These tests are named after Ulisse Dini and Rudolf Lipschitz.
[1] Let f be a function on [0,2π], let t be some point and let δ be a positive number.
The global modulus of continuity (or simply the modulus of continuity) is defined by With these definitions we may state the main results: For example, the theorem holds with ωf = log−2(1/δ) but does not hold with log−1(1/δ).
In particular, any function that obeys a Hölder condition satisfies the Dini–Lipschitz test.
For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that there exists a function f such that and the Fourier series of f diverges at 0.