EXPANSIONS OF DISTRIBUTIONS IN TERM OF GENERALIZED HEAT POLYNOMIALS AND THEIR APPELL TRANSFORMS

This paper is concerned with expansions of distributions in terms of the generalized heat polynomials and of their Appell transforms. Two different techniques are used to prove theorems concerning expansions of distributions. A theorem which provides an orthogonal series expansion of generalized functions is also established. It is shown that this theorem gives an inversion formula for a certain generalized integral transformation.

and pansions based upon the ideas different from those of the Schwartz-Sobolev approach [3], and from those of Ehrenpreis [4].This theory did not include the notion of convergence for pansions as there exists such notions for distributions.
In fact, the theory is essentially in algebraic in nature.It is also shown that distributions which are finite order derivatives of certain functions growing no (ex2) i faster than exp < are uniquely determined by their Hermite series.However, no topology was introduced there, and hence there is no way of expressing the generalized function as a functional in terms of the Hermite coefficients.
In two papers [5][6], Haimo studied the expansion of functions in terms of heat polynomials and their Appell transforms.This work is an extension of some results of Rosenbloom and Widder [7] on expansions in terms of heat polynomials and associ- ated functions.Despite these works, no attention is given to the expansions of distributions in terms of heat polynomials and their Appell transforms.This paper is devoted to the study of expansions of distributions in terms of the generalized heat polynomials and their Appell transforms.A theorem which pro- vides an orthogonal series expansion of generalized functions is also proved.It is shown that this theorem gives an inversion formula for a certain generalized integral transformation.

PRELIMINARY RESULTS ON THE GENERALIZED HEAT POLYNOMIALS.
For real values of x and t, the generalized heat polynomials, Pn (x,t) and its (x t) are defined by Appell transform, nW-, Pn,v Wn,v(x,t) t G(x,t) Pn, where n 0,1,2,... 9 is a fixed positive number and G(x,t) is given by G(x,t) (2t) where V2n(X,t is the ordinary heat polynomial of even order defined by Rosenbloom and Widder [7, p. 222], and H2n is the Hermite polynomial of even order given in Erdelyi's book [8]. It can readily be verified that for < x, t < , P (x,t) satifies the n, X -- where the operator A X 22 2v =---+ (--) x with some fixed positive number x It is noted that Wn,(x,t) is also a solution of (2.6).Frther, nW,v(x't) satifies the following operator formulas W (x,t) 8)   x n,9 With the help of the following results reported by Erdelyi [8] 2 P (x,-t) ILn(X) < A exp(), n, where L(x) is the Laguerre polynomial of degree n and order > -i and n 1 2 (e+l) G(x,y; s + t)= Z b W (y,s) Pn (x,t).
n n,9 ,9 n=o The source solution of (2.6) is given by G(x,t) in the for (2.17) and I (x) is the modified Bessel function of imaginary argument of order r; and r G(x,0;t) S(x,t).

THE TEST FUNCTION SPACE U
We denote the open interval (0,) by I.A complex valued infinitely differentiable function (x) defined over I belongs to the space U (I) if 2 yk() A_ sup lexp('') A k (x) <" x O<x< (3.1) for any fixed k, where k assumes the values 0,i,2,...; c is a positive real number, 1 >and A is the differential operator involved in section 2. x The topology in the space U is induced by the collection of seminorms {y }k== Since 7o is a norm, the collection of seminorms is separating as in- if for each k, yk(r ) tends to zero as r / (R).A sequence {r} with each r--I _r(X) belonging to Uc,v(1) is a Cauchy sequence in U (I) if yk(r__ _s / 0 as r,s / independently of each other for every fixed k, where k 0,1,2, It is noted here that U (I) is a locally convex, sequentially complete, Hausdorff topological linear space.Its dual space U' (I) is the space of generalized functions under consideration.c 1 From the estimate (2.10), it follows that for x > 0, s > and > -, (x) P s) U (I).Using results (2.4) (2 5) and (2.8) it can also be n, (x)-, seen that W (x,t) U (I) for x > 0 and 0 < t < .
In order to make this paper self-contalned to some extent, we now llst a few properties of the above spaces: (1) (I) denotes the space of infinitely dlfferentiable functions defined over I with a compact support.The dual space D' (I) is the space of Schwartz distributions [9] on I.It can easily be shown that D(1) is a subspace of the space U (I) and that the topology of (I) is stronger than that induced on it by U (I).
(ii) The space U (I) is a dense subspace of g(1) which is the space of all complex-valued smooth functions on I.The topology of U is stronger than the topology induced on U by 8(I).So 8' (I) can be identified with a subspace of i (ill) The space S' of tempered distributions is a subspace of U' 2 [Pandey,i0].
(iv) For each f U' there exists a non-negatlve integer r and a positive constant C such that < f, >I < C max yk() for every ( U where r and C depend on f but not on [8,p19].and o > 0.Then, for 0 < o/2 < t < o, N i f(x), (x) > lim lim knan(t) <(X)Pn (x'-s)' (x) >, (4.1) (4.3)We shall need the following lammas for the proof of this theorem.We arrive at the desired estimate using result (2.11).

