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 functions can Limit (mathematics), converge to a particular function. It is weaker than uniform convergence, to which it i ...
of
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 ''p ...
, introduced by .
Statement
Dini's criterion states that if 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 desc ...
' 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 0 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