In mathematics, the Dini–Lipschitz criterion is a

sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...

for the

Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...

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 d ...

to converge uniformly at all

real numbers In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

. 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 :

\lim_\omega(\delta,f)\log(\delta)=0

where

\omega

is the

modulus of continuity In mathematical analysis, a modulus of continuity is a function ω :

, ∞ The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...

→

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 :, ...

of ''f'' with respect to

\delta

References

* * {{DEFAULTSORT:Dini-Lipschitz criterion Fourier series Theorems in Fourier analysis