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.
Let be a function on [0,2{{pi}}], let be some point and let be a positive number. We define the local modulus of continuity at the point by
\left.\right.\omegaf(\delta;t)=max|\varepsilon||f(t)-f(t+\varepsilon)|
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
\omegaf(\delta)=maxt\omegaf(\delta;t)
With these definitions we may state the main results:
Theorem (Dini's test): Assume a function satisfies at a point that
| \pi | |
| \int | |
| 0 |
| 1 | |
| \delta |
\omegaf(\delta;t)d\delta<infty.
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
| \omega | ||||
|
\right)-1.
Then the Fourier series of converges uniformly to .
In particular, any function that obeys a Hölder condition satisfies the Dini–Lipschitz test.
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.
| \omega | ||||
|
\right)-1.
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
| \pi | |
| \int | |
| 0 |
| 1 | |
| \delta |
\Omega(\delta)d\delta=infty
there exists a function such that
\omegaf(\delta;0)<\Omega(\delta)
and the Fourier series of diverges at 0.