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]

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

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

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

${\displaystyle \omega _{f}(\delta )=\max _{t}\omega _{f}(\delta ;t)}$

With these definitions we may state the main results:

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

${\displaystyle \int _{0}^{\pi }{\frac {1}{\delta }}\omega _{f}(\delta ;t)\,\mathrm {d} \delta <\infty .}$

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

${\displaystyle \omega _{f}(\delta )=o\left(\log {\frac {1}{\delta }}\right)^{-1}.}$

Then the Fourier series of f converges uniformly to f.

In particular, any function that obeys a Hölder condition satisfies the Dini–Lipschitz test.

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.

${\displaystyle \omega _{f}(\delta )=O\left(\log {\frac {1}{\delta }}\right)^{-1}.}$

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

${\displaystyle \int _{0}^{\pi }{\frac {1}{\delta }}\Omega (\delta )\,\mathrm {d} \delta =\infty }$

there exists a function f such that

${\displaystyle \omega _{f}(\delta ;0)<\Omega (\delta )}$

and the Fourier series of f diverges at 0.

• Convergence of Fourier series
• Dini continuity
• Dini criterion