LEMMA 4 2
Let (x) E U Then, for 0 < 0/2 < s < t < , , exp(y2/40) IA k n=N where C 1 is another positive constant.Clearly the last term tends to zero as N / for s < t.
LEMMA 4.3.For 0 < o/2 < s < t < , let G(x,y;t-s) be the function defined by (2.14) and let (x) 6 U Then there exist e > 0 and a large positive constant Al(n) such that for y > Al(n), PROOF.Proceeding as in the proof of Lemma 4.2, we write exp(y2/4) I Aky {G(x,y,t-s)) (x) d(x) i) F( + + k + n) Hence for t < , the last expression tends to zero as y .

PROOF. If (x). U
then proceeding as in [i0, p846], it follows that i for >-2 (k)(x) 0(e-CX), c > 0, x / (k) (0+) exist finitely, Using these orders of (k) (x) and the identity A G(x,y,t) A G(x,y,t) and integrating by parts, we obtain x y by {G(x,y,t-s)} (x) d(x) G(x,y,t-s) A x (x) d(x).We next use the standard technique [I0] and remember that (x) is not an element of 9(1) although it does belong to U to show exp(y2/4) G(x,y,t-s) {k(X) k(y) d(x) < e uniformly for all y satisfying 0 < y <_ A I We are now prepared to prove the main expansion theorem 4.1.
where C I is a certain positive constant.This proves the existence of the limit N+, when s < t.
Using the llnearlty of U and U' we can write N an kn <(x) Pn,(x'-s)' (x)> n--o N n=o < f(Y)' Wn,9(y't) > (x) kn Pn,9(x'-s)' #(x) > N n=o f(Y)' Wn'(y't) > < (x) kn pn,(x'-s)' The corresponding infinite series equals G(x,y,t-s).We observe that (x) G(x,y,t-s) E U'G,9 and (x) GN(X,y,t,s), E U'G, for s > Furthermore, as a function of y, < (x) G(x,y,t-s), (x)> is an element of U From Lemma 4.2, we know that in U Hence To complete the proof we have to show that lim <(x) G(x,y,t-s), (x) In other words, we need to show that for all k and e, there exists (n,e) such that, for 0< t-s < 6(n,e), exp(y2/4o) A k {G(x,y,t-s)} (x) d(x)   k(y) < e Y Since (y), U we have for y > A(n), exp(y2/4o) 0 k(y) < From Lemma 4.3 we know that there exists A I > A such that for y > A l(n) and t < t o (Al(n) independent of t),  x I (0,) and t is a fixed real number.
A complex-valued smooth function (x) belong to the space V (I where k 0,1,2, and for each n,k (k,n) (,kn), (5.3)Then V (I) is a linear space, and {Bk}'=0K is a multinorm on V The topology over V (I) is a subspace of L2(1) when we identify each function in V (I) with the corresponding equivalence class in L2(1).Also convergence in V (I) implies convergence in L2(1).The space V (I) is complete and therefore is Frechet The dual space of V (I) is denoted by V' (I).It can also be shown that V'  This is consistent with the inkier proguct notaticm of L 2 (I).The multiplication by a complex number a is given by (af,) <af,) f,a) a(f,). (5.10) The following two lemmas are useful in the subsequent analysis LEMMA 5.1.If $(x) 6 V (I) then for o < t < o, (x) where the series converges in V (I).
PROOF.By (5 2) fk x,t o(x) is in L 2(I) for each nonnegative Integer k.Hence by [5 pp. 470-471]we may expand k x,t O(x) into a series of orthonormal functions n(X't)" Thus for fixed t, o< t < , Consequently, for each k, x, t nmo t (x), n(X't) n(x't) ( (x) fk x, t n (x' t)) n (x, t) by E ((x), (-%n)k n(X,t)) n ((x), n(X't))(-Xn )k n(X,t) (5.12) (5.3)Using (5.3), (5.12) and the fact that the inner product is continuous with repect to each of its arguments, for any two members and of V we obtain (x,t (x)' X(x)) ((x), fix,t X(x)) (5.14) Therefore the operator fl is a self adjoint operator.xt LEMMA 5.2.Let a denote complex numbers.Then converges in V (I) if and only if converges for every non- n=o negative integer k.
PROOF The proof is similar to that of Zemaniar ,, p 255] and hence it is omitted.

