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.