286 CHAPTER 11. FUNDAMENTAL TRANSFORMS

Proof: An application of Minkowski’s inequality to Borel representatives yields(∫Rn

(∫Rn|h(x−y)| | f (y)|dy

)2

dx

)1/2

≤ ∥ f∥1 ∥h∥2 . (11.16)

Hence∫|h(x−y)| | f (y)|dy < ∞ a.e. x and x→

∫h(x−y) f (y)dy is in L2 (Rn). Let Er ↑

Rn, mn (Er)< ∞. Thus,hr ≡XEr h ∈ L2 (Rn)∩L1 (Rn),

and letting φ ∈ G ,∫

F (hr ∗ f )(φ)dx

≡∫

(hr ∗ f )(Fφ)dx

= (2π)−n/2∫ ∫ ∫

hr (x−y) f (y)e−ix·tφ (t)dtdydx

= (2π)−n/2∫ ∫ (∫

hr (x−y)e−i(x−y)·tdx)

f (y)e−iy·tdyφ (t)dt

=∫

(2π)n/2 Fhr (t)F f (t)φ (t)dt.

Since φ is arbitrary and G is dense in L2 (Rn),

F (hr ∗ f ) = (2π)n/2 FhrF f .

Now by Minkowski’s Inequality, hr ∗ f → h ∗ f in L2 (Rn) and also it is clear that hr → hin L2 (Rn) ; so, by Plancherel’s theorem, you may take the limit in the above and concludeF (h∗ f )= (2π)n/2 FhF f . The assertion for F−1 is similar and 11.15 follows from 11.16. ■

11.6.5 The Schwartz ClassThe problem with G is that it does not contain C∞

c (Rn). I have used it in presenting theFourier 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 11.6.21 f ∈S, the Schwartz class, if f ∈C∞(Rn) and for all positiveintegers N, ρN( f )< ∞ where

ρN( 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}< ∞ (11.17)

for all multi indices α and β .