In mathematics, Dini's criterion is a condition for the
pointwise convergence
In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
Definition
Suppose that X is a set and ...
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 '' ...
, 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 d ...
' 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