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 ,
Let , , , be a solution of (1.3), and both and are linearly independent. Then, , , , , , where , , , a smooth solution of (1.1).
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:
for , . We first prove that are smooth functions and equal to for all . Let
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
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
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 , we have the following corollary as a consequence of the above result.
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.