32.3. FOURIER TRANSFORMS OF JUST ABOUT ANYTHING 1111

32.3.4 The Schwartz Class

The problem with G is that it does not contain C∞c (Rn). I have used it in presenting the

Fourier transform because the functions in G have a very specific form which made sometechnical details work out easier than in any other approach I have seen. The Schwartzclass is a larger class of functions which does contain C∞

c (Rn) and also has the same niceproperties as G . The functions in the Schwartz class are infinitely differentiable and theyvanish very rapidly as |x|→∞ along with all their partial derivatives. This is the descriptionof these functions, not a specific form involving polynomials times e−α|x|2 . To describe thisprecisely requires some notation.

Definition 32.3.20 f ∈S, the Schwartz class, if f ∈C∞(Rn) and for all positive integersN,

ρN( f )< ∞

whereρN( f ) = sup{(1+ |x|2)N |Dα f (x)| : x ∈ Rn , |α| ≤ N}.

Thus f ∈S if and only if f ∈C∞(Rn) and

sup{|xβ Dα f (x)| : x ∈ Rn}< ∞ (32.3.13)

for all multi indices α and β .

Also note that if f ∈S, then p( f ) ∈S for any polynomial, p with p(0) = 0 and that

S⊆ Lp(Rn)∩L∞(Rn)

for any p ≥ 1. To see this assertion about the p( f ), it suffices to consider the case of theproduct of two elements of the Schwartz class. If f ,g ∈S, then Dα ( f g) is a finite sum ofderivatives of f times derivatives of g. Therefore, ρN ( f g)< ∞ for all N. You may wonderabout examples of things in S. Clearly any function in C∞

c (Rn) is in S. However there areother functions in S. For example e−|x|

2is in S as you can verify for yourself and so is any

function from G . Note also that the density of Cc (Rn) in Lp (Rn) shows that S is dense inLp (Rn) for every p.

Recall the Fourier transform of a function in L1 (Rn) is given by

F f (t)≡ (2π)−n/2∫Rn

e−it·x f (x)dx.

Therefore, this gives the Fourier transform for f ∈ S. The nice property which S has incommon with G is that the Fourier transform and its inverse map S one to one onto S.This means I could have presented the whole of the above theory in terms of S rather thanin terms of G . However, it is more technical.

Theorem 32.3.21 If f ∈S, then F f and F−1 f are also in S.