### 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]}

## Contents

## 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

- $\backslash left.\backslash right.\backslash omega\_f(\backslash delta;t)=\backslash max\_\{|\backslash varepsilon|\; \backslash le\; \backslash delta\}\; |f(t)-f(t+\backslash varepsilon)|$

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

- $\backslash left.\backslash right.\backslash omega\_f(\backslash delta)\; =\; \backslash max\_t\; \backslash omega\_f(\backslash delta;t)$

With these definitions we may state the main results

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

- $\backslash int\_0^\backslash pi\; \backslash frac\{1\}\{\backslash delta\}\backslash omega\_f(\backslash delta;t)\backslash ,d\backslash delta\; <\; \backslash infty.$

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

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

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

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

*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 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.

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

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

- $\backslash int\_0^\backslash pi\; \backslash frac\{1\}\{\backslash delta\}\backslash Omega(\backslash delta)\backslash ,d\backslash delta\; =\; \backslash infty$

there exists a function *f* such that

- $\backslash left.\backslash right.\backslash omega\_f(\backslash delta;0)\; <\; \backslash Omega(\backslash delta)$

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