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

<math>\left.\right.\omega_f(\delta;t)=\max_{|\epsilon| < \delta} |f(t)-f(t+\epsilon)|.<math>

Notice that we consider here f to be a periodic function, e.g. if t = 0 and ε is negative then we define <math>f(\epsilon)=f(2\pi + \epsilon)<math>.

The global modulus of continuity (or simply the modulus of continuity) is defined by

<math>\omega_f(\delta) = \max_t \omega_f(\delta;t)<math>

With these definitions we may state the main results

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

<math>\int_0^\pi \frac{1}{\delta}\omega_f(\delta;t)\,d\delta < \infty.<math>

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

For example, the theorem holds with <math>\omega_f=\log^{-2}(\delta^{-1})<math> but does not hold with <math>\log^{-1}(\delta^{-1})<math>.

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

<math>\omega_f(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}.<math>

Then the Fourier series of f converges uniformly to f.

In particular, any function of a Convergence of Fourier series.