The Fourier transform is a fundamental tool in many parts of
mathematics. This is even more so when one looks at various
natural generalizations of it. This article contains brief
descriptions of the Fourier transform in various contexts and
links to articles about its use. Basic analysis, complex
numbers. Different kinds of Fourier transform Periodic functions
and functions defined on . Let be a function such that for every
. Then the th Fourier coefficient is given by the formula . The
function is called the Fourier transform of . Periodic functions
are naturally thought of as functions defined on the circle. If
we write for the unit circle and have a function , then the
formula for becomes . In the other direction, let be a function
from to . We can create a periodic function by defining it to
equal . Under some circumstances, and with suitable notions of
convergence, one can show that this inverts the previous
operation: that is, the sum converges to the function . If we
express as a function defined on ,