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 $f$ be a function on [0,2$\pi$], let $t$ be some point and let $δ$ be a positive number. We define the local modulus of continuity at the point $t$ by


 * $$\left.\right.\omega_f(\delta;t)=\max_{|\varepsilon| \le \delta} |f(t)-f(t+\varepsilon)|$$

Notice that we consider here $f$ to be a periodic function, e.g. if $t = 0$ and $ε$ is negative then we define $f(ε) = f(2π + ε)$.

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 $f$ satisfies at a point $t$ that
 * $$\int_0^\pi \frac{1}{\delta}\omega_f(\delta;t)\,\mathrm{d}\delta < \infty.$$
 * Then the Fourier series of $f$ converges at $t$ to $f(t)$.

For example, the theorem holds with $ω_{f} = log^{−2}(1⁄δ)$ but does not hold with $log^{−1}(1⁄δ)$.


 * Theorem (the Dini–Lipschitz test): Assume a function $f$ satisfies
 * $$\omega_f(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}.$$
 * Then the Fourier series of $f$ converges uniformly to $f$.

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 $f$ with its modulus of continuity satisfying the test with $O$ instead of $o$, i.e.


 * $$\omega_f(\delta)=O\left(\log\frac{1}{\delta}\right)^{-1}.$$

and the Fourier series of $f$ diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that


 * $$\int_0^\pi \frac{1}{\delta}\Omega(\delta)\,\mathrm{d}\delta = \infty$$

there exists a function $f$ such that


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

and the Fourier series of $f$ diverges at 0.