Editing Fourier transform (section)

== Units ==
{{see also|Spectral density#Units}}

The frequency variable must have inverse units to the units of the original function's domain (typically named <math>t</math> or <math>x</math>). For example, if <math>t</math> is measured in seconds, <math>\xi</math> should be in cycles per second or [[hertz]]. If the scale of time is in units of <math>2\pi</math> seconds, then another Greek letter <math>\omega</math> is typically used instead to represent [[angular frequency]] (where <math>\omega=2\pi \xi</math>) in units of [[radian]]s per second. If using <math>x</math> for units of length, then <math>\xi</math> must be in inverse length, e.g., [[wavenumber]]s. That is to say, there are two versions of the real line: one which is the [[Range of a function|range]] of <math>t</math> and measured in units of <math>t,</math> and the other which is the range of <math>\xi</math> and measured in inverse units to the units of <math>t.</math> These two distinct versions of the real line cannot be equated with each other. Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.

In general, <math>\xi</math> must always be taken to be a [[linear form]] on the space of its domain, which is to say that the second real line is the [[dual space]] of the first real line. See the article on [[linear algebra]] for a more formal explanation and for more details. This point of view becomes essential in generalizations of the Fourier transform to general [[symmetry group]]s, including the case of Fourier series.

That there is no one preferred way (often, one says "no canonical way") to compare the two versions of the real line which are involved in the Fourier transform—fixing the units on one line does not force the scale of the units on the other line—is the reason for the plethora of rival conventions on the definition of the Fourier transform. The various definitions resulting from different choices of units differ by various constants.

In other conventions, the Fourier transform has {{mvar|i}} in the exponent instead of {{math|−''i''}}, and vice versa for the inversion formula.  This convention is common in modern physics<ref>{{harvnb|Arfken|1985}}</ref> and is the default for [https://www.wolframalpha.com Wolfram Alpha], and does not mean that the frequency has become negative, since there is no canonical definition of positivity for frequency of a complex wave. It simply means that <math>\hat f(\xi)</math> is the amplitude of the wave &nbsp;<math>e^{-i 2\pi \xi x}</math>&nbsp; instead of the wave &nbsp;<math>e^{i 2\pi \xi x}</math> (the former, with its minus sign, is often seen in the time dependence for [[Sinusoidal plane-wave solutions of the electromagnetic wave equation]], or in the [[Wave function#Time dependence|time dependence for quantum wave functions]]).  Many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involve {{math|''i''}} have it replaced by {{math|−''i''}}.  In [[Electrical engineering]] the letter {{math|''j''}} is typically used for the [[imaginary unit]] instead of {{math|''i''}} because {{math|''i''}} is used for current.

When using [[dimensionless units]], the constant factors might not even be written in the transform definition. For instance, in [[probability theory]], the characteristic function {{mvar|Φ}} of the probability density function {{mvar|f}} of a random variable {{mvar|X}} of continuous type is defined without a negative sign in the exponential, and since the units of {{mvar|x}} are ignored, there is no 2{{pi}} either:
<math display="block">\phi (\lambda) = \int_{-\infty}^\infty f(x) e^{i\lambda x} \,dx.</math>

(In probability theory, and in mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because so many random variables are not of continuous type, and do not possess a density function, and one must treat not functions but [[Distribution (mathematics)|distributions]], i.e., measures which possess "atoms".)

From the higher point of view of [[character theory|group characters]], which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on a [[locally compact abelian group|locally compact Abelian group]].