Fourier transform of the product

1. What is Fourier transform of a product?
2. Can you multiply Fourier transforms?
3. What is the formula for Fourier transform?
4. What is the Fourier transform of the convolution?

What is Fourier transform of a product?

A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency.

Can you multiply Fourier transforms?

Linearity. The Fourier Transform is linear. The Fourier Transform of a sum of functions, is the sum of the Fourier Transforms of the functions. Also, if you multiply a function by a constant, the Fourier Transform is multiplied by the same constant.

What is the formula for Fourier transform?

As T→∞, 1/T=ω0/2π. Since ω₀ is very small (as T gets large, replace it by the quantity dω). As before, we write ω=nω0 and X(ω)=Tc_n. A little work (and replacing the sum by an integral) yields the synthesis equation of the Fourier Transform.

What is the Fourier transform of the convolution?

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 ) .