Math 6420 • PDE

For a function \(f(x)\) ``sufficiently smooth'' and
``sufficiently fast decaying'' as \(x\to\pm\infty\), the Fourier transform of \(f\) is defined
as \(g(k)=\int_{-\infty}^{\infty}f(x)e^{ikx}\,dx\). Suppose that the function \(f(x)\) is
normalized such that its \(L_2\)-norm is 1. Let
\[
  E_x=\int_{-\infty}^{\infty}x^2|f(x)|^2\,dx,\qquad
  E_k=\int_{-\infty}^{\infty}k^2|g(k)|^2\,dk.
\]
Prove and interpret the inequality
\[
  E_xE_k\ge\frac{1}{4}.
\]
(Hint: Apply Schwartz's inequality to \(xf(x)f'(x)\); use Plancherel's equality.)
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