ORTHOGONAL SERIES EXPANSION OF GENERALIZED FUNCTIONS.
The following theorem provides an orthonormal series expansion of enerallzed functions belonging to V' which in turn yields an inversion formula for a certain generalized integral transformation.THEOREM 6.1.Let F(n) be the generalized integral transformation of f V' defined for fixed t by Then F(n) __A (f(x), Sn(X,t)) =-T[f] f(x) F(n) n(X,t), n=o (6.1) (6.2) where the series converges in V' (I).PROOF.By Lemma 5.1, for any (x), V (I), we obtain (f,) (f, E((x), n(X,t)) n(X,t)) (f, n (#, n n--'o E (f' n)(n ')   Fn(n(X,t) (x,t)).n=o n=o This proves that the series on the right of (6.1) converges in V' (I).THEOREM 6.2 (UNIQUENESS).Let f and g be elements of V' and let their transforms F(n) and G(n) satisfy F(n) G(n) for all n, then f g in the sense of equallty in V' PROOF.We have f-g (f-g' n)n [(f' n (g' n )] n 0.
7. CHARACTERIZATION THEOREMS.The following theorem gives a characterization of the transform F(n) of f6 V' Its proof being similar to that of Theorem 9.6-1, p 261 in [i] and heRce it is omitted.
THEOREM 7.1.Let b denote complex numbers.Then for fixed, t, o < t < o; n b n(X t) n (7.1) converges in V' (I) if and only if there exists a nonnegative integer q such that the series absolutely integrable on 0 < x < =, then f(x) generates a regular generalized function f E U' defined by the integral < f'*> I f(x) 0(xx) be a locally integrable function defined for x > 0 such that f(x) O(x p) O(e x2 p + i > 0,x/0+ 0 < 6 < i/(4), x / Then, clearly f U' Let A denote Zemanlan's test function space ([I], p252). )(x) dx does not exist.Hence, f(x) satisfying (2 4) does not belong to A'([I] p258) the boundedness property of generalized functions there exist a positive constant C and a non-negative integer r k W (y,t)P n (x-s) (x) d(x) -0 Y 0 n=N n n, , as Nuniformly for all y > O.PROOF.Using the estimates (2.10) and (2.11) we can write IJl--exp(y2/4o) IA k k W (y,t) Pn (x-s) (x) d(x) y,t) IP n (x -s) l,(x)[ d(x) /8s) l(x) d(x) Since (x) U there exists a positive constant C such that ConsequentlyA k {G(x,y,t-s)} d(x)-k(y)= G(x,y,t-s)[k(X) k(y)] du(x), Y where we have made use of the fact that d(x) i.