Questions tagged [sequence-of-function]

Use this tag only when your query is about sequences of functions. Don't use this tag for any other sequence such as sequences of real numbers or sequences of complex numbers etc.

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Is there any $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ such that $\frac{d}{dt}\|K_t^{\epsilon}\|^2\to 1$?

I am looking for conditions or at least one example such that the following holds true. Let $K_t^{\epsilon}:[0,T]\to\mathbb R$ be an smooth function such that $K_t^{\epsilon}\to \mathbf{1}_{[0,t]}$ in ...
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Compute sup norm of sequence of functions.

Consider the operator $T: C^2[0,1] \subset C^1[0,1] \to C^1[0,1]$ defined by $Tf=f'+f''$. Compute $\| T e^{-nx } \|_{\infty }$ and $\| T x^n \|_{\infty }$. My attempt. First I tried to compute $\| e^{...
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Why is the definition of lub failing in the sequence $\left\{ \frac{1}{n}\right\}$?

The sequence $\left\{ \frac{1}{n}\right\}$ is bounded since $0\lt \frac{1}{n}\leq 1$ and $1$ is its lest upper bound (lub) of the sequence $\left\{ \frac{1}{n}\right\}$. According to definition of ...
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Intuition behind the proof that pointwise limit of measurable function is measurable.

I am self-studying measure theory.There is a very important theorem in measure theory which says that a pointwise limit of measurable functions is measurable.Now,I understand the proof but do not get ...
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Convergence of derivatives.

We have a course on complex analysis in the current semester.Our professor introduced a theorem in class which is as follows: Theorem Let $f_n:\Omega\to \mathbb C$ be a sequence of holomorphic ...
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Predictive numerical sequence in ADPCM decoding

I was inquiring about ADPCM type audio decoding, decoding where a predictive formula is used that I cannot find, despite having checked several articles and sites. If a prediction phase is added, in ...
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0 answers
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Question about final sequence in the diagonal trick.

currently I am reviewing the diagonal trick regarding real analysis, i.e. one can prove that for example if $V=\{F: \mathbb{R} \to \mathbb R : \text{ right cont. , monotone increasing and bounded} \}$ ...
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1 answer
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Sequence of Lipschitz continuous functions approaching Holder continuous function

Let $\{f_n\}$ be a sequence of uniformly bounded Lipschitz functions with Lipschitz constant $L_n$, i.e. for each $n\in\mathbb{N}$ $$ |f_n(x) - f_n(y)| \leq L_n |x-y| $$ It is easy to see if the ...
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1 answer
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Why $P(\bigcap_{n\geq 1}A_n) = P(|X_n - X| \geq \varepsilon , \forall n \geq 1)$?

Let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of random variables that converges almost surely to the variable X. Prove that $X_n$ converges to $X$ in probability. My attempt Let $\varepsilon >0$. ...
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2 answers
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Definition of $\lim_{x \to x_0} f_n (x)$ and related questions

While studying uniform convergence of sequences of functions, I stumbled upon the theorem that, under uniform convergence hypothesis of $f_n$ to $f$ as $n \to \infty$, one can interchange the limits: $...
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Transitivity of convergence of sequence in $L^1$ and $BV$ [closed]

Let $(f_n)_n\in BV([0,T])$ a sequence of functions of bounded variation converging to some $f\in BV([0,T])$, that is to say $$\lim_{n\rightarrow+\infty}\parallel f_n-f\parallel_{BV([0,T])} = 0.$$ and ...
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1 vote
1 answer
41 views

Uniform convergence of sequence of functions that converges pointwise to an unbounded function

I have the following sequence of functions before me: $f_n(x)=\dfrac{1-x^n}{1+x},-1<x<1$ I have to determine whether above sequence of functions is uniformly convergent or not. First of all, I ...
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1 vote
1 answer
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Sequences of Functions, continuity and Uniform convergence.

I need help verifying my solution to a problem, it seemed really simple but that's often when I make mistakes so I'd appreciate some help. Let $D \subset \mathbb{C}$ and $f_n:D\rightarrow \mathbb{C},n\...
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Question related to approximation of measurable functions by simple functions

I am self-studying measure theory from the book by Sheldon Axler.There I found a theorem before integration is introduced: Let $(X,\mathcal S)$ be a measurable space and $f:X\to [-\infty,\infty]$ be a ...
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Issues with theorem 7.15 from Rudin's PMA.

