Fourier transform

The Fourier transform associated to each function $$f \in L^1(\mathbb R)$$ a new function $$\hat f:\mathbb R\to C$$ given by (D.7.1.1 )

$$\hat f(\gamma) := \int_{-\infty}^\infty f(x) e^{-2\pi i x \gamma} dx, \,\,\gamma \in \mathbb R$$.

The Fourier transform is also denoted as an operator: $$(\mathcal F f)(\gamma) := \hat f(\gamma)$$