I nternat. J. Math. & Math. Si.
Vol. 3 No. 4 (1980) 773- 787
773
EXPANSIONS OF DISTRIBUTIONS IN TERM OF GENERALIZED HEAT POLYNOMIALS
AND THEIR APPELL TRANSFORMS
R. S. PATHAK
Department of Mathematics, Banaras Hindu University
Varanasi, India
and
LOKENATH DEBNATH
Mathematical Institute, University of Oxford
Oxford, England
(Received November 13, 1978 and in revised form July 15, 1980)
ABSTRACT.
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.
KEY WORDS AND PHRASES. Epansion of distributions (or generalized func,ons ),
Generalized heat polynomials and their Appell tra forms, Orthogonal Si
expansion of generalized functions and generalized integral transformations.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i.
46F12, 46F10, 44A05.
INTRODUCTION.
Zemanian [i] has developed a procedure for expanding certain Schwartz distri-
butions (or generalized functions) in a series of the Fourier type.
He has also
shown that every distribution of polynomial growth (or tempered distribution) ca6
be expanded in a series of Hermite functions.
Some general results concerning the
orthogonal series expansion of generalized functions are also available in recent
literature including reference listed by Zemanian [i].
In a paper, Korevaar [2] has developed a general theory of Fourier transforms
774
R.S. PATHAK AND L. DEBNATH
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
faster than exp (ex
2)
<
i
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
This work is an extension of some results
polynomials and their Appell transforms.
of Rosenbloom and Widder [7] on expansions in terms of heat polynomials and associDespite these works, no attention is given to the expansions of
ated functions.
distributions in terms of heat polynomials and their Appell transforms.
This paper is devoted to the study of expansions of distributions in terms of
A theorem which pro-
the generalized heat polynomials and their Appell transforms.
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.
2.
PRELIMINARY RESULTS ON THE GENERALIZED HEAT POLYNOMIALS.
For real values of x and t, the generalized heat polynomials,
Appell transform,
nW-, (x
Pn
(x,t) and
t) are defined by
n
22r ()
Pn,v (x,t)
i
F( + )
F( +
1
x
+
n
2(n-r)
t
r
(2.1)
r)
r=0
Wn,v(x,t)
where n
0,1,2,...
9
t
-2n
Pn, (x,-t),
t > 0,
is a fixed positive number and
G(x,t)
When
G(x,t)
(2t)
-( +
1/2)
G(x,t)
(2.2)
is given by
2
exp(-
)
(2.3)
0, it follows from (2.1) that
P
P
no
no
(x,t)
(x ,-i)
V2n(X, t),
H2n(),
(2.4)
(2.5)
its
EXPANSIONS OF DISTRIBUTIONS IN HEAT POLYNOMIALS
where
V2n(X,t
775
is the ordinary heat polynomial of even order defined by Rosenbloom
and Widder [7, p. 222], and
is the Hermite polynomial of even order given in
H2n
Erdelyi’s book [8].
-
It can readily be verified that for
< x, t <
generalized heat equation
A
where the operator A
It is noted that
=---22
X
x
X
Wn,(x,t)
x
satifies the
U
u(x t)
2v
+ (--)
, Pn, (x,t)
(2 6)
with some fixed positive number
Frther,
(2.6).
is also a solution of
nW,v(x’t)
satifies the following operator formulas
W
n,
22n
(x,t)
A
n
x
G(x,t),
(2.7)
with
A
k
W
(x t)
x n,9
-2k
2
(x t).
Wn+k,
(2 8)
With the help of the following results reported by Erdelyi
P
n,
(x,-t)
(-I)
n
22n
ILn(X)
where
L(x)
n
in1’2.
[
n(
(e)n
L (-1/2)
n
n
(t)
t > 0,
exp(),
n,
F (n+)
F()
(e+l)n
> -i and
0]
>
-+/-,.(+l)n
n
(2.9)
(2.10)
is the Laguerre polynomial of degree n and order
A
with
< A
n. t
[8]
2
-i < e <
(2.11ab)
it can easily be shown that for t > 0
IP n, (x -t)
(x t)
<
<
n’
22n--1/2
22n
A
n,9-1/2
n’. A n, -1/2
t
n
exp
x
2
()
t-(n+v)
exp(-
(2.12)
x
2
St’"
(2.13)
The differential d(x) is defined in a paper of Haimo [6] as
d(x)
2
(1/2-)
It is interesting to observe that P
0 < x < =.
