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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14
votes
1answer
796 views

Theorems similar to Dini's Theorem and Egoroff's Theorem

Dini's Theorem states that Given a sequence of real-valued continuous functions $(f_n)$ on a compact set $E\subseteq \mathbb{R},$ if $(f_n)$ decreases to a continuous function $f$ pointwise on $E$, ...
8
votes
4answers
742 views

Counterexample around Dini's Theorem

"Give an example of an increasing sequence $(f_n)$ of bounded continuous functions from $(0,1]$ to $\mathbb{R}$ which converge pointwise but not uniformly to a bounded continuous function $f$ and ...
7
votes
3answers
86 views

Pointwise convergence of sequence $(f_n)_n$ of functions to $f$ and changing limits

My analysis notes contains the following question: if $(f_n)_n$ is a sequence of functions of $A \subset \mathbb{R} \to \mathbb{R}$ and $a \in \mathbb{R} \cup \{-\infty, +\infty\}$ an accumulation ...
7
votes
1answer
133 views

Does $f_{n}(z) = \frac{1}{1+n^{2}|z-e^{in}|}$ converge pointwise or uniformly?

I want to check whether the sequence below converges pointwise or uniformly $$f_{n}(z):\{z\in\mathbb{C}:|z| = 1\}\to\mathbb{R}$$ $$\qquad\qquad f_{n}(z) = \frac{1}{1+n^{2}|z-e^{in}|}$$ I have tried ...
7
votes
1answer
140 views

Criteria for smoothness of the pointwise limit of a sequence of functions

Let $\#$ denote cardinality, and fix $p\in[1,\infty]$. Let $(f_n)_{n\in\mathbb{N}}$ be a sequence of functions from $[0,1]$ to $\mathbb{R}$ with the properties $f_n\in C^{n-1}([0,1])$ $f^{(n)}_n$ is ...
6
votes
2answers
108 views

Does an integral inequality imply a pointwise inequality?

$\newcommand{\R}{\mathbb{R}}$ Let $(f_n)_n \subset L^1(\R^N)$. Suppose that for any nonnegative function $\phi \in C_c^{\infty}(\R^N)$, we have: $$ 0 \leq \liminf_{n \rightarrow \infty} \int f_n \, \...
6
votes
1answer
94 views

Show uniform convergence and pointwise convergence for $\sum_{n=1}^ \infty \frac{z^ {n-1}}{(1-z^n)(1-z^ {n+1})}$

Consider the series: $$\sum_{n=1}^ \infty \frac{z^ {n-1}}{(1-z^n)(1-z^ {n+1})}$$ show this converges to: (a) $\frac{1}{(1-z)^2}$ for $|z|<1$ (b) $\frac{1}{z(1-z)^2}$ for $|z|>1$ ...
6
votes
1answer
457 views

Proof for Sturm Liouville eigenfunction expantion pointwise convergence theorem

In "Elementary Partial Differential Equation" by Berg and McGregor, the following theorem is given without proof: Let $f(x)$ be piecewise smooth on the interval $[a,b]$ and let $\{\varphi_n(x)\}$ ...
6
votes
0answers
306 views

Pointwise limit of continuous functions, but not Riemann integrable.

I am trying to find a simple example of a function $f:[0,1]\rightarrow\mathbb{R}$ which is a pointwise limit of continuous functions, but is not Riemann integrable. I know the classical example where ...
5
votes
4answers
472 views

What is the main difference between pointwise and uniform convergence as defined here?

I have a little confusion here. I have seen the following several times and seem to be a bit confused as to differentiating them. Let $E$ be a non-empty subset of $\Bbb{R}$. A sequence of functions $\{...
5
votes
4answers
669 views

Pointwise convergence to a uniform continuous function

What can we say about a sequence of functions that is pointwise convergent (over $R$) to a uniform continuous function? Does it converge uniformly? I have tried it using the definition but can't get ...
5
votes
1answer
793 views

Pointwise convergence of $x^n$ in $[0,1]$

I am reading a book where they state that $f_n(x) = x^n$ converges pointwise to $f(x) = \chi_{\{1\}}(x)$ with respect to the norm $\sup_{x\in[0,1]}|f_n(x)-f(x)|$ and two pages later they state that it ...
5
votes
1answer
1k views

