERALIZED DUAL FOCK SPACE
We now examine the dual of the inductive limit space, "rr'0)', by first analyzing the dual to each space, rs'r'l'.The dual is presented in the spirit of ,reference [7].In our present environment a member, F, belonging to the dual, " \rs'r'l'), will also enjoy a sequence representation, F (Fo,F1,...,Fq,...) where F o is a scalar and Fq, q _> are tempered distributions of order _< m.Moreover, F where F o and o are scalars and Fq, Hq q _> are the already respectively(, defined tempered distributions and rapid descent test functions.We can now equip, (Fr'Ml') with a projective limit.Again projective limits are extensively developedin reference [9].
PROOF.We decompose the q-dimensional orthant, Eq, into the portion contained in the q-dimensional unit sphere, Sq {0 < Ct+...+tl < 1} and its complement, CSq {1 < +... + tl < oo}.We then have Eq Sq o CSq.We will estimate the integrals over Sq and CSq.The estimate over Sq will use the formula (18, pg 75] for the volume of a unit sphere R q.
COROLLARY 6.2.The Hankel transform y./,/ >-1/2 is an automorphism on the space, rr'0.PROOF Since we apply our Hankel transform, :y./, t >-1/2 to each component, Cq(tl,..., tq), of the vector, q, e rr''%, we can apply the classical theorem [4, pg 141] to each component.As in that "result the Hankel transform is its own inverse namely %/ %1 for/ > -1/2 on each of our components.Since we have equipped, rr'*0, with an inductive limit topology in s we have for each s and s' such that our Hankel transform is one to one and 9nto between r s'r'*0 and rs''r'o.The theorem 6.1 in this paper proves the continuity in both directions making it an automorphism on rr''%. 7. THE HANKEL TRANSFORM OF THE GENERALIZED DIRAC DELTA FUNCTIONAL.
One of the principal distributions utilized by physicists is the celebrated Dirac delta functional.Clearly in a contemporary setting the Dirac functional must be admitted into a generalized Fock space.There are several applications where this is beneficial and we merely select the application of annihilation and creation requirements as put forth in reference [8]. 0 We select for > 0 our generalized Fock functional, 1 t,0.6t (7.1) where o (R) (R) 6to is the tensor product of q-copies of the translated Dirac delta functional already defined in xpression (4.8).We immediately verify that for p > 1, (P) o0 sq IIl-[s,r,0] q 0 q 0 making the generalized Delta given in expression (7.1) a member of ,Fr'"tt'/.
DEFINITION 7.1.The Hankel transform on the space," \(I'r'Ml'), is defined by the formula, ((HF, O>> a__ <<F, HtO>> for/ _> 2" We see from reference [15] that this definition for the generalized Hankel transform applied to the generalized Dirac functional given in expression (7.1) results in the vector, t/Ju(ty,) Again recalling tfi6-, J/(ty.)=O(y+'/') as y,-*0+ and as y.-, oo, 1 < 7 < q, it follows that the vector in expression (7.2) is a member 0) as a regular generalized Fock functional.The term regular has the obvious definition extended from the notion of regular distribution.
An alternate method must be selected for t9 when t91 =0.This is because our q- dimensional orthant, Eq, does not contain the origin, lThus the delta functional concentrated at the origin is not a member of H.If we include the origin and consider the closed q-dimensional orthant, Eq, then the Hankel transform does not have unique inverses.For further investigations surrounding this difficulty we refer the reader to reference [14].
To circumvent this difficulty we take an alternate definition for the distributional Hankel transform of *(k)(m2+p) given in reference [10].It is based upbn distributions on surfaces developed in reference [11].
It defines the Hankel transform of a test function, $(t), to be n-2 Hn-{(t)}(y) 1 / 2 I(t) -Y-Rn-(v/fi)dt 0 where Rm(W)a_ Jm(w) and Jm(w) is the Sessel function of the first kind given in expression w m (a.1).

=1
The Hankel transform is then extended to the distributional setting using the same technique as indicated in equation (3.2).Then the Hankel transform of the tensor product of q-copies of the Dirac delta concentrated at the origin becomes 1 (y,.y2...., yq)-n-.(2n/2)q[r()] q Clearly for n 2 > 0 and in particular if is an integer, we have a polynomial in [Yl so our Hankel transform functional becomes a regular distribution and once again in our setting ( Y becomes a member of r r''0 However the functional is still not a member of r r''% given in section 6. Therefore we must use a domn space as in reference [17].This technique would provide procedures leading to excellent computational results. space, F s'r'Ag, ) 1, if the norm, LEMMA 4.1.Conditions (M1) and (M3) on the sequence, r {mq}qeN0 imply that for any real number, t, we have Itlq (4.6) mq< c.
d k e Iq, hen the condition, ..,tl _a_ aq [St' "'"Styli for every test function, (t) e %(Mp).As was shown in reference [S], each Cq(tl,...,tt defined in expression (4.7)is a rapid descent test function.Thus for each C e--(Fr"AL), we have an alternate representation, o ,