Indeed,
n,
i -i
[F( + )
x2dx.
(x,t) form a biorthogonal system
(2.14)
in
R.S. PATHAK AND L. DEBNATH
776
0
Wn__
(x,t)
Pro, v (x,-t)
nm
d(x)
b
(2.15)
t > 0,
n
where
b
n
i
F(v + )/[
24n n.’
I
r(v + n + )]
(2.16)
An important consequence of (2.15)-(2.16) is the bilinear generating function
for the biorthogonal set P
and W
n,9
n,9
G(x,y; s + t)=
as
Z
Pn
b W
(y,s)
(x,t).
n n,9
,9
n=o
The source solution of (2.6) is given by G(x,t) in the for
(2t) -() exp[-(x 2 +
G(x,y;t)
(x)
2 9-1/2 F(9
y2)/4t]
(2.17)
(2t),
(2.18)
x1/2- 19_1/2 (x),
1
+ y)
(2 19)
and I (x) is the modified Bessel function of imaginary argument of order r; and
r
S(x,t).
G(x,0;t)
THE TEST FUNCTION SPACE U
3.
We denote the open interval (0,) by I.
A complex valued infinitely
differentiable function (x) defined over I belongs to the space U
2
-
yk()
A_
sup
O<x<
lexp(’’)
A k (x)
x
(I) if
(3.1)
<"
for any fixed k, where k assumes the values 0,i,2,...; c is a positive real number,
>
1
and A
x
is the differential operator involved in section 2.
The topology in the space U
{y k }k==
Since
dicated by
Zemania
k--0
if for each k,
_r(X)
r,s
/
is a norm, the collection of seminorms is separating as in-
7o
yk(r
belonging to
is induced by the collection of seminorms
[i, p8]
A sequence
) tends
Uc,v(1)
{r- }I "is" said
to zero as r
/ (R).
to converge to
{r} r--I
A sequence
is a Cauchy sequence in U
(I) if
yk(r__
independently of each other for every fixed k, where k
is noted here that U
(I)
in U
c’
with each
_s
/
0 as
0,1,2,
It
(I) is a locally convex, sequentially complete, Hausdorff
topological linear space.
Its dual space U’
functions under consideration.
(I) is the space of generalized
777
EXPANSIONS OF DISTRIBUTION IN HEAT POLYNOMIALS
From the estimate (2.10), it follows that for x > 0, s > c and
U
(x) P n, (x)- s)
seen that W
(x,t)
, (I).
Using results (2.4)
(2 5) and (2.8)
(I) for x > 0 and 0 < t <
U
>
-,1
it can also be
.
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
The dual space
with a compact support.
[9] on I.
D’ (I)
is the space of Schwartz distributions
It can easily be shown that D(1) is a subspace of the space U
that the topology of (I) is stronger than that induced on it by U
(ii) The space U
(I) and
(I).
(I) is a dense subspace of g(1) which is the space of all
The topology of U
complex-valued smooth functions on I.
by 8(I).
topology induced on U
is stronger than the
So 8’ (I) can be identified with a subspace of
(ill) The space S’ of tempered distributions is a subspace of U’
o’))
i
for 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
for every
(v)
( U
yk()
(3.2)
[8,p19].
If f(x) is a locally integrable for x > 0 such that f(x) exp(- x
2
)
function f E
U’
=, then f(x)
is
generates a regular generalized
defined by the integral
<
(vi)
max
0<k<r
where r and C depend on f but not on
o’))
absolutely integrable on 0 < x <
where
< C
(x)
U
I
f’*>
f(x) 0(x) dx
(3.3)
0
(I)
Let f(x) be a locally integrable function defined for x > 0 such that
O(xp)
f(x)
O(e
Then, clearly f
U’
Let
A
x2
p
+
i > 0,
x/0+
0 < 6 < i/(4), x
/
denote Zemanlan’s test function space ([I], p252).
R.S. PATHAK AND L. DEBNATH
778
e-X/2 Ln () (x)
1/2
(x)
Then
x
E A.
f(x) (x) dx does not exist.
But
satisfying (2 4) does not belong to
A’([I]
p258)
Since D(1) is contained in both the spaces U
spaces U’
4.
A’
and
and
A,
A’
it follows that the
A’.
U’
over-lap but
U’O,
Therefore
Hence, f(x)
EXPANSION OF DISTRIBUTIONS.
