Expository Notes on Introduction to Harmonic Analysis

1. (f̂ + g)(ξ) = f̂(ξ) + ĝ(ξ). 2. For any α ∈ C, (α̂f)(ξ) = αf̂(ξ). 3. Let f̄ denote the complex conjugate of f . Then ˆ̄ f(ξ) = f̂(ξ). 4. For λ ∈ R \ {0}, denote fλ(x) = λf(λx). Then f̂λ(ξ) = f̂( ξ λ ). 5. For y ∈ R let (τyf)(t) = f(t− y). Then τ̂yf(ξ) = f̂(ξ)e−2πiξy. 6. |f̂(ξ)| ≤ ‖f‖L1(R). Definition 1.3. The Schwartz space, S(R), is the space of C∞ functions that are rapidly decreasing, in the sense that