The following is from Stein and Shakarchi's Complex Analysis:

For each $a>0$ we denote by ${\mathcal F}_a$ the class of all functions $f$ that satisfy the following two conditions:

  1. The function $f$ is holomorphic in the horizontal strip $$ S_a=\{z\in{\Bbb C}:|Im(z)|<a\} $$
  2. There exists a constant $A>0$ such that $$ |f(x+iy)|\leq\frac{A}{1+x^2}\quad\text{for all}\quad x\in{\Bbb R}, |y|<a. $$

Denote by ${\mathcal F}$ the class of all functions that belong to ${\mathcal F}_a$ for some $a$. The the Fourier inversion holds in this class.

Here are my questions:

  • Is there a name for this class?
  • Does it have anything to do with the Schwartz space on which the Fourier transform is a linear isomorphism?
  • $\begingroup$ I don't think there's a canonical name for this class, nor any relation to the Schwartz space. $\endgroup$ – Potato Feb 21 '13 at 1:28

For the sake of an answer, this is done on MO.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.