,
The expansion of f E
Appell transforms W
n,
U’
in terms of heat polynomials P
(x,-t) and their
(x,t) is given in the following theorem:
Let f6U’
THEOREM 4.1.
n,
i f(x), (x)
where
Then, for 0 < o/2 < t < o,
N
>
where
> i and o > 0.
a
K
lim
s/t-
N-> n=o
n
lim
knan(t) <(X)Pn (x’-s)’
(t) A If(y)
r(v+
n
i
W
n,
-) / [2
4n
-
(x)
(y,t)>,
n. F(
+
i
>,
(4.1)
(4.2)
+ n)].
(4.3)
We shall need the following lammas for the proof of this theorem.
For f 6 U’
LEMMA 4.1.
a
Then for
n
> 0 and o >
where
(t) A If(y)
W
n
, (y,t)>
0, we define
0,1,2
(4.4)
+ 2(n+r)’.],
(4.5)
n
0 < t < o,
n
<_ C(4/t) [( +
lan(t)
i
)n+r
where C and r are independent of n.
PROOF.
By the boundedness property of generalized functions there exist a
positive constant C and a non-negative integer r such that
lan
(f(y)
<_
C
W
max
0<k<r
n,
(y,t)>
Yk (Wn
(y’ t)
2
C
max
0<k<r
sup
0<y<
12-2k ey / (4)
Wn+k
’
(y, t)
We arrive at the desired estimate using result (2.11).
LEMMA 4 2
Let (x) E U
exp(y2/40) IA kY
,
Then, for 0 < 0/2 < s < t <
(y,t)P
k W
n
n n,
0 n=N
as N- uniformly for all y > O.
, (x-s)
(x) d(x)
,
-
0
EXPANSIONS OF DISTRIBUTIONS IN HEAT POLYNOMIALS
PROOF.
779
Using the estimates (2.10) and (2.11) we can write
IJl-- exp(y2/4o) IA ky
Pn , (x-s)
k W
(y,t)
n n,
(x) d(x)
0 n=N
exp(y2/4o)
<
k
n
0
2
exp[-
<
1
n=N
)
4(
)
1
+
F
)
l,(x)[
d(x)
I
F( +
i
IWn+k,(y,t) IP n ’) (x -s)
I
F (v-I-
(n+k)
+
An+k,_1/2 An,_1/2
n
2
2k--1/2 X
n=N
Xt
Since
-2k--1/2 (s/t) n
(x)
U
exp(x2/8s) l(x)
d(x)
there exists a positive constant C such that
IJl
<_
C
E
F( + i + k + n)
(n + I)
n=N
<_ C 1
n
(s/t)n
+k-1/2 (s/t) n
n=N
where C
N
1
Clearly the last term tends to zero as
is another positive constant.
for s < t.
/
For 0 < o/2 < s < t <
LEMMA 4.3.
(2.14) and let (x) 6 U
Al(n)
such that for y >
exp(y
2/4)
A
0
PROOF.
,
let G(x,y;t-s) be the function defined by
Then there exist e > 0 and a large positive constant
Al(n),
k
{G(x,y,t-s)}
y
(x)
d(x)]
<
0,1,2,
k
Proceeding as in the proof of Lemma 4.2, we write
I Aky
exp(y2/4)
<_
C
2
exp[-y
2
{G(x,y,t-s)) (x) d(x)
i
/4(-ci)
F(
n=o
c3
exp[-y2/4( !)](
for appropriate constants C
Hence for t <
,
2
and C
+
+ k + n)
r(n + )
s
)
3.
the last expression tends to zero as y
.
R.S.
780
.
Let A I,
LEMMA 4.4.
let
i
>
PATHK
AND L. DEBNATH
, G(x,y,t-s) and (x)
be the same as in
Lemma 4.3 and
Then for t-s < (n,),
exp(y2/4)
{G(x, Y, t-s)} (x) d(x)
k(y)
<
0
kY
k (y)
where
PROOF.
(0,AI).
(y), uniformly for all y
If (x).
then proceeding as in [i0, p846], it follows that
U
i
>2
for
(k)(x) 0(e-CX),
(k)
(0+)
A x G(x,y,t)
by
A
y
x
/
exist finitely,
Using these orders of
0,1,2,
for each k
c > 0,
(k)
(x) and the identity
G(x,y,t) and integrating by parts, we obtain
G(x,y,t-s) A x (x) d(x).
{G(x,y,t-s)} (x) d(x)
Consequently
A
k
Y
{G(x,y,t-s)}
G(x,y,t-s)[k(X)
d(x)-k(y)=
k(y)]
du(x),
where we have made use of the fact that
d(x)
i.
