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

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

With these definitions we may state the main results:

Theorem (Dini's test): Assume a function satisfies at a point that

:

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

:

Then the Fourier series of converges uniformly to .

In particular, any function that obeys a Hölder condition 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.

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

there exists a function such that

and the Fourier series of diverges at 0.

See also

• Convergence of Fourier series
• Dini continuity
• Dini criterion

References