Editing Fourier transform (section)

=== On other ''L''<sup>''p''</sup> ===
For <math>1<p<2</math>, the Fourier transform can be defined on <math>L^p(\mathbb R)</math> by [[Marcinkiewicz interpolation]], which amounts to decomposing such functions into a fat tail part in {{math|''L''<sup>2</sup>}} plus a fat body part in {{math|''L''<sup>1</sup>}}. In each of these spaces, the Fourier transform of a function in {{math|''L''{{isup|''p''}}('''R'''<sup>''n''</sup>)}} is in {{math|''L''{{isup|''q''}}('''R'''<sup>''n''</sup>)}}, where {{math|1=''q'' = {{sfrac|''p''|''p'' − 1}}}} is the [[Hölder conjugate]] of {{mvar|p}} (by the [[Hausdorff–Young inequality]]). However, except for {{math|1=''p'' = 2}}, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in {{math|''L''{{isup|''p''}}}} for the range {{math|2 < ''p'' < ∞}} requires the study of distributions.{{sfn|Katznelson|2004}} In fact, it can be shown that there are functions in {{math|''L''{{isup|''p''}}}} with {{math|''p'' > 2}} so that the Fourier transform is not defined as a function.<ref name="Stein-Weiss-1971" />