In mathematics, Dini's criterion is a condition for the
pointwise convergence
In mathematics, pointwise convergence is one of Modes of convergence (annotated index), various senses in which a sequence of function (mathematics), functions can Limit (mathematics), converge to a particular function. It is weaker than uniform co ...
of
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 ...
, introduced by .
Statement
Dini's criterion states that if 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 ...
has the property that
is
locally integrable
In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
near
, then the Fourier series of
converges to
at
.
Dini's criterion is in some sense as strong as possible: if
is a positive continuous function such that
is not locally integrable near
, there is a continuous function
with
whose Fourier series does not converge at
.
References
*
*{{SpringerEOM, id=Dini_criterion&oldid=28457, title=Dini criterion, first=B. I., last= Golubov
Fourier series