Dini test

The Dini test is a criterion for convergence of a Fourier series at a point. It applies when the function’s local oscillation around that point has a finite integral after division by distance.

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Fourier series: A representation of a periodic function as a sum of sine and cosine waves. The Dini test gives a sufficient condition for this series to converge at a specified point.
Fourier coefficient: A coefficient measuring the contribution of a particular frequency to a Fourier series. The test concerns the convergence of the partial sums formed from these coefficients.
Ulisse Dini: An Italian mathematician known for work in real analysis and the foundations of calculus. The convergence criterion bears his name through his contributions to Fourier analysis.
Dirichlet–Jordan test: A Fourier-series convergence criterion for functions of bounded variation. It uses global variation control rather than the Dini test’s local integral condition.
Dini continuity: A continuity condition whose modulus of continuity has a finite integral when weighted by the reciprocal of its argument. This local regularity condition supplies a common form of the test’s integrability hypothesis.
Fourier partial sum: The finite sum of Fourier terms through a chosen maximum frequency. Convergence means these finite approximations approach the target value at the point.
Dirichlet's theorem: A classical theorem giving conditions under which a Fourier series converges to the midpoint of a function’s one-sided limits. It is a foundational earlier convergence criterion against which Dini-type conditions are understood.
Lipschitz condition: A bound requiring function differences to be at most a constant times the distance between inputs. A Lipschitz bound near a point is a familiar, stronger route to the Dini integrability condition.
Modulus of continuity: A function measuring the largest change in a function between points separated by a given distance. It quantifies the local oscillation used to formulate the Dini condition.
Continuity: A property of a function whose values change by arbitrarily small amounts near each point. The Dini condition strengthens local continuity with a quantitative oscillation bound.