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
:<math>\left.\right.\omega_f(\delta;t)=\max_{|\varepsilon| \le \delta} |f(t)-f(t+\varepsilon)|</math>
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
:<math>\omega_f(\delta) = \max_t \omega_f(\delta;t)</math>
With these definitions we may state the main results:
:Theorem (Dini's test): Assume a function satisfies at a point that
::<math>\int_0^\pi \frac{1}{\delta}\omega_f(\delta;t)\,\mathrm{d}\delta < \infty.</math>
: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
::<math>\omega_f(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}.</math>
:Then the Fourier series of converges uniformly to .
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.
:<math>\omega_f(\delta)=O\left(\log\frac{1}{\delta}\right)^{-1}.</math>
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
:<math>\int_0^\pi \frac{1}{\delta}\Omega(\delta)\,\mathrm{d}\delta = \infty</math>
there exists a function such that
:<math>\omega_f(\delta;0) < \Omega(\delta)</math>
and the Fourier series of diverges at 0.
See also
- Convergence of Fourier series
- Dini continuity
- Dini criterion