In mathematics, the Dini–Lipschitz criterion is a
sufficient condition for the
Fourier series of a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
to
converge uniformly at all
real numbers. It was introduced by , as a strengthening of a weaker criterion introduced by . The criterion states that the Fourier series of a periodic function ''f'' converges uniformly on the real line if
:
where
is the
modulus of continuity In mathematical analysis, a modulus of continuity is a function ω : , ∞→ , ∞used to measure quantitatively the uniform continuity of functions. So, a function ''f'' : ''I'' → R admits ω as a modulus of continuity if and only if
:, f(x)-f ...
of ''f'' with respect to
.
References
*
*
{{DEFAULTSORT:Dini-Lipschitz criterion
Fourier series
Theorems in Fourier analysis