Dini's 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 ,2 let be some point and let be a positive number. We define the local modulus of continuity at the point by

:$\left.\right.\omega_f(\delta;t)=\max_ , f(t)-f(t+\varepsilon),$

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

:$\omega_f(\delta) = \max_t \omega_f(\delta;t)$

With these definitions we may state the main results:

:Theorem (Dini's test): Assume a function satisfies at a point that
::$\int_0^\pi \frac\omega_f(\delta;t)\,\mathrm\delta < \infty.$
: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
::$\omega_f(\delta)=o\left(\log\frac\right)^.$
: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.

:$\omega_f(\delta)=O\left(\log\frac\right)^.$

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

:$\int_0^\pi \frac\Omega(\delta)\,\mathrm\delta = \infty$

there exists a function such that

:$\omega_f(\delta;0) < \Omega(\delta)$

and the Fourier series of diverges at 0.

See also

* Convergence of Fourier series
* Dini continuity
* Dini criterion

References

{{reflist

Fourier series
Convergence tests