# Questions tagged [pointwise-convergence]

For questions about pointwise convergence, a common mode of convergence in which a sequence of functions converges to a particular function. This tag should be used with the tag [convergence].

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### Convergence of integrals without the dominated convergence theorem

Let $t>0$ and $\{f_n\}_{n \geq 1}$ be a sequence of functions that do not converge pointwise to any integrable function, but where: \begin{equation} \int_0^t f_n(s) ds \rightarrow F(t) < \...
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### problem on $L^2$ (pointwise) convergence and Carleson's Theorem

I am having some problem on my arguments and I want to see where it fails. It deals with pointwise convergence in $L^2[-\pi,\pi]$: pointwise convergence is given by Carleson's Theorem (so it is a hard ...
1 vote
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### $\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$

How can I show $\lim\limits_{n\to\infty}\int_0^2 \frac{nx^{n-1}e^{-x^n}}{1+x}dx=1-\lim\limits_{n\to\infty}\int_0^2\frac{e^{-x^n}}{(1+x)^2}dx$? It is $\frac{d}{dx}x^n=nx^{n-1}$ so I tried to solve it ...
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### There exists such result for any convergence of sequence definiton

Assume that $L(x_n)$ stands for the limit of the sequence $(x_n)$ in some sense, not necessarily the usual limit. For example, could be the almost convergence limit or the statistical limit. I ...
1 vote
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### If {fn} converges pointwise to {f} and each {fn} is bounded in a closed interval [a,b], is {f} also bounded in [a,b]?

Is it possible to say that if $f_n$ converges pointwise to $f$ and each $f_n$ is bounded in the closed interval $[a,b]$, then $f$ is also bounded in $[a,b]$? As far as I know, boundness is not ...
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### Comparing different kinds of convergence

I want to gain a better understanding of the convergence of a sequence of functions $(u_n:I\to \mathbb R)_{n\in \mathbb N}$ in different norms, I know some results, for example that if $u_n(x)\to u(x)$...
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Can anyone clear up the misconception in my argumentation. If a sequence of functions $f_k(x)$ is pointwise convergent towards some limit $f(x)$ for every $x$, then at every point $x$ we can choose an ...
Definition 1. Suppose that $\left(f_{n}\right)$ is a sequence of functions $f_{n}: A \rightarrow \mathbb{R}$ and $f: A \rightarrow \mathbb{R}$. Then $f_{n} \rightarrow f$ pointwise on $A$ if, for ...
### If $(f_n')$ converges uniformly on an interval, does $(f_n)$ converge?
Let $(f_n)$ be a sequence of functions that are all differentiable on an interval A, and suppose the sequence of derivatives $(f_n')$ converges uniformly on A to a limit function $g$. Does it follow ...