In mathematics, the Dini–Lipschitz criterion is a
sufficient condition for the
Fourier series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems ...
of a
periodic function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the t ...
to
converge uniformly at all
real numbers
In mathematics, a real number is a number that can be used to measurement, measure a continuous variable, continuous one-dimensional quantity such as a time, duration or temperature. Here, ''continuous'' means that pairs of values can have arbi ...
. 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 of ''f'' with respect to
.
References
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{{DEFAULTSORT:Dini-Lipschitz criterion
Fourier series
Theorems in Fourier analysis