Editing Fourier transform (section)

=== Lebesgue integrable functions ===
{{see also|Lp space#Lp spaces and Lebesgue integrals}}

A [[measurable function]] <math>f:\mathbb R\to\mathbb C</math> is called (Lebesgue) integrable if the [[Lebesgue integral]] of its absolute value is finite:
<math display="block">\|f\|_1 = \int_{\mathbb R}|f(x)|\,dx < \infty.</math>
If <math>f</math> is Lebesgue integrable then the Fourier transform, given by {{EquationNote|Eq.1}}, is well-defined for all <math>\xi\in\mathbb R</math>.{{sfn|Stade|2005|pp=298-299}} Furthermore, <math>\widehat f\in L^\infty\cap C(\mathbb R)</math> is bounded, [[uniformly continuous]] and (by the [[Riemann–Lebesgue lemma]]) zero at infinity.

The space <math>L^1(\mathbb R)</math> is the space of measurable functions for which the norm <math>\|f\|_1</math> is finite, modulo the [[Equivalence class|equivalence relation]] of equality [[almost everywhere]].  The Fourier transform on <math>L^1(\mathbb R)</math> is [[Bijection, injection and surjection|one-to-one]].  However, there is no easy characterization of the image, and thus no easy characterization of the inverse transform.  In particular, {{EquationNote|Eq.2}} is no longer valid, as it was stated only under the hypothesis that <math>f(x)</math> decayed with all derivatives.

While {{EquationNote|Eq.1}} defines the Fourier transform for (complex-valued) functions in <math>L^1(\mathbb R)</math>, it is not well-defined for other integrability classes, most importantly the space of [[square-integrable function]]s <math>L^2(\mathbb R)</math>. For example, the function <math>f(x)=(1+x^2)^{-1/2}</math> is in <math>L^2</math> but not <math>L^1</math> and therefore the Lebesgue integral {{EquationNote|Eq.1}} does not exist. However, the Fourier transform on the dense subspace <math>L^1\cap L^2(\mathbb R) \subset L^2(\mathbb R)</math> admits a unique continuous extension to a [[unitary operator]] on <math>L^2(\mathbb R)</math>. This extension is important in part because, unlike the case of <math>L^1</math>, the Fourier transform is an [[automorphism]] of the space <math>L^2(\mathbb R)</math>.

In such cases, the Fourier transform can be obtained explicitly by [[Regularization (mathematics)|regularizing]] the integral, and then passing to a limit. In practice, the integral is often regarded as an [[improper integral]] instead of a proper Lebesgue integral, but sometimes for convergence one needs to use [[weak limit]] or [[Cauchy principal value|principal value]] instead of the (pointwise) limits implicit in an improper integral. {{harvtxt|Titchmarsh|1986}} and {{harvtxt|Dym|McKean|1985}} each gives three rigorous ways of extending the Fourier transform to square integrable functions using this procedure.  A general principle in working with the <math>L^2</math> Fourier transform is that Gaussians are dense in <math>L^1\cap L^2</math>, and the various features of the Fourier transform, such as its unitarity, are easily inferred for Gaussians.  Many of the  properties of the Fourier transform, can then be proven from two facts about Gaussians:{{sfn|Howe|1980}}
* that <math>e^{-\pi x^2}</math> is its own Fourier transform; and
* that the Gaussian integral <math>\int_{-\infty}^\infty e^{-\pi x^2}\,dx = 1.</math>

A feature of the <math>L^1</math> Fourier transform is that it is a homomorphism of Banach algebras from <math>L^1</math> equipped with the convolution operation to the Banach algebra of continuous functions under the <math>L^\infty</math> (supremum) norm.  The conventions chosen in this article are those of [[harmonic analysis]], and are characterized as the unique conventions such that the Fourier transform is both [[Unitary operator|unitary]] on {{math|''L''<sup>2</sup>}} and an algebra homomorphism from {{math|''L''<sup>1</sup>}} to {{math|''L''<sup>∞</sup>}}, without renormalizing the Lebesgue measure.<ref>{{harvnb|Folland|1989}}</ref>