In mathematics, the Dini and Dini–Lipschitz tests are highly precise tests that can be used to prove that the Fourier series of a function converges at a given point. These tests are named after Ulisse Dini and Rudolf Lipschitz. Definition Let be a function on [0,2], let be some point and let be a positive number. We define the local modulus of continuity at the point by Notice that we consider here to be a periodic function, e.g. if and is negative then we define . The global modulus of continuity (or simply the modulus of continuity) is defined by With these definitions we may state the main results: Theorem (Dini's test): Assume a function satisfies at a point that Then the Fourier series of converges at to . For example, the theorem holds with but does not hold with . Theorem (the Dini–Lipschitz test): Assume a function satisfies Then the Fourier series of converges uniformly to . In particular, any function of a Hölder class satisfies the Dini–Lipschitz test. Precision Both tests are the best of their kind. For the Dini-Lipschitz test, it is possible to construct a function with its modulus of continuity satisfying the test with instead of , i.e. and the Fourier series of diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that there exists a function such that and the Fourier series of diverges at 0. See also Convergence of Fourier series Dini continuity Dini criterion References Fourier series Convergence tests