We next use the standard technique [I0] and remember that (x) is not an element
of 9(1) although it does belong to U
exp(y2/4)
to show
G(x,y,t-s)
uniformly for all y satisfying 0 < y <_ A
{k(X)
k(y)
d(x)
< e
I
We are now prepared to prove the main expansion theorem 4.1.
PROOF OF THEOREM 4.1.
Let t be a fixed number such that 0 <
Then using the estimates (2.10) and (4.5), we obtain
< s < t < o
EXPANSIONS OF DISTRIBUTIONS IN HEAT POLYNOMIALS
781
Knlan]l((x) en, (x,-s), (x) >I
n--o
(v +
F( +
n=o
F(
++n’
1/2)n+r +
r
+ n)
1
+
2(n + r)’.
n
n, 9 -1/2
r)
n
n
<n +n. r)’.
+ 2C
i
( >
n--o
n=o
<__C I
where C
I
is a certain positive constant.
This proves the existence of the limit
N+, when s < t.
Using the llnearlty of U
N
n--o
and
an kn <(x) Pn,(x’-s)’
N
n=o <
n=o
U’
we can write
(x)>
f(Y)’
Wn,9(y’t) >
kn Pn,9(x’-s)’
#(x)
>
f(Y)’
Wn’(y’t) > < (x) kn pn,(x’-s)’
(x)
>
(x)
>>
N
(x)
N
E (x)
n--o
f(Y)
< f(y), < p(x)
Pn ,9 (x -s)
(x) > >
(y,t)
k W
n n,9
GN(X,y,t,s),
N
GN(X,y,t
where
s)--
k
n--o
n
W
n,9
(y,t)
en
,9
The corresponding infinite series equals G(x,y,t-s).
(x) G(x,y,t-s) E
function of y,
U’G,9
< (x)
and (x)
G(x,y,t-s),
GN(X,y,t,s),
(x)>
E
U’G,
(x-s)
We observe that
for s >
is an element of U
we know that
in U
Hence
To complete the proof we have to show that
Furthermore, as a
From Lemma 4.2,
R.S. PATHAK AND L. DEBNATH
782
<(x) G(x,y,t-s), (x)
lim
s-t-
(y)
in U
In other words, we need to show that for all k and e, there exists (n,e) such
6(n,e),
<
that, for 0< t-s
exp(y2/4o)
(y),
Since
U
A
k
Y
{G(x,y,t-s)} (x) d(x)
we have for y >
k(y)
< e
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),
exp(y2/4)
]0
Y
{g(x,y,t-s)} (x) d(x)
<
e
2
An application of Lemma 4.4 completes the proof of the theorem.
5.
(I) AND ITS DUAL.
THE TEST FUNCTION SPACE V
x,t
Suppose
denotes the differential operator given by
x, t
x
2
16t
dx
2
+ (-i)
(5.1)
x
(0,) and t is a fixed real number.
I
x
where
-
2
d
4t[-2
A complex-valued smooth function (x) belong to the space V
i
ak
8k(0) =A
(x)] 2
dx]
2
<
(I) if
,
(5.2)
0
where
0,1,2,
k
and for each
n,k
(k,n) (,kn),
Then V
(I)
over V
(I) is a subspace of
is a linear space, and
L2(1)
{Bk}’=0K
L2(1).
The dual space of V
The space V
L2(1).
The topology
Also convergence in V
(I) is complete and therefore
(I) is denoted by V’
is sequentially complete.
is a multinorm on V
when we identify each function in V
the corresponding equivalence class in
convergence in
(5.3)
(I).
(I) with
(I) implies
is Frechet
It can also be shown that V’
(I)
783
EXPANSIONS OF DISTRIBUTIONS IN HEAT POLYNOMIALS
We note that
(5.4)
x,t n(X’t) =-%n n(x,t)
where
4n
with
n(X,t),
x[G(x,t)]2 Pn,9(x’-t)
(5.5)
+ 2 +
(5.6)
{F(v)}2
n(X,t)
Therefore
i
i
21/2_
i
(I) for fixed t.
as a function of x, is a member of V
Following are important properties of these spaces: (i) for fixed t,
nature of the operator
operator
x,t
on
x,t
V’
f(x)
for fixed t, where f 6
(I)
(x)) A (f(x)
V’
g(I)
(I), and the convergence
V’
in
D(1) implies convergence
(I) to D(1) is in ’(I).
