Editing Dirichlet–Jordan test (section)

== Jordan test for Fourier integrals ==
For the [[Fourier transform]] on the real line, there is a version of the test as well.<ref>{{citation|author=[[E. C. Titchmarsh]]|title=Introduction to the theory of Fourier integrals|year=1948|page=13|publisher=Oxford Clarendon Press}}.</ref>  Suppose that <math>f(x)</math> is in <math>L^1(-\infty,\infty)</math> and of bounded variation in a neighborhood of the point <math>x</math>.  Then
<math display="block">\frac1\pi\lim_{M\to\infty}\int_0^{M}du\int_{-\infty}^\infty f(t)\cos u(x-t)\,dt = \lim_{\varepsilon\to 0}\frac{f(x+\varepsilon)+f(x-\varepsilon)}{2}.</math>
If <math>f</math> is continuous in an open interval, then the integral on the left-hand side converges uniformly in the interval, and the limit on the right-hand side is <math>f(x)</math>.

This version of the test (although not satisfying modern demands for rigor) is historically prior to Dirichlet, being due to [[Joseph Fourier]].<ref name="Fourier series and Fourier integrals">{{citation|author=[[Jaak Peetre]]|title=On Fourier's discovery of Fourier series and Fourier integrals|year=2000|url=https://web.archive.org/web/20221201121132/https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=d72e7ff6baf9008d523a192bab2e3400982389d3}}</ref>