3.12 In one dimension, d = 1, consider a stationary Gaussian random field X(t)

with the exponential covariance function

????(h) = exp(−|h|).

Consider the indicator test function

????(u) = I[|u| ≤ 1∕2].

(a) Show that ????(u) has the Fourier transform

????̃(????) = {(2∕????)sin(????∕2), ???? ≠ 0, 1, ???? = 0.

(b) Let ????(u) = ???? ∗ ????̌ (u) and show that

????(u) =

⎧

⎪

⎪

⎨

⎪

⎪

⎩

0, u ≤ −1, 1 + u, −1 ≤ u ≤ 0, 1 − u, 0 ≤ u ≤ 1, 0, u ≥ 1.

Confirm that ????(u) is a compactly supported function with a tent-like shape.