Changes

Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.

Often, functions which are difficult to analyze in one topological group become much easier to analyze when transformed to another topological group.

The formulas usually favored by mathematicians are the "normalized" form.  Given a function f(t) defined on the entire real line, its Fourier transform g(x) is given by:

The formulas usually favored by mathematicians are the "normalized" form.  Given a function f(k) defined on the entire real line, its Fourier transform g(k) is given by:

:<math>g(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(t) e^{-ixt}\, dt</math>

:<math>g(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x) e^{-ixk}\, dx</math>

The inverse transform, that recovers the original function, is:

The inverse transform, that recovers the original function, is:

:<math>f(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(x) e^{ixt}\, dx</math>

:<math>f(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} g(k) e^{ixk}\, dk</math>

 

The constants in front of the integrals are arbitrary, so long as their product is <math>1/2\pi</math>. In order to make the forward and inverse transforms as similar as possible, an oft-used convention is to set them both equal to <math>1/\sqrt{2\pi}</math>.

==Discrete Fourier transformation==  

==Discrete Fourier transformation==