Английская Википедия:Dini test

Содержание

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

Текст

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

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

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

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

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

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

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

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

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

In particular, any function of a Hölder classШаблон:Clarify 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 Шаблон:Mvar with its modulus of continuity satisfying the test with [[Big O notation|Шаблон:Mvar instead of Шаблон:Mvar]], i.e.

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

and the Fourier series of Шаблон: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 Шаблон:Mvar such that

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

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

See also

• Convergence of Fourier series
• Dini continuity
• Dini criterion