Dirichlet–Jordan test

The Dirichlet–Jordan test states that a function of bounded variation has a Fourier series converging at each point to the average of its one-sided limits.

Fourier series: A representation of a periodic function as a sum of sine and cosine waves, or complex exponentials. The test specifies pointwise convergence of these expansions under bounded variation.
Total variation: The supremum of summed absolute increments of a function over finite ordered partitions of its domain. Finite total variation is the precise condition imposed on the function.
Peter Gustav Lejeune Dirichlet: A German mathematician whose work established foundational results in number theory and Fourier analysis. Dirichlet's convergence work supplied a classical foundation for the test.
Dini's test: A Fourier-series convergence criterion based on an integrability condition on local oscillation. It uses a local condition near a point rather than global bounded variation.
Dirichlet kernel: The trigonometric kernel that expresses Fourier partial sums as integrals against the original function. Its oscillatory integral underlies the convergence proof.
Pointwise convergence: A sequence of functions converges pointwise when its values converge separately at every point. The test guarantees convergence point by point, rather than uniformly.
Camille Jordan: A French mathematician known for work in analysis, algebra, and the theory of bounded variation. Jordan extended Fourier convergence results to functions of bounded variation.
Carleson's theorem: A theorem that Fourier series of square-integrable functions converge almost everywhere. It guarantees convergence almost everywhere, unlike this test's pointwise conclusion under stronger regularity.
Bounded variation: A function has bounded variation when the total accumulated change over its domain is finite. This is the regularity condition that guarantees convergence in the test.
Jump discontinuity: A discontinuity where finite left and right limits exist but differ. At a jump, the Fourier series converges to the midpoint of the two limits.

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Dini testCompared with: It uses global variation control rather than the Dini test’s local integral condition.