Pointwise convergence and convergence in product topology

Let $A$ be an index set, $X$ a topological space. Define $X^A$ to be the product $\displaystyle\prod_{\alpha \in A}X_\alpha$ where $X_\alpha = X, \forall \alpha \in A$. We can think the elements of $X^...
5
votes
1answer
145 views

Sequence that converges in $L^{2}$ on the real line but nowhere pointwise

Give a sequence of functions $f_{n}:\mathbb{R}\to\mathbb{R}$ that converge in $L^{2}$ but pointwise nowhere. I know that on bounded intervals such as $[0,1]$, the typical example of a sequence ...
5
votes
1answer
2k views

Examples of some Pointwise Convergent Sequences of Functions

I have recently come across pointwise/uniformly convergent sequences of functions, and I am hoping if someone could give some examples of certain sequences of functions so that I could understand the ...
4
votes
3answers
218 views

Uniformly bounded sequence of Riemann integrable functions

Let {$f_n$} be a uniformly bounded sequence of Riemann int'ble functions on $[a,b]$.If $f_n\rightarrow 0$ pointwise then does it follow that $\int _{[a,b]}f_n\rightarrow0$? My thoughts: The result ...
4
votes
1answer
92 views

Does pointwise converge imply uniform convergence in some interval?

I was wondering if $f_n$ converges pointwise to $f$ on a closed interval, $E$, must there be closed interval $E' \subseteq E$ such that $f_n|_{E'}$ converges uniformly to $f|_{E'}$. Currently, with ...
4
votes
2answers
105 views

Finding a pointwise convergent sequence that does not converge in the square mean

I am having a tough time finding a function sequence $(f_{n})_{n\in\mathbb N}$ of continuous functions of $[0,1] \to \mathbb K$, whereby $\mathbb K \in \{\mathbb R,\mathbb C\}$, such that $(f_{n})_{n\...
4
votes
1answer
736 views

Under what metric spaces are pointwise and uniform convergence equivalent?

In exercise 1 of Charles Chapman Pugh's Real Mathematical Analysis, is a question of when pointwise and uniform convergence are 'equivalent' for metric spaces $M$ and $N$ and a sequence of functions $...
4
votes
1answer
96 views

Pointwise convergence of Fourier series of function $\sqrt{|x|}$

I am trying to solve the following exercise: Let $f(x) = \sqrt{|x|}$, $x\in[-\pi,\pi]$. Show that the Fourier series $s_n(0)$ converges to $f(0)$. The hint is that one should consider the ...
4
votes
1answer
72 views

Convergence on $D$ vs. convergence on $\partial D$

Let $f_n(z)$ be a sequence of holomorphic functions on some bounded domain $D$, continuously extendable to the boundary $\partial D$, which converges locally uniformly in the interior. Let us also ...
4
votes
1answer
596 views

Uniform Convergence of $\sin^2 (x +1/n )$

For $n\geq 1$ and let $g_n(x) = \sin^2(x+ \frac{1}{n}) , x\in [0,\infty)$ and $f_n(x)=\int_{0}^{x}g_n(t)dt$ , then {$f_n(x)$} converges point wise to a function on $[0,\infty)$ , but does not ...
4
votes
1answer
724 views

Pointwise sup of step functions is lower semicontinuous (a.e.)

I've found this problem while I was reading a paragraph about Riemann integration on some notes a mate gave me a long time ago. Let $f \colon [a,b] \to \mathbb R$ be a bounded function. Suppose ...
4
votes
1answer
248 views

Pointwise Limit of a Sequence of Measurable Functions

Let $X$ be a measurable space and $Y$ a topological space. I am trying to show that if $f_n : X \to Y$ is measurable for each $n$, and the pointwise limit of $\{f_n\}$ exists, then $f(x) = \lim_{n \to ...
4
votes
0answers
366 views

Weak topology = topology of pointwise convergence?

