# 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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### Limits of functions in $L^p$ spaces and Hölder inequality

I have a severe problem understanding $L^p$ spaces and everything related. For example, see my thoughts on the following exercise: Let $f_n \in L^1(0,1) \cap L^2(0,1)$ for $n = 1, 2, 3, \ldots$ and ...
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### Study the series of functions $\sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2 + n}$ regarding pointwise and uniform convergence. [closed]

I'm studying Real Analysis and I have stumbled upon the following question. Study the series of functions $$\sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2 + n}$$ regarding pointwise and uniform convergence ...
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### Study the pointwise convergence of the series $\sum_{n=1}^{\infty} \frac{n}{1+(x-1)^2 n^3}, x \in \mathbb{R}$.

I was studying Real Analysis and I was doing the following exercise. Study the pointwise convergence of the series of functions $\sum_{n=1}^{\infty} \frac{n}{1 + (x-1)^2 n^3}, x \in \mathbb{R}.$ I'm ...
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### Show $\lim\limits_{n \to \infty}\frac{f\left(nx\right)}{n^2}=0$ a.e. $x\in\mathbb{R}$ if $f \in L^1\left([0,T]\right)$ and periodic.

Show $\lim\limits_{n \to \infty}\frac{f\left(nx\right)}{n^2}=0$ a.e. $x\in\mathbb{R}$ if $f \in L^1\left([0,T]\right)$ and periodic in $\mathbb{R}$, where $T>0$ is the period. My idea is as follows:...
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### Is the set of functions with positive derivative a.e. sequentially compact over the topology of pointwise convergence?

Consider the following set $H = \{ f :[0,1]\to[0,1] | \frac{df}{dx} \geq 0 \text{ a.e.} \}$, I was wondering if this set could be a sequentially compact set over the pointwise convergence topology, ...
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### Sequence of functions in $C^\infty ([0, 1])$ which satisfy $\int_0^1|f_i(t)|dt = 1$ and converge pointwise to 0

I think taking bump functions supported in neighborhoods of $1/n$ should work, but I wonder if there's a more explicit construction. Here's my answer so far: For every $n\in\mathbb{N}$ and real ...
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### Uniform convergence impies pointwise convergence. Am I missing something?

I'm reading 'Calculus 3rd Ed.' by M. Spivak, specifically a Corollary in Chapter 24. In it, the following is stated: Let $\sum_{n=1}^\infty f_n$ converge uniformly to $f$ on $[a,b]$. (1) If each $f_n$...
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### Uniform convergence of a series of functions depending upon a parameter

I'm unable to prove if the following series converge uniformly on $[0,+\infty)$ for the values of the parameter $\beta$ between $2$ and $3$. $$\sum_{n\geq 1}\frac{n^\beta x}{x^4+n^4}$$ The maximum ...
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### What is the limit function of $f_n(x) := \begin{cases} (x-n+1)(n+1-x): n-1 < x < n+1\\ 0: x \leq n-1 \lor x \geq n+1 \end{cases}$?

I have a question about the convergence of $(f_n)_{n \in \mathbb{N}}$ with $$f_n(x) := \begin{cases} (x-n+1)(n+1-x): n-1 < x < n+1\\ 0: x \leq n-1 \lor x \geq n+1 \end{cases}.$$ Does this ...
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### Rate of convergence of a threshold defined with sequences of functions

Let $f$ be a real decreasing function (resp. $g$ a real increasing function), and $(f_n)$ be a sequence of real decreasing functions (resp. $(g_n)$ be a sequence of real increasing functions) such ...
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