The Fourier Transform

The Fourier Transform is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions.

 To know more about Fourier transform click on the link below:

http://www.thefouriertransform.com/

Application in signal field
In signal field, the application of the Fourier transform allows to convert between the time and frequency spaces.

To learn more about this subject check the following links:

http://cns-alumni.bu.edu/~slehar/fourier/fourier.html
http://www.cs.otago.ac.nz/cosc453/student_tutorials/fourier_analysis.pdf
https://users.cs.cf.ac.uk/Dave.Marshall/Vision_lecture/node14.html
http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/OWENS/LECT4/node2.html#SECTION00020000000000000000
https://www.youtube.com/watch?v=vQLH7qTeJRM

Application in solid state physics
In solid state physics, the application of the Fourier transform allows to convert between the time and wave vector spaces to facilitate the numerical calculations.

The time space represents the real space while the wave vector space represents the reciprocal space which is the in reality the diffraction pattern space.

To get detailed information about Fourier transform and its different applications check the book in the following link :

https://see.stanford.edu/materials/lsoftaee261/book-fall-07.pdf