This is an extension to my previous post A short note about tempered distributions. Our objective is to discuss the Fourier transform of distributions and give some “little” applications. Before that let us review some basics.
We shall use the multi-index notation. Denote by the space of compactly supported smooth functions on which is an open set in .
Definition 1 A linear map is called a distribution if for any compact set there are and a nonnegative integer such that
for any and . Denote by the space of distributions.
Example 1 (Delta distribution on ) .
Example 2 Let be the space of locally integrable functions, that is, functions which are integrable on any compact subset of . If , then induces a distribution given by
For example, is locally integrable on so it defines a distribution.
Definition 2 (Restriction of a distribution) If is open, then given any one can define the restriction of to by: given any , we view by zero extension, then define
Example 3 (p.v. on ) Define
By mean value theorem one can check defines a distribution. Clearly its restriction to is the distribution .
Definition 4 (distributional derivative) Given one defines
One can see that this definition comes from integration by parts formula: whenever one of has compact support in .
Example 4 .
Example 5 Define the Heaviside function by if and if . Then .
Definition 5 (support of a distribution) Given one defines the support of by
complement of the largest open set such that the restriction of to that set is 0.
Theorem 1 (characterization of zero support distribution) If be such that , then there are a nonnegative integer and complex numbers such that
Definition 6 Given . Define the Fourier transform of by .
Example 6 .
Definition 7 (Schwartz class) A function is called rapidly decreasing if for all , . Denote by the space of such functions, and call this collection the Schwartz class. The above seminorms generate a complete metrizable topology (see Frechet space) of .
Theorem 2 The space is dense in .
Definition 8 (tempered distribution) A linear map is called a tempered distribution if there is a constant and a nonnegative integer such that
for all . Denote by the space of tempered distributions.
Proposition 3 A locally integrable function on of polynomial growth (that is, there exist and non-negative integer such that
induces a tempered distribution on .
Proof. Since is locally integrable, it induces a distribution. Next we check is integrable for . Indeed for such , which is integrable as is rapidly decreasing. So is integrable. Thus is well defined on .
It remains to establish a semi-norm estimate for in order to show it is continuous on . For each ,
where is some constant, and is some nonnegative integer.
Definition 9 The Fourier transform of is given by .
One idea of defining Fourier transform like this comes from the fact that
for all . This fact can be proved by Fubini’s Theorem.
Theorem 4 The Fourier transform is a continuous isomorphism , so does its inverse.
Example 7 and . Also .
Example 8 Show that
are tempered distributions on . Also show that , where denotes the Fourier transform.
Solution. For the first part, a direct check would work. Or it is just an application to some general results. Let be the Dirac delta distribution. For any two distributions and on , one can define the tensor product
Using semi-norm estimates one can check is a tempered distribution on whenever and are tempered distributions on . In our case, while . By the above proposition is a tempered distribution on . It is known that is tempered too. Thus is a tempered distribution on , and so is . The second part follows from the following facts which can be checked by computations (on ):
In our case, .
Proposition 5 Let be a linear differential operator with constant coefficients. Denote . If whenever and satisfies , then is a polynomial.
Proof. Taking Fourier transform on both sides of one gets (Note ). The assumption on implies . So by Theorem 1,
which is the Fourier transform of for some constants (cf. Example 7). Then by Theorem 4, is a polynomial.
Corollary 6 (Liouville’s Theorem in complex variables) Let be an entire function satisfying
for some constant and nonnegative integer . Then is a polynomial of degree at most .
Proof. By Proposition 3 one can view . Let . Clearly $P$ is nonzero if . And . Since is entire, it satisfies the Cauchy-Riemann equations so . Thus . By the previous proposition is a polynomial. By the given estimate has degree at most .
Corollary 7 Let be such that
for some constant and nonnegative integer . If is a polynomial then is a polynomial.
Proof. Let . Taking Fourier transform on both sides and following the same argument as the proof of Proposition 5, one sees that so is a polynomial.
The following is a direct consequence.
Corollary 8 If is a harmonic function on satisfying
for some constant and nonnegative integer , then is a polynomial of degree at most .
For the case , this result can be proved using tools in complex analysis say Poisson integral formula and Cauchy estimate.
(To be continued…)