In a book, I found the following theorem (slightly simplified here): Consider $\mathbb{R}$ with the Borel $\sigma$-field. Let $B$ be the set of all measurable, bounded, real-valued functions on $\...
3
votes
4answers
1k views

Does pointwise convergence to a continuous function on a closed interval imply uniform convergence? [duplicate]

Let's suppose that $(f_n)_{n}$ is a sequence of continuous functions that converges pointwise to a continuous function $f(x)$ on a closed interval $[a, b]$. Is then the convergence uniform, too? If ...
3
votes
5answers
210 views

Language around Pointwise convergence v. Uniform convergence

I've heard uniform convergence described loosely as convergence "occurring at the same rate at every point." Is that really accurate? If the absolute difference between a member of a sequence of ...
3
votes
2answers
92 views

Does $L_1$ convergence of continuous functions imply pointwise convergence?

Suppose that $(f_n)$ is a sequence in $C[0,1]$ which is convergent with respect to the $L_1$ norm. Then is $(f_n(x))$ necessarily convergent for all $x\in[0,1]$? I'm pretty sure the answer is no, ...
3
votes
2answers
73 views

Weak convergence + pointwise convergence on dense subspace of $L^2$

Let $H$ be a dense subspace of $L^2(X,\mu)$ ($X$ being a separable and complete metric space with finite measure $\mu$) and $(f_n)_{n\in\mathbb N}$ be a sequence in $H$ that converges to $f\in L^2(X,\...
3
votes
2answers
423 views

Continuous function as pointwise limit but not as uniform limit of a sequence of continuous functions on $[0,1]$

I recentaly find an article where it is said that there is a sequence of continuous functions $\{f_n:[0,1]\rightarrow\Bbb R\}_{n\in \Bbb N}$ that converges pointqise almost everywhere to zero function ...
3
votes
2answers
97 views

uniform convergence of the composition of two sequence of functions

Let $(f_{n}:\mathbb{R}\to\mathbb{R})$ be a sequence of continuous functions with uniform limit $f:\mathbb{R}\to\mathbb{R}$. Now, for each $n\in\mathbb{N}$, define $g_{n}:\mathbb{R}\to\mathbb{R}$ by $$...
3
votes
1answer
178 views

The limit of a pointwise converging sequence of polynomial is smooth

Let $(P_n)_{n\geqslant 0}$ being a sequence of real polynomials with non negative coefficients, that converge pointwise on $\mathbb{R}$ to a functoin $f$. Then: $f\in \mathcal{C}^{\infty}$. How ...
3
votes
1answer
2k views

What is the difference between pointwise boundedness and boundedness?

In my textbook, the Arzela-Ascoli Theorem states that for a compact set $A$ in a metric space $M$ and $B\subset\mathcal{C}\left(A, N\right)$, where $N$ is another metric space, then $B$ is compact if ...
3
votes
2answers
42 views

Will we have uniform convergence in this case?

Assume that $f: [a,b] \rightarrow \mathbb{R}$ is continuously differentiable. Then we know that for each $t \in [a,b]$ we have that $$\frac{f(t+\Delta t)-f(t)}{\Delta t}-f'(t)$$ will converge to ...
3
votes
1answer
32 views

Prove that $\lim\limits_{n\rightarrow \infty}P_n(A)=P(A)$ implies $\lim\limits_{n\rightarrow \infty} \int f ~dP_n = \int f ~dP$

Let $P_n, n \in \mathbb{N}$ and $P$ be probability measures on the measurable space $(\Omega,\mathfrak{S})$ and assume $\forall A \in \mathfrak{S}: \lim\limits_{n\rightarrow \infty}P_n(A)=P(A)$. I ...
3
votes
1answer
145 views

Uniform convergence of $\sum\limits_{n=1}^∞n^{-x}(e^{\frac{x}{n^2}}-1)$

Pointwise and uniform convergence of the following series of functions: $$\sum_{n=1}^{\infty} n^{-x}\left(e^{\frac{x}{n^2}}-1\right).$$ Now, the series of function converges pointwise as $x \in (-1,...
3
votes
1answer
62 views

$f \in L^1(\mathbb{R})$ and $\lim_{|x| \to +\infty} f(x) = 0$ implies $\lim_{|x| \to +\infty} \int_{\mathbb{R}} f(x-y) d\mu(y) = 0$

Let $f \in L^1(\mathbb{R})$ be a Lebesgue integrable function on $\mathbb{R}$. Assume that $$ \lim_{|x| \to +\infty} f(x) = 0. $$ Let $\mu$ be a non-singular (i.e., if $\mu = \mu_a + \mu_s$, where $\...
3
votes
3answers
33 views

