Dini test

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

:$\backslash left.\backslash right.\backslash omega\_f(\backslash delta;t)=\backslash max\_\{|\backslash varepsilon|\; \backslash le\; \backslash delta\}\; |f(t)-f(t+\backslash varepsilon)|$

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

:$\backslash omega\_f(\backslash delta)\; =\; \backslash max\_t\; \backslash omega\_f(\backslash delta;t)$

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

::

:Then the Fourier series of {{mvar|f}} converges at {{mvar|t}} to {{math|f(t)}}.

For example, the theorem holds with {{math|1=ω_f = log⁻²({{sfrac|1|δ}})}} but does not hold with {{math|log⁻¹({{sfrac|1|δ}})}}.

:Theorem (the Dini–Lipschitz test): Assume a function {{mvar|f}} satisfies

::$\backslash omega\_f(\backslash delta)=o\backslash left(\backslash log\backslash frac\{1\}\{\backslash delta\}\backslash right)^\{-1\}.$

: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.

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, i.e.

:$\backslash omega\_f(\backslash delta)=O\backslash left(\backslash log\backslash frac\{1\}\{\backslash delta\}\backslash right)^\{-1\}.$

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

:$\backslash int\_0^\backslash pi\; \backslash frac\{1\}\{\backslash delta\}\backslash Omega(\backslash delta)\backslash ,\backslash mathrm\{d\}\backslash delta\; =\; \backslash infty$

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

:

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

• Convergence of Fourier series
• Dini continuity
• Dini criterion

{{reflist}}

Category:Fourier series

Category:Convergence tests