Expressing Exponential-Type Conditions in Terms of Derivative Growth Rates

When I talked about my alternative proof of Helgason’s support theorem in class, I asserted that our function $\hat{f}$ had compact support in $B_A(0)$ if and only if its partial derivatives satisfied $|\partial^{\alpha}_\xi\hat{f}(0)| \leq CA^{|\alpha|}$ for any multi-index $\alpha$ .

In other words, a function is of exponential type if its partial derivatives grow like those of an exponential function.

This is true, but the claims I stated on the board were not correct — I did not mention the fact that we needed to use extra information about $f$ , in particular the fact that $f \in \mathcal{S}(\mathbb{R}^n)$ is all we need to complete our proof.

It is possible to state much more general results, but to get the idea across let me state the following result in one dimension:

Claim Assume $f \in \mathcal{S}(\mathbb{R})$ then $\text{supp} f \subset [-A, A]$ if and only if $\hat{f}$ is entire analytic and for all $n$

$\big |\frac{d^n\hat{f}}{dz^n}(0) \big | \leq C A^n$

for some constant $C$ .

To prove this, we note the following

Lemma Assume $f \in \mathcal{S}(\mathbb{R})$ then $\text{supp} f \subset [-A, A]$ if and only if $\hat{f}$ is entire analytic and for all $z$

$| \hat{f}(z) | \leq Ce^{A|z|}$

for some constant $C$ . In this case we will say that $\hat{f}$ is analytic of exponential type.

Pointer to Proof I may elaborate on these ideas in a later post, but for now I will point you to references that prove more general results.

In Rudin’s Real and Complex Analysis, chapter 19, Theorem 19.3 we see that if

$f$ is entire
$\int_{-\infty}^{\infty}|f(x)|^2dx < \infty$ (It is an $L^2$ function on the real line)
$|f(z)| < Ce^{A|z|}$ for all $z \in \mathbb{C}$

Then $f$ is the Fourier transform of a function in $L^2([-A,A])$ .

Similarly, in Complex Analysis, p. 122, Theorem 3.3, Stein and Shakarchi show that if $f$ is continuous with “moderate decrease” on $\mathbb{R}$ , then $f$ can be extended to an entire function on the complex plane satisfying $|f(z)| \leq Ce^{A|z|}$ if and only if $\hat{f}$ is supported in $[-A,A]$ . (Their statement looks a bit different because they use a different definition for the Fourier transform, I restated the result for consistency). Thanks for pointing this out, Huy!

Both of these results directly imply the lemma.

Having let Stein and Rudin do the heavy lifting, we can prove our claim using the direct estimates I put on the board in class. Here is a recap:

Step 1: If $f$ has compact support, the derivatives of $\hat{f}$ satisfy our bounds and $\hat{f}$ is analytic.

Specifically, let $f \in C_0^{\infty}, \text{supp} f \subset [-A,A]$ . Then

$|\frac{d^n}{dz^n}\hat{f}(0)| = |\int_{-A}^{A}f(x)\frac{d^n}{dz^n}e^{-ixz}dx$

$\leq \int_{-A}^{A}|f(x)||(-ix)^ne^{-ixz}|dx$

$\leq \int_{-A}^{A}|f(x)|A^ndx = \|f\|_1A^n$

So we have our derivative bound. The standard Paley-Wiener theorem tells us that $\hat{f}$ is analytic of exponential type.

Now we need to go the other direction

Step 2: If $\hat{f}$ is analytic and the derivatives of $\hat{f}$ satisfy our bounds, then $f$ has compact support

This result is straightforward as well. We will prove that $latex \hat{f}$ is of exponential type as defined in our lemma, i.e. $|\hat{f}(z)| < Ce^{A|z|}$ .