Proving this pointwise limit via definition

For $f_n(x) = \frac{nx + x^2}{n^2}$, this converges pointwise to the function $f(x) = 0$. How would I prove this formally? This is my attempt: $$|f_n(x) - f(x)| = |\frac{nx + x^2}{n^2}| \leq |\frac{...
3
votes
1answer
70 views

Pointwise convergence of $f_n(x) = (x+1)\arctan(x^n)$

I want to study the pointwise convergence of $f_n(x) = (x+1)\arctan(x^n)$ on $R$ but I have trouble establishing pointwise convergence on the interval $I = [-\infty, -1)$. My reasoning is the ...
3
votes
2answers
57 views

Sequence of measurable $\&$ continuous functions defined on $[0,1]$

Let $\{f_n\}$ be a sequence of measurable $\&$ continuous functions from $[0,1]$ to $[0,1]$. Assume $f_n \rightarrow f$ pointwise. Is it true/false that, $f$ is Riemann integrable $\& \int ...
3
votes
1answer
435 views

Limits in Probability [closed]

Let $X_1,X_2,X_3,...,X_n$ be a sequence of random variables which are defined on the same sample space $\Omega$. Show that limits in probability are unique almost surely. That is, if $X_n \...
3
votes
1answer
45 views

Sequence of functions that converges strongly in $L^2(\mathbb R)$ but not pointwise

I am trying to find a sequence of functions $\{f_j\}$ and a function $f$ such that $f_j\to f$ strongly in $L^2(\mathbb R)$ but $f_j$ does not converge to $f$ pointwise. The definition of strong ...
3
votes
1answer
26 views

Proof Verification for Uniform Convergence on Sequence of Functions

just looking for a verification on a proof. Thanks in Advance Let $f_n$ be a sequence of functions such that $f_n=\frac{x^{2n}}{1+x^{2n}}$ defined on $[-2,2]$. Prove or Disprove Uniform Convergence ...
3
votes
1answer
109 views

Pointwise convergence of average of continuous functions

Suppose that $(f^n)$ is a sequence of continuous functions from a compact metric space $A$ into a convex compact subset of the reals $B$. Is there a subsequence, $(f^{n_k})$, such that the sequence $...
3
votes
1answer
389 views

Pointwise limit of continuous functions is continuous on a dense set

I'm stuck in understanding the proof of the following theorem given during a course: Let $X$ be a Baire space, and $(Y,d)$ a metric space. Let $f_n:X\to Y$ be a sequence of continuous function, ...
3
votes
1answer
114 views

Uniform convergence of $n\sin\sqrt{4\pi^2n^2+x^2}$ on $[0,a]$ and $\mathbb{R}.$

Let $f_n(x)=n\sin\sqrt{4\pi^2n^2+x^2}$. Prove that $(f_n)$ is uniformly convergent on $[0,a]$ for every $a>0.$ Is the convergence uniform on $\mathbb{R}$? Attempt. For $x\in \mathbb{R}$ constant ...
3
votes
2answers
22 views

Convergence of Sequence of Indicator Functions

With $I \subset \mathbb{R}$, let $\chi_{i}$ denote the indicator function of I. Define a sequence of functions $\{g_{n}\}_{n=1}^{\infty}$ on $[0,1]$ as: $g_{1}(x) = \chi_{[0,1]}(x),\quad g_{2}(x) = \...
3
votes
0answers
38 views

Finding the Pointwise Limit of a Function

If I have a sequence of functions $f_n[0,2] \rightarrow \mathbb{R}$ where $f_n(x) = \frac{x^n}{2^n+n}$. If I attempt to find the pointwise limit, I work out that by taking $x \in [0,2]$: We can ...
3
votes
0answers
100 views

Prove the compactness and second countability of a subset of $\Bbb{R}^{L^1(G)\times D}$

I'm trying to fulfill the details in a proof whose aim is to guarantee the existence of a compact space with certain properties, but in this case I had no success. Let $X$ be a metric space (actually,...
3
votes
0answers
65 views

Pointwise convergence vector valued continuous functions

Let $(X,\|\cdot\|)$ be a (infinite dimensional) Banach space and $f_{n}:[0,1]\longrightarrow (X,\|\cdot\|)$ continuous, for each integer $n\geq 1$, a sequence of mappings. There is some "pointwise ...

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