Editing Dirichlet–Jordan test (section)

== Dirichlet conditions in signal processing ==
In [[signal processing]], the test is often retained in the original form due to Dirichlet:<ref name='sands'>{{cite book|last1= Alan V. Oppenheim|last2= Alan S. Willsky|last3= Syed Hamish Nawab|year= 1997|title= Signals & Systems|url= https://books.google.com/books?id=O9ZHSAAACAAJ&q=signals+and+systems|publisher= Prentice Hall| isbn= 9780136511755|page= 198}}</ref>{{sfn|Khare|Butola|Rajora|2023|p=9}}{{sfn|Proakis|Manolakis|1996|p=234}} a piecewise monotone bounded periodic function <math>f</math> (having a finite number of monotonic intervals per period) has a convergent Fourier series whose value at each point is the arithmetic mean of the left and right limits of the function.  The condition of piecewise monotonicity stipulates having only finitely many local extrema per period, which implies <math>f</math> is of bounded variation (though the reverse is not true).<ref name="Fourier series and Fourier integrals"/> (Dirichlet required in addition that the function have only finitely many discontinuities, but this constraint is unnecessarily stringent.{{sfn|Lanczos|2016|p=46}})  Any signal that can be physically produced in a laboratory satisfies these conditions.<ref>{{citation|author=B P Lathi|title=Signal processing and linear systems|year=2000|publisher=Oxford}}</ref>

As in the pointwise case of the Jordan test, the condition of boundedness can be relaxed if the function is assumed to be [[absolutely integrable]] (i.e., <math>L^1</math>) over a period, provided it satisfies the other conditions of the test in a neighborhood of the point <math>x</math> where the limit is taken.{{sfn|Lanczos|2016|p=48}}