1140 CHAPTER 33. FOURIER ANALYSIS IN Rn

Then changing the variables in 33.4.48,,

∫3π|y|−1<|x|≤R

Kε (x)e−ix·ydx

=12

∫3π|y|−1<|x|≤R

Kε (x)e−ix·ydx

−12

∫3π|y|−1<|x−z|≤R

Kε (x− z)e−ix·ydx.

Thus

∫3π|y|−1<|x|≤R

Kε (x)e−ix·ydx =

12∫|x|≤R

Kε (x)e−ix·ydx− 12

∫|x−z|≤R

Kε (x− z)e−ix·ydx

+ 12

∫|x−z|≤3π|y|−1

Kε (x− z)e−ix·ydx− 12

∫|x|≤3π|y|−1

Kε (x)e−ix·ydx.

(33.4.49)

Since |z| = π/ |y|, it follows |z| = π

|y| <3π

|y| < R and so the following picture describes

the situation. In this picture, the radius of each ball equals either R or 3π |y|−1 and eachintegral above is taken over one of the two balls in the picture, either the one centered at 0or the one centered at z.

0 z

To begin with, consider the integrals which involve Kε (x− z).

∫|x−z|≤R

Kε (x− z)e−ix·ydx

=∫|x|≤R

Kε (x− z)e−ix·ydx

−∫

|x−z|>R,|x|<RKε (x− z)e−ix·ydx

+∫

|x−z|<R,|x|>RKε (x− z)e−ix·ydx.

(33.4.50)