Fourier Transform

1D Continuous Time Fourier Transform

Basis functions (complex sinusoidal or exponential)

$$\phi(x,u)=e^{j2\pi ux}, u \in (-\infty,+\infty)$$ $u$ = frequency = # cycles per unit of $x$, $\omega=2\pi u$ (radian freq.)

Inverse transform

$$f(x)=F^{-1}\{F(u)\}=\int_{-\infty}^\infty F(u)e^{j2\pi ux}\, du$$

Forward Transform: $F(u) =\langle f(x), \phi(x,u) \rangle$

$$F(u)=F\{f(x)\} = \int_{-\infty}^\infty f(x) e^{-j2\pi e x} \, dx$$ Here we use frequency (rather then radian frequency) to define FT. Inverse transform does not need the factor of $1/2\pi$

Representation of FT

#incomplete