Duality of the continuous-time Fourier transform - derivation and notation

1. What is duality of Fourier transform?
2. How do you find the continuous time of a Fourier transform?
3. What happens if you Fourier transform twice?

What is duality of Fourier transform?

The Duality Property tells us that if x(t) has a Fourier Transform X(ω), then if we form a new function of time that has the functional form of the transform, X(t), it will have a Fourier Transform x(ω) that has the functional form of the original time function (but is a function of frequency).

How do you find the continuous time of a Fourier transform?

Continuous time Fourier transform of x(t) is defined as X(ω)=∫−∞+∞x(t)e−jωtdt and discrete time Fourier transform of x(n) is defined as X(ω)=Σ∀nx(n)e−ωn.

What happens if you Fourier transform twice?

That is, if we apply the Fourier transform twice, we get a spatially reversed version of the function.