Number Theory: An Introduction to Mathematics

336 VIII The Geometry of Numbers
∫

Π

|φ(x)|^2 dx=

∑

w∈Zn

|cw|^2 ,

where

cw=

∫

Π

φ(x)e−^2 πiw

tx

dx.

But

cw=

∫

Π

∑

z∈Zn

Ψ(x+z)e−^2 πiw

tx

dx

=

∫

Π

∑

z∈Zn

Ψ(x+z)e−^2 πiw

t(x+z)

dx,

sincee^2 kπi=1 for any integerk. Hence

cw=

∫

Rn

Ψ(y)e−^2 πiw

ty

dy.

On the other hand,

∫

Π

|φ(x)|^2 dx=

∫

Π

∑

z′,z′′∈Zn

Ψ(x+z′)Ψ(x+z′′)dx

=

∫

Π

∑

z,z′∈Zn

Ψ(x+z′)Ψ(x+z′+z)dx

=

∫

Rn

∑

z∈Zn

Ψ(y)Ψ(y+z)dy=

∫

Rn

Ψ(y)φ(y)dy.

Substituting these expressions in Parseval’s equality, we obtain the result.

Suppose, in particular, thatΨtakes only real nonnegative values. Then so also does
φand

∫

Rn

Ψ(x)φ(x)dx≤sup

x∈Rn

φ(x)

∫

Rn

Ψ(x)dx.

On the other hand, omitting all terms withw=0 we obtain

∑

w∈Zn

∣

∣

∣

∣

∫

Rn

Ψ(x)e−^2 πiw

tx

dx

∣

∣

∣

∣

2

≥

(∫

Rn

Ψ(x)dx

) 2

.