Examples discussion 08

[08.3] Compute e−πx 2 ∗ e−πx2 and sinx x ∗ sinx x . (Be careful what you assert: sin x x is not in L (R).) Discussion: The idea is to invoke f ∗ g = (f̂ · ĝ)̂ for even functions f, g ∈ L, since for even functions the inverse Fourier transform is the same as the forward Fourier transform. Conveniently, Gaussians are in L ∩ L, and have Fourier transforms which are again Gaussians: ̂ e−πax(ξ) = 1 √ a e−πξ /a (for a > 0)