We employ a function
such that
Let
and
,
. Then, for each
,
is well defined. We call
a regularizing function of the distribution
, since
is a smooth function of
satisfying
in the sense of distributions, that is, for every
,
Theorem 3.1.
Let
,
,
, be a solution of (1.3), and both
and
are linearly independent. Then,
,
,
,
,
, where
,
,
, a smooth solution of (1.1).
Proof.
By convolving the tensor product
in each side of (1.3), we have, for
,
where
,
,
. Similarly we have for
,
Thus (1.3) is converted to the following functional equation:
where
for
,
. We first prove that
are smooth functions and equal to
for all
. Let
Then,
is a smooth function of
for each
,
, and
is linearly independent. We may choose
,
such that
. Then, it follows from (3.5) that
where
,
. Putting (3.9) in (3.5), we have
where
Since
is a smooth function of
for each
,
, it follows from (3.11) that
is a smooth function of
for each
,
. Also, since
is linearly independent, it follows from (3.12) that
is linearly independent. Thus we can choose
,
such that
. Then, it follows from (3.10) that
where
,
. Putting (3.15) in (3.10), we have
where
By continuing this process, we obtain the following equations:
for all
, where
,
,
,
for all
, and
By the induction argument, we have for each
,
is a smooth function of
for each
,
. Thus, in view of (3.20),
is a smooth function. Furthermore,
converges to
locally uniformly, which implies that
in the sense of distributions, that is, for every
,
In view of (3.19) and the induction argument, for each
, we have
is a smooth function and
for all
. Changing the roles of
and
for
, we obtain, for each
,
is a smooth function and
. Finally, we show that for each
,
is equal to a smooth function. Letting
in (3.5), we have
For each fixed
,
, replacing
by
, multiplying
and integrating with respect to
, we have
where
for all
,
. Letting
in (3.27), we have
It is obvious that
is a smooth function. Also it follows from (3.27) that each
,
, converges locally and uniformly to the function
as
, which implies that the equality (3.28) holds in the sense of distributions. Finally, letting
and
in (3.5) we see that
,
are smooth solutions of (1.1). This completes the proof.
Combined with the result of Aczél and Chung [1], we have the following corollary as a consequence of the above result.
Corollary 3.2.
Every solution
,
,
, of (1.3) for the dimension
has the form of exponential polynomials.
The result of Theorem 3.1 holds for
,
,
. Using the following
-dimensional heat kernel,
Applying the proof of Theorem 3.1, we get the result for the space of Gelfand generalized functions.