Dini test

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, 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 is defined by
With these definitions we may state the main results:
For example, the theorem holds with but does not hold with.
In particular, any function that obeys a Hölder condition 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.