A function 
is called a Schwartz function if it goes to zero as 
faster than any inverse power of 
, as do all its derivatives. That is, a function is a Schwartz
function if there exist real constants 
such that
 |
where multi-index notation has been used for 
and 
.
The set of all Schwartz functions is called a Schwartz space and is denoted by 
. It can also be proven that the Fourier
transform gives a one-to-one and onto correspondence between 
and 
, where the pointwise product is taken into the convolution
product and vice versa. The Fourier transform has a fixed point in 
, which is the function 
, the Gaussian function. Its image under the
Fourier transform is the function 
(times some factors of 
).
Instead of 
,
one can also consider 
. It consists of functions 
that go to zero, as 
, faster than any inverse power of 
(
). It is well known that the Fourier transform carries
onto 
,
where 
is the 
-torus,
defined as the direct product of 
copies of the circle 
.
