Questions tagged [pointwise-convergence]

Question 1

Let $f:[-1,1]\longrightarrow[-1,1]$ be a continuous function satisfying the following condition : $$\tag{1}\forall x\in[-1,1]\!\setminus\!\{0\},\quad|f(x)|<|x|.$$ Is that condition sufficient to ...

Question 2

I am trying to prove that the following sequence of density functions $$f_{n}(x) = \mathbb{1}_{\left(0,2^{n}\right)}(x) \cdot \left(\dfrac{\left(n\ln(2) - \ln(x)\right)^{n}}{2^{n}\cdot n!} \right)$$ ...

Question 3

I wonder if it is possible to describe pointwise convergence set-theoretically. More specifically, let $(f_n)_{n\geq 1}$ be a sequence of functions from $\mathbb{R}$ to $\mathbb{R}$, and let $f$ be ...

Question 4

Inspired by this question. Let $\tilde{f}:[0,1)\to \mathbb{R}$ be a bounded, non-constant continuous function. Let $f$ be the $1$-periodic extension of $\tilde{f}$ to $\mathbb{R}$. Let $g:[0,1]\to\...

Question 5

Consider the space of probability distributions on $(0,\infty)$ denoted through their cdf $F$. Let $F_0$ be a probability distribution having a point mass at $0$, so that $F_0$ has a step at $0$ (say ...

Question 6

I have a background in mathematical statistics and now am trying to self-study topology (from John Kelley's General Topology). I read about convergence classes and how convergence classes generate ...

Question 7

Question Is the everywhere‑divergent series(Steinhaus) $$ \sum_{k=2}^{\infty} \frac{\sin\bigl(k(x-\ln\ln k)\bigr)}{\ln k} $$ a genuine Fourier series? H.Steinhaus: A divergent trigonometrical ...

Question 8

The question below is from N.L Carother's Real Analysis, "The Space of Continuous Functions" - Problem 53: Let $X$ be a compact metric space, and let $(f_n)$ be a sequence in $C(X)$. If $(...

Question 9

Let $$S_a(x)=\sum^{\infty }_{n=1}\frac{x}{n^a({1+nx^2})}$$ for $\ x\in \mathbb{R}$. I know $S_a(x)$ is pointwise convergent if $a>0$. I know $S_a(x)$ is uniformly convergent if $a>1/2$ (by the ...

Question 10

So I am a little confused about a statement on on Lemma 4.2 Gilbarg-Trudinger (p.55). Here is a picture of the statement. In the preceding Lemma (that's Lemma 4.1) they assume that $f$ is bounded and ...

Question 11

Let $(X,A,\mu)$ a measure space. Assume a sequence $(f_n)$ of positive (i.e. $\geq 0$) measurable functions from $X$ to $\mathbb{R}$ converges to $f$ in $L^p$. Then is $f$ positive ? My idea is the ...

Question 12

Let $N>1$ be a natural number. Let $C:\mathbb R^N\times\mathbb R^N\rightarrow\mathbb R$ be a $L^1(\mathbb R^N\times\mathbb R^N)$ function such that $C(x,y)=C(y,x)$ (this is a condition that ...

Question 13

As a technical detail in my work, I need a concrete formula for a sequence of functions $f_n:[0, \infty) \to (a, \infty)$, as simple as possible, that satisfies all of the following conditions : $f_n(...

Question 14

Let $(X_n)$ be a sequence of real-valued random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. In many classical results, such as the strong law of large numbers, we have almost ...

Question 15

Let $(X,d)$ and $(Y,\rho)$ be metric spaces. For all $n\in \mathbb{N}$, $f, f_n: (X,d) \to (Y,\rho)$ be functions. I want to find an example satifies the following conditions that is (i) $f_n\to f$ (...

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Questions tagged [pointwise-convergence]

Pointwise convergence of a sequence of iterations of a continuous function

Pointwise limit of a sequence of density functions

set theoretic definition of pointwise convergence

Pointwise convergence of series of scalings of continuous, periodic, non-constant function.

Closure of probability distributions on an open set vs probability distributions on the closure of an open set

Apparent contradiction in topologies induced by probabilistic convergence modes

Is the everywhere‑divergent series $\sum_{k=2}^{\infty} \frac{\sin\bigl(k(x-\ln\ln k)\bigr)}{\ln k}$ a genuine Fourier series?

Pointwise monotonic convergence implies equicontinuity

Proving $\sum^{\infty }_{n=1}\frac{x}{n^a({1+nx^2})}$ is not uniformly convergent if $0<a\leq 1/2$

Well definedness of the newtonian potential when $f$ is bounded and locally Holder continuous (Gilbarg-Trudinger Lemma 4.2)

$L^2$ limit of a positive sequence

On convergence on $L^1$ of a "weird" sequence of functions.

Concatenating $x^{\frac{1}{n}}$ with $1$

Why is uniform convergence over $\omega$ not possible for sequences of random variables?

Counterexample for uniformly equicontinuous function sequence with non uniformly continuous pointwise limit function

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