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

${\omega }_f(\delta ;t)=\underset{|\epsilon |\le \delta }{\max }|f(t)-f(t+\epsilon )|$

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

${\omega }_f(\delta )=\underset{t}{\max }{\omega }_f(\delta ;t)$

With these definitions we may state the main results:

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

$\underset{0}{\overset{\pi }{\int }}\frac{1}{\delta }{\omega }_f(\delta ;t)\mathrm{d}\delta <\mathrm{\infty }.$

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

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

Then the Fourier series of f converges uniformly to f.

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

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

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

$\underset{0}{\overset{\pi }{\int }}\frac{1}{\delta }\mathrm{\Omega }(\delta )\mathrm{d}\delta =\mathrm{\infty }$

there exists a function f such that

${\omega }_f(\delta ;0)<\mathrm{\Omega }(\delta )$

and the Fourier series of f diverges at 0.

See also

• Convergence of Fourier series
• Dini continuity
• Dini criterion