Dini test
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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.
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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_{|\varepsilon| < \delta} |f(t)-f(t+\varepsilon)|.</math>
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
- <math>\left.\right.\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 Hölder class satisfies the Dini-Lipschitz test.
Precision
Both tests are 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.
- <math>\omega_f(\delta)=O\left(\log\frac{1}{\delta}\right)^{-1}.</math>
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
- <math>\int_0^\pi \frac{1}{\delta}\Omega(\delta)\,d\delta = \infty</math>
there exists a function f such that
- <math>\left.\right.\omega_f(\delta;0) < \Omega(\delta)</math>
and the Fourier series of f diverges at 0.