Editing 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 (mathematics)|function]] converges at a given point. These tests are named after [[Ulisse Dini]] and [[Rudolf Lipschitz]].<ref>{{citation|title=Introduction to Partial Differential Equations and Hilbert Space Methods|first= Karl E.|last=Gustafson|author-link=Karl Edwin Gustafson|year=1999|publisher=Courier Dover Publications|pages=121 |url=https://books.google.com/books?id=uu059Rj4x8oC&pg=PA121&dq=%22Dini+test%22|isbn=978-0-486-61271-3}}</ref>

== Definition ==

Let {{mvar|f}} be a function on [0,2{{pi}}], let {{mvar|t}} be some point and let {{mvar|δ}} be a positive number. We define the '''local modulus of continuity''' at the point {{mvar|t}} by

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

Notice that we consider here {{mvar|f}} to be a periodic function, e.g. if {{math|1=''t'' = 0}} and {{mvar|ε}} is negative then we define {{math|1=''f''(''ε'') = ''f''(2π + ''ε'')}}. 

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 {{mvar|f}} satisfies at a point {{mvar|t}} that
::<math>\int_0^\pi \frac{1}{\delta}\omega_f(\delta;t)\,\mathrm{d}\delta < \infty.</math>
:Then the Fourier series of {{mvar|f}} converges at {{mvar|t}} to {{math|''f''(''t'')}}.

For example, the theorem holds with {{math|1=''ω<sub>f</sub>'' = log<sup>−2</sup>({{sfrac|1|''δ''}})}} but does not hold with {{math|log<sup>−1</sup>({{sfrac|1|''δ''}})}}.

:'''Theorem (the Dini–Lipschitz test):''' Assume a function {{mvar|f}} satisfies
::<math>\omega_f(\delta)=o\left(\log\frac{1}{\delta}\right)^{-1}.</math>
:Then the Fourier series of {{mvar|f}} converges uniformly to {{mvar|f}}.

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 {{mvar|f}} with its modulus of continuity satisfying the test with [[Big O notation|{{mvar|O}} instead of {{mvar|o}}]], i.e.

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

and the Fourier series of {{mvar|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)\,\mathrm{d}\delta = \infty</math>

there exists a function {{mvar|f}} such that 

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

and the Fourier series of {{mvar|f}} diverges at 0.

==See also==

* [[Convergence of Fourier series]]
* [[Dini continuity]]
* [[Dini criterion]]

==References==
{{reflist}}

[[Category:Fourier series]]
[[Category:Convergence tests]]