Modulation operator

The modulation operator $$E_b$$ is defined on $$L^(\mathbb R)$$ (see Def. 6.2.1 ; not sure if it can be defined for other $$L^p(\mathbb R)$$ spaces too). By definition, for a given $$b$$ this operator acts as:

$$(E_b f)(x):=e^{-2\pi ibx} f(x), \,x\in \mathbb R.$$

$$E_b$$ is a unitary operator and it holds that
 * $$E_b^{-1} = E_{-b} = (E_b)^*$$.