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.<ref>Template:Citation</ref>

Definition

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

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

Notice that we consider here Template:Mvar to be a periodic function, e.g. if Template:Math and Template:Mvar is negative then we define Template: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 Template:Mvar satisfies at a point Template:Mvar that

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

Then the Fourier series of Template:Mvar converges at Template:Mvar to Template:Math.

For example, the theorem holds with Template:Math but does not hold with Template:Math.

Theorem (the Dini–Lipschitz test): Assume a function Template:Mvar satisfies

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

Then the Fourier series of Template:Mvar converges uniformly to Template:Mvar.

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 Template:Mvar with its modulus of continuity satisfying the test with [[Big O notation|Template:Mvar instead of Template:Mvar]], i.e.

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

and the Fourier series of Template:Mvar diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that

<math>\int_0^\pi \frac{1}{\delta}\Omega(\delta)\,\mathrm{d}\delta = \infty</math>

there exists a function Template:Mvar such that

<math>\omega_f(\delta;0) < \Omega(\delta)</math>

and the Fourier series of Template:Mvar diverges at 0.

See also

• Convergence of Fourier series
• Dini continuity
• Dini criterion