Does convolution violate duality of a Fourier Transform

1. How is convolution related to Fourier transform?
2. What is the duality property of Fourier transform?
3. Is the Fourier transform a convolution?
4. What happens if you Fourier transform twice?

How is convolution related to Fourier transform?

The convolution theorem (together with related theorems) is one of the most important results of Fourier theory which is that the convolution of two functions in real space is the same as the product of their respective Fourier transforms in Fourier space, i.e. f ( r ) ⊗ ⊗ g ( r ) ⇔ F ( k ) G ( k ) .

What is the duality property 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).

Is the Fourier transform a convolution?

It states that the Fourier Transform of the product of two signals in time is the convolution of the two Fourier Transforms.

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.