pointwise convergence

As is usual in analysis, the most interesting questions arise when one discusses pointwise convergence.

In this example one can easily see that pointwise convergence does not preserve differentiability or continuity.

In 1975, Natterer proved pointwise convergence of finite element methods.

In such cases, pointwise convergence and uniform convergence are two prominent examples.

This expression asserts the pointwise convergence of the empirical distribution function to the true cdf.

It is therefore plain that uniform convergence implies pointwise convergence.

This is the topology of pointwise convergence.

That means pointwise convergence almost everywhere.

This theorem does not hold if uniform convergence is replaced by pointwise convergence.

We have pointwise convergence, but not uniform convergence.

So in some sense pointwise convergence is atypical, and for most continuous functions the Fourier series does not converge at a given point.

It is more natural too, since it is simply the topology of pointwise convergence for an operator.

With this topology, X becomes a topological vector space, called the space of pointwise convergence.

Indeed, it coincides with the topology of pointwise convergence of linear functionals.

This is the type of stochastic convergence that is most similar to pointwise convergence known from elementary real analysis.

"Pointwise absolute convergence" is then simply pointwise convergence of .

This establishes pointwise convergence.

- Uniform convergence both pointwise convergence and uniform Cauchy-convergence.

- Pointwise convergence almost everywhere convergence.

Pointwise convergence (of partial sum sequence)

Pointwise convergence implies pointwise Cauchy-convergence, and the converse holds if the space in which the functions take their values is complete.

In mathematics, Dini's criterion is a condition for the pointwise convergence of Fourier series, introduced by .

In mathematics, pointwise convergence is one of various senses in which a sequence of functions can converge to a particular function.

This is the notion of pointwise convergence of sequence functions extended to sequence of random variables.

Pointwise absolute-convergence (pointwise convergence of )