Define Cf = ‑ Rd ωF(ω)dω (see (16.25)). Let f(x) = g(x), i. e., f is a radial function. Show that Cf = Vd ‑ ∞ 0 rd|Fˆ(r)|dr, where Vd is the volume of d−1-dimensional unit sphere in Rd. Prove that if...

Define Cf = ‑ Rd ωF(ω)dω (see (16.25)). Let f(x) = g(x), i. e., f is a radial function. Show that Cf = Vd ‑ ∞ 0 rd|Fˆ(r)|dr, where Vd is the volume of d−1-dimensional unit sphere in Rd. Prove that if f(x) = exp(−x2/2), i. e., f is the Gaussian function then Cf ≤ d1/2.