Some prerequisite definitions and theorems. Given a metric space $X$, let $$ \mathscr{C}(X) = \{f: X \rightarrow \mathbb{C} \; | \; f \text{ bounded and continuous.}\}. $$ Next we have the familiar ...
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Rudin's PMA theorems 7.11 and 7.12

Theorem 7.11: Suppose $f_n \to f$ uniformly on a set $E$ in a metric space. Let $x$ be a limit point of $E$, and suppose that $$\lim_{t\to x} f_n(t) = A_n \qquad (n \in N).$$ Then $\{A_n\}$ converges, ...
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1 answer
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Pointwise convergence of $\ln(\ln(...\ln(x+1)...+1)+1)$ on $[0,\infty)$

Let $\{f_n\}$ be the sequence of functions defined by this recurrence relation \begin{cases} f_0(x) & = & x \\ f_{n+1}(x) & = & \ln(f_n(x) + 1) \end{cases} . Is it true that $f_n$ ...
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2 votes
1 answer
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What's the name of this theorem: $\int_a^b|u_n-u|^pdt\rightarrow0\wedge\int_a^b|u_n'-v|^pdt\rightarrow0\Rightarrow u\in W_p^1(a,b) \ : \ u'=v$?

My Professor gave me this as an exercise to prove, that: $$\int_a^b|u_n(t) -u(t)|^pdt\rightarrow0\wedge\int_a^b|u_n(t)'-v(t)|^pdt\rightarrow0\Rightarrow u(t)\in W_p^1(a,b) \ : \ u'(t)=v(t)$$ given, ...
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Component wise convergence of a sequence of complex harmonic functions

It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In ...
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3 votes
2 answers
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Investigate whether the sequence $\frac{nx^n + 3\sin{2n\pi x}}{n}$ is uniformly convergent

Let $(f_n)$ be a sequence of functions defined by \begin{equation*} f_n(x) = \frac{nx^n + 3\sin{(2n\pi x)}}{n} \end{equation*} for all $x \in [0,1]$ and $n \in \Bbb N$. Show that $f_n$ is converges ...
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3 votes
3 answers
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Finding "limit-function" of $f_n=\frac{x}{1+n^2x^2}$, Pointwise/Uniform convergence of ${f_n'}$

I am preparing for my exam and need help with the following tasks: Let ${f_n}:[-1,1]\to \mathbb{R}$ be defined as $f_n=\frac{x}{1+n^2x^2}$ Is ${f_n}$ pointwise/uniformly convergent? Specify the "...
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0 votes
1 answer
37 views

Proof explanation (Complete subspace)

While studying functional analysis, more specifically that the subspace $Y=\{x \in \mathcal{C}[a, b] \mid x(a)=x(b)\} \subset \mathcal{C}[a, b]$ is complete, I came across a very simple question I ...
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1 vote
1 answer
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Choice of $N$ for $g_n(x) = \frac{x}{1+x^n}$

Show that the convergence cannot be uniform on $[0, \infty)$ for $$g_n(x) = \frac{x}{1+x^n}$$ Here's what I have tried: We know it's pointwise convergent on $g(x) = x$ on the limit $[0, \infty)$. $$|...
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0 votes
2 answers
45 views

Choice of $N$ for $\frac{nx}{1+nx^2}$

Let $$f_x(x) = \frac{nx}{1+nx^2}$$ Is the convergence uniform on $(0, \infty)$? We know that it's pointwise convergent onf $f(x)= \frac{1}{x}$ as we take the limit $\lim_{n \to \infty} f_n(x)$. ...
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0 votes
1 answer
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Prove sequence $B[t]=\sum_{k=1}^{t}\sum_{i=1}^{k}S[i]$ for $S=\{0,1\}^T$ contains at most one element equal to $1$ at the same index as in $S$

I have the following problem. Given is a sequence of bits $S=\{0,1\}^T$ and functions $$A[t]=\sum_{k=1}^{t}S[k]$$ $$B[t]=\sum_{k=1}^{t}A[k]$$ Now for sequence $\{B[0], B[1], \dots, B[T]\}$ I would ...
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-1 votes
1 answer
28 views

