1118 CHAPTER 32. FOURIER TRANSFORMS

13. Let u ∈ G . Then Fu ∈ G and so, in particular, it makes sense to form the integral,∫R

Fu(x′,xn

)dxn

where (x′,xn) = x ∈ Rn. For u ∈ G , define γu(x′) ≡ u(x′,0). Find a constant suchthat F (γu)(x′) equals this constant times the above integral. Hint: By the dominatedconvergence theorem∫

RFu(x′,xn

)dxn = lim

ε→0

∫R

e−(εxn)2Fu(x′,xn

)dxn.

Now use the definition of the Fourier transform and Fubini’s theorem as required inorder to obtain the desired relationship.

14. Recall the Fourier series of a function in L2 (−π,π) converges to the function inL2 (−π,π). Prove a similar theorem with L2 (−π,π) replaced by L2 (−mπ,mπ) andthe functions {

(2π)−(1/2) einx}

n∈Z

used in the Fourier series replaced with{(2mπ)−(1/2) ei n

m x}

n∈Z

Now suppose f is a function in L2 (R) satisfying F f (t) = 0 if |t|> mπ . Show that ifthis is so, then

f (x) =1π

∑n∈Z

f(−nm

)sin(π (mx+n))

mx+n.

Here m is a positive integer. This is sometimes called the Shannon sampling theo-rem.Hint: First note that since F f ∈ L2 and is zero off a finite interval, it followsF f ∈ L1. Also

f (t) =1√2π

∫ mπ

−mπ

eitxF f (x)dx

and you can conclude from this that f has all derivatives and they are all bounded.Thus f is a very nice function. You can replace F f with its Fourier series. Thenconsider carefully the Fourier coefficient of F f . Argue it equals f

(−nm

)or at least

an appropriate constant times this. When you get this the rest will fall quickly intoplace if you use F f is zero off [−mπ,mπ].