(I) implies convergence in D’ (I).
g (I) and since D(I) is dense in g(I), V
(I)
Furthermore
if {qb
}oo
m .,1
.
converges in V
converges in g(I) to the some limit
(iv) We imbed
(5.7)
(x))
Consequently, restriction of any f 6
Moreover, convergence in V’
(iii) V
x,t
6 V
and
is a subspace of V
(I).
in V
[see (5.1)] we define the generalized differential
by
(x,t
(ii)
In view of the self adjoint
(I) into itself.
continuous linear mapping of V
L2(I)
is a
x,t
(and therefore
defining the number that f, (:
L2(I)
(f,,)
(I) to the limit q
assigns to any qb
A__
i0
then
(I)
c_
V
(I) as
L2(I
into
V’
also
(I).
(I) by
(5.8)
f(x) ,(x) dx.
This imbedding of
Then f is linear and continuous on V
{m }
V’
The space g’ (I) is a subspace of
(I) since V
V
(I) is also dense in
L2(1)
into
V’
(I) is
one to one.
(v) If f(x)
kx,t
g(x)
for fixed t and g
Instead of working with the number
.
L2(1)
(f,)that
and some k
f 6
V’
then f
assigns to
, (I).
V’
6 V
it is more convenient to work with the number that f assigns to the complex conjugate
of
We write
R.S. PATHAK AND L. DEBNATH
784
(f,)
This is consistent with the inkier
i
f,>
(5.9)
proguct notaticm of L 2 (I).
The multiplication
by a complex number a is given by
f,a)
<af,)
(af,)
(5.10)
a(f,).
The following two lemmas are useful in the subsequent analysis
LEMMA 5.1.
If
$(x) 6 V
(I) then for o < t < o,
((x),
(x)
(5.11)
n(X,t)) n(X,t)
n--o
(I).
where the series converges in V
PROOF.
by [5
By (5 2)
fkx,t
o(x) is in L 2(I) for each nonnegative Integer k.
pp. 470-471] we may expand
n(X’t)"
kx,t
Thus for fixed t, o< t <
x, t
(x),
n(X’t) n(x’t)
( (x)
fkx, t
n (x’ t)) n (x, t)
((x),
(-%n)k n(X,t))
((x),
n(X’t))(-Xn )k n(X,t)
n
Z
n=o
a series of orthonormal functions
,
t
nmo
En
O(x) into
Hence
(,n) flkx,t
by
n
(5.12)
Consequently, for each k,
N
k[(x)-
(,
n n
/
0
as N
(5.3)
/
n--o
This proves the lemma.
Using (5.3), (5.12) and the fact that the inner product is continuous with
and
repect to each of its arguments, for any two members
(x,t
(x)’ X(x))
Therefore the operator fl
LEMMA 5.2.
converges in V
Let a
xt
((x),
fix,t
of V
we obtain
(5.14)
X(x))
is a self adjoint operator.
denote complex numbers.
Then
converges for every non-
(I) if and only if
n=o
negative integer k.
PROOF
The proof is similar to that of Zemaniar
,,
p 255] and hence it is omitted.
785
EXPANSIONS OF DISTRIBUTIONS IN HEAT POLYNOMIALS
ORTHOGONAL SERIES EXPANSION OF GENERALIZED FUNCTIONS.
6.
The following theorem provides an orthonormal series expansion of
V’
functions belonging to
enerallzed
which in turn yields an inversion formula for a
certain generalized integral transformation.
Let F(n) be the generalized integral transformation of f
THEOREM 6.1.
V’
defined for fixed t by
F(n)
__A
Sn(X,t))
(f(x),
(6.1)
=-T[f]
Then
F(n)
f(x)
n=o
V’
where the series converges in
(6.2)
n(X,t),
(I).
PROOF.
By Lemma 5.1, for any (x),
(f,)
(f,
(I), we obtain
V
E((x), n(X,t)) n(X,t))
E
n
(f,
n
(f’
n)(n ’)
n--’o
n=o
(#,
n=o
Fn(n(X,t)
(x,t)).
This proves that the series on the right of (6.1) converges in
THEOREM 6.2 (UNIQUENESS).
Let f and g be elements of V’
transforms F(n) and G(n) satisfy F(n)
(f-
CHARACTERIZATION THEOREMS.
g’
n)n
g in the sense
f6 V’
p 261 in [i] and
it is omitted.
