A Schwartz function is a C∞ function 
that goes to zero as 
faster than any inverse power of 
, as do all its derivatives.
That is, for all 
and 
with nonnegative integer
components, there is a real constant 
such
that
 |
where multi-index notation is used for 
and 
, and 
is the standard l2-norm.
The set of all Schwartz functions is called a Schwartz space and is denoted by 
. The Fourier transform
gives a one-to-one correspondence between
and itself, taking the pointwise product into the convolution product and vice versa (Hunter 2014, p. 169).
With the convention having integral kernel 
, the Gaussian
function 
is a fixed point of the Fourier
transform.
Analogous to 
,
the class 
consists of functions 
that go to zero as 
faster than any inverse
power of 
.
Taking Fourier coefficients gives a one-to-one
correspondence from 
onto 
, where 
is the 
-torus, defined as the direct
product of 
copies of the circle 
.
