When I talked about my alternative proof of Helgason’s support theorem in class, I asserted that our function had compact support in if and only if its partial derivatives satisfied for any multi-index .

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 , in particular the fact that 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 then if and only if is entire analytic and for all

for some constant . Continue reading