THEOREM 7.1.
heRce
Let b
n
n
(g’
n )] n
0.
Its proof being similar to that of Theorem 9.6-1,
denote complex numbers.
b n n(X
converges in V’
[(f’
The following theorem gives a characterization of
the transform F(n) of
the series
and let their
We have
f- g
7.
(I).
V’
of equallty in
PROOF.
G(n) for all n, then f
V’
t)
Then for fixed, t, o < t <
o;
(7.1)
(I) if and only if there exists a nonnegative integer q such that
786
R,S.
PATHAK AND L. DEBNATH
l%n l-2q Ibn 12
V’,v
Furthermore, if f denotes the sum in
converges.
of (7.1) then
n )"
(f’
bn
The next theorem is analogous to that of Theorem 9.6-2, p 262 [I] and gives a
V’
characterization of elements of
A necessary and sufficient condition for f to be a member of
THEOREM 7.2.
V’
(I).
(I) is that there be some nonnegative integer q and a
fixed t,
q
f (x)
where c
which X
8.
are certain complex
n
g(x) +
x,t
.
E n n(X,t),
constantsLa-d
%
O; these are finite
n
in number.
g L2(1)
o < t
such that for
(7.2)
<_
denotes a summation on those n for
n=o
AN OPERATIONAL CALCULUS.
Since
x,t
is a continuous linear mapping of
,
V’
for every f
V’
o,v
we can
write
kx,t
f(x)
k
x,t
n (x’t)
(f(x)
n(X
(f(x),
n(X,t))(-Xn)k n(X,t)
t))
n=o
(8.1)
n--o
Using this fact we can solve the differential equation
P(x,t
(8.2)
g(x)
u(x)
where P is a polynomial and the given g and unknown u are required to be in
V’
Applying the transformation T defined by (6.1) we obtain
P(% n
If
P(Xn)
# 0 for every n,
U(n)
G(n),
U-- T u
we can divide P(X
u
=o
P(n
n
G
T g.
and apply T
-1
to obtain
n
(8.3)
,
By Theorems 6.2 and 7.1, this solution exists, and is unique in V’
for some %
n
say, for %
only if G(%
n
0 for k
(k
,
l,...,m), then a solution exists in V’
k
1,2
,m.
k=l
nk
n
if and
In this case solution (8.3) is no longer
unique and we may add to it any complementary solution
m
a
where a are arbitrary constants,
k
k
Uc
If P(%
(8.4)
0
EXPANSIONS OF DISTRIBUTIONS IN HEAT POLYNOMIALS
787
REFERENCES
i.
ZEMANIAN, A.H.
(1968).
2.
KOREVAAR, J.
3.
GELFAND, I.M. and SILOV, G.E.
4.
EHRENPREIS, L. Analytic Functions and the Fourier Transform of Distributions
I, Ann. Math. Vol. 63 (1956) pp. 129-159.
5.
HAIMO, D.T.
6.
HAIMO, D.T.
7.
ROSENBLOOM, P.C. & WIDDER, D.V.
8.
ERDELYI, A.
(1953).
9.
SCHWARTZ, L.
i0.
PANDEY, J.N.
ii.
PANDEY, J.N.
Generalized Integral Transformation, Interscience, New York
Pansions and the Theory of Fourier Transforms, Trans. Amer. Math.
Soc. 91 (1959), 53-101.
Fourier Transforms of Rapidly Increasing Functions
and Questions of Uniqueness of the Solution of Cauchy’s Problem, Uspehi.
Mat. Nauk. (N.S.) Vol. 8 (58) (1963) pp. 3-54.
2
L expansions in Terms of Generalized Heat Polynomials and of
their Appell Transforms, Pacific J. Math. 15 (1965), pp. 865-875.
Expansions in Terms of Generalized Heat Polynomials and of their
Appell Transforms, J. Math. Mech. 15 (1966), 735-758.
Expansions in Terms of Heat Polynomials and
Associated Functions, Trans. Amer. Math. Soc. 92 (1959), 220-266.
Higher Transcendental Functions, Vol. II, McGraw-Hill, New York
Theorie des Distributions, Tomes
I, II, Hermann
Paris 1950, 1951.
Complex Inversion for the Generalized Hankel Convolution
Transformation, SlAM J. Appl. Math. 17 (5) (1969), pp. 835-848.
The Generalized Weierstrass Hankel Convolution Transform, SlAM
J. Appl. Math. 20 (I) (1971), pp. 110-123.