3.11 Let D be a bounded region in d and let ????(t) = I[t D] be...

3.11 Let D be a bounded region in ℝd and let ????(t) = I[t ∈ D] be an indicator function on D. Show that the Fourier transform ????̃(????) of ???? satisfies the following:

(a) ????̃(????) is bounded for all ????.

(b) ∫ |????̃(????)|

2d???? < ∞.

Hint:

(a) |????̃(????)| = | ∫ exp(ihT????)????(h) dh| ≤ ∫ |????(t)| dt ≤ volume of D.

(b) Use the identity ∫ |????(t)|

2 dt = (2????)

−d ∫ |????̃(????)|

2 d???? and note that

∫ |????(t)|

2 dt = volume of D.