# 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.^{[1]}

## Definition[edit]

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

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

With these definitions we may state the main results:

**Theorem (Dini's test):**Assume a function f satisfies at a point t that- 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- Then the Fourier series of f converges uniformly to f.

In particular, any function of a Hölder class^{[clarification needed]} satisfies the Dini–Lipschitz test.

## Precision[edit]

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.

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

there exists a function f such that

and the Fourier series of f diverges at 0.

## See also[edit]

## References[edit]

**^**Gustafson, Karl E. (1999),*Introduction to Partial Differential Equations and Hilbert Space Methods*, Courier Dover Publications, p. 121, ISBN 978-0-486-61271-3