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Concept
Dirichlet–Jordan test
Summary
In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real-valued, periodic function f to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity). It is one of many conditions for the convergence of Fourier series.
The original test was established by Peter Gustav Lejeune Dirichlet in 1829, for piecewise monotone functions. It was extended in the late 19th century by Camille Jordan to functions of bounded variation (any function of bounded variation is the difference of two increasing functions).
The Dirichlet–Jordan test states that if a periodic function is of bounded variation on a period, then the Fourier series converges, as , at each point of the domain to
In particular, if is continuous at , then the Fourier series converges to . Moreover, if is continuous everywhere, then the convergence is uniform.
Stated in terms of a periodic function of period 2π, the Fourier series coefficients are defined as
and the partial sums of the Fourier series are
The analogous statement holds irrespective of what the period of f is, or which version of the Fourier series is chosen.
There is also a pointwise version of the test: if is a periodic function in , and is of bounded variation in a neighborhood of , then the Fourier series at converges to the limit as above
For the Fourier transform on the real line, there is a version of the test as well. Suppose that is in and of bounded variation in a neighborhood of the point . Then
If 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 .
This version of the test (although not satisfying modern demands for rigor) is historically prior to Dirichlet, being due to Joseph Fourier.
Official source
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