Sequence of continuous functions $f_n $ with $\lim_{n \rightarrow +\infty} f_n(x)=f(x)$ and property

Let be $(X, d)$ a complete metric space, $f_n : X \rightarrow \mathbb{R}$ a sequence of continuous functions and a function $f: X \rightarrow \mathbb{R}$ such that: $$\lim_{n \rightarrow +\infty} f_n(...
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1 vote
0 answers
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Proving that $B(E)$ is closed subalgebra of $C(K,\mathbb{C})$

$C(K,\mathbb{C})$ is the space of all continuous functions $f$ on a compact set $K$ of $\mathbb{C}$. Let $B(E)$ be the collection of $f$ in $C(K,\mathbb{C})$ such that there is sequence of rational ...
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1 vote
1 answer
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Does this proof of the Boundedness Theorem contain a mistake?

My course notes (mathematics BSc, second-year module in real analysis, unpublished) have a proof of the Boundedness Theorem which begins: But does that sequence work? Here's my reasoning. Let \begin{...
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1 vote
0 answers
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Prove that the sequence of functions $f_n$ is equicontinuos, but with not uniformly convergent subsequence in $(0,1)$.

Question: Prove that the sequence of functions $f_n:(0,1)\to\mathbb{R}$ below is equicontinuous and simply converges to $f\equiv\dfrac{1}{2}$, but with not uniformly convergent subsequence in $(0,1)$. ...
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0 answers
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Find the $\lim\limits_{n \to \infty} \int_{0}^{1}g_n(x)dx$ for these functions

The sequence of the continuous functions $f_n$ on the $I= [-1,1]$ satisfies $(a)$~$(c)$ [Here $\mathbb{N}$ is the natural number set] $(a)$ $f_n(x) \geq 0 $ for $\forall x \in I$, $\forall n \in \...
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1 vote
0 answers
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Uniform convergence of a sequence of functions on $[0,1]$ as follows [duplicate]

Let $f_n:[0,1]\to \Bbb R$ be a sequence of functions defined by \begin{align*} f_n(x):= \begin{cases} nx, 0\le x \le \frac1n \\ 1, \frac1n \le x \le 1 \end{cases}. \end{align*} Show that $(f_n)$ is ...
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0 votes
1 answer
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Show that the sequence of functions $(f_n)$ is bounded uniformly. [duplicate]

I am trying to show that the sequence $$f_n(x)=\sum_{m=1}^n sin(mx)$$ is uniformly bounded on $[a,2\pi-a]$ with $0<a\le\pi$. From playing around with the graph of $f$ its pretty clear that the ...
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1 vote
2 answers
64 views

Uniform convergence and rigor proof of pointwise convergence of $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$ for all $x \in (0,\infty)$.

Let $(f_n)$ be a sequence of functions where $f_n:(0,\infty) \to \Bbb R$ defined by $f_n(x)=n\left(\sqrt{x+\frac1n}- \sqrt{x}\right)$. Show that $f_n$ does not converge uniformly on $(0,\infty)$. My ...
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  • 1,951
0 votes
1 answer
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Sequence of functions $f_n(x) = \frac{\sin nx}{1+nx}$ converges to $0$.

Let $(f_n)$ be a sequence of functions with $f_n : [a,\infty) \to \Bbb R$ defined by $f_n(x) = \frac{\sin nx}{1+nx}$, for all $x \in [a,\infty)$ and $n \in \Bbb N$. Show that $f_n$ converges pointwise ...
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  • 1,951
1 vote
2 answers
68 views

Find $\sum f_n(x)$

Problem For non-negative integer $n$, define $$f_n\colon [0, 1]\to\mathbb{R}, \quad f_n(x)=\int_0^x f_{n-1}(t)dt\quad (n>0),\quad f_0(x) = e^x$$ Find $g(x)=\displaystyle\sum_{n=0}^\infty f_n(x)$ ...
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  • 1,765
2 votes
1 answer
35 views

Extraction of pointwise convergenct subsequence using Arzela-Ascoli theorem

