Some uses are: Signals Analysis, DSP, cryptography, steganography, and image editing.
Okan K. Ersoy has written: 'Fourier-related transforms, fast algorithms, and applications' -- subject(s): Fourier transformations
J. Zorn has written: 'Methods of evaluating Fourier transforms with applications to control engineering'
Athanasios Papoulis has written: 'The Fourier integral and its applications' -- subject(s): Fourier series 'Circuits and Systems' -- subject(s): Electric circuits, Electric networks 'Solutions manual to accompany Probability, random variables and stochastic processes' 'Systems and transforms with applications in optics' -- subject(s): Optics, System analysis, Transformations (Mathematics)
Fritz Oberhettinger has written: 'Tables of Laplace transforms' -- subject(s): Laplace transformation 'Tabellen zur Fourier Transformation' -- subject(s): Mathematics, Tables, Fourier transformations 'Tabellen zur Fourier Transformation' -- subject(s): Mathematics, Tables, Fourier transformations 'Tables of Bessel transforms' -- subject(s): Integral transforms, Bessel functions 'Anwendung der elliptischen Funktionen in Physik und Technik' -- subject(s): Elliptic functions
Folke Bolinder has written: 'Fourier transforms in the theory of inhomogeneous transmission lines' -- subject(s): Electric lines, Fourier series
Charles Tong has written: 'Ordered fast Fourier transforms on a massively parallel hypercube multiprocessor' -- subject- s -: Fourier transformations, Multiprocessors
J. F. James has written: 'A student's guide to Fourier transforms' -- subject(s): Fourier transformations, Mathematical physics, Engineering mathematics
Roger Clifton Jennison has written: 'Fourier transforms and convolutions for the experimentalist'
There is a beautiful paper by Ales Cerny entitled "Introduction to Fast Fourier Transform in finance", which gives many interesting examples.
You can graph both with Energy on the y-axis and frequency on the x. Such a frequency domain graph of a fourier series will be discrete with a finite number of values corresponding to the coefficients a0, a1, a2, ...., b1, b2,... Also, the fourier series will have a limited domain corresponding to the longest period of your original function. A fourier transforms turns a sum into an integral and as such is a continuous function (with uncountably many values) over the entire domain (-inf,inf). Because the frequency domain is unrestricted, fourier transforms can be used to model nonperiodic functions as well while fourier series only work on periodic ones. Series: discrete, limited domain Transform: continuous, infinite domain.
Fourier transform and Laplace transform are similar. Laplace transforms map a function to a new function on the complex plane, while Fourier maps a function to a new function on the real line. You can view Fourier as the Laplace transform on the circle, that is |z|=1. z transform is the discrete version of Laplace transform.
Daniel Huybrechts has written: 'Fourier-Mukai Transforms in Algebraic Geometry (Oxford Mathematical Monographs)' 'The geometry of moduli spaces of sheaves' -- subject(s): Sheaf theory, Moduli theory, Algebraic Surfaces 'The geometry of moduli spaces of sheaves' -- subject(s): Algebraic Surfaces, Moduli theory, Sheaf theory, Surfaces, Algebraic 'Fourier-Mukai transforms in algebraic geometry' -- subject(s): Algebraic Geometry, Fourier transformations, Geometry, Algebraic