### Dini-Lipschitz 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.[1]

## Definition

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

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

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

$\left.\right.\omega_f\left(\delta\right) = \max_t \omega_f\left(\delta;t\right)$

With these definitions we may state the main results

Theorem (Dini's test): Assume a function f satisfies at a point t that

$\int_0^\pi \frac\left\{1\right\}\left\{\delta\right\}\omega_f\left(\delta;t\right)\,d\delta < \infty.$

Then the Fourier series of f converges at t to f(t).

For example, the theorem holds with $\omega_f=\log^\left\{-2\right\}\left(\delta^\left\{-1\right\}\right)$ but does not hold with $\log^\left\{-1\right\}\left(\delta^\left\{-1\right\}\right)$.

Theorem (the Dini-Lipschitz test): Assume a function f satisfies

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

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 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.

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

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

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

there exists a function f such that

$\left.\right.\omega_f\left(\delta;0\right) < \Omega\left(\delta\right)$

and the Fourier series of f diverges at 0.