Answer to Question #95684 in Calculus for Rachel

Question

Let f be a nonnegative function in L^1(R^n). Prove that
|f̂(ξ)| ≤ f̂(0), ξ∈R^n

Answer 1

$|f̂(ξ)|=|\int_{-\infty}^{\infty}e^{-2\pi i x ξ}f(x)dx|\le$

$\int_{-\infty}^{\infty}|e^{-2\pi i x ξ}||f(x)|dx=$

$\int_{-\infty}^{\infty}1\cdot f(x)dx=$

$\int_{-\infty}^{\infty}|e^{-2\pi i x \cdot0}|f(x)dx=$  f̂(0)

And therefore |f̂(ξ)| ≤ f̂(0).