Lecture 4 Applications of Harmonic Analysis February 4 , 2005 Lecturer : Nati Linial

Most of our applications of harmonic analysis to computer science will involve only Parseval’s identity. Theorem 4.1 (Parseval’s Identity). ‖f‖2 = ‖f̂‖2 Corollary 4.2. 〈f, g〉 = 〈f̂ , ĝ〉. Proof. Note that 〈f + g, f + g〉 = ‖f + g‖2 = ‖f̂ + g‖2 = ‖f̂ + ĝ‖2. Now as 〈f + g, f + g〉 = ‖f‖2 + ‖g‖2 + 2〈f, g〉, and similarly ‖f̂ + ĝ‖2 = ‖f̂‖2 + ‖ĝ‖2 + 2〈f̂ , ĝ〉, applying Parseval to ‖f‖2 and ‖g‖2 and equating finishes the proof. The other basic identity is the following. Lemma 4.3. f̂ ∗ g = f̂ · ĝ Proof. We will show this for the unit circle T, but one should note that it is true more generally. Recall that by definition h = f ∗ g means that h(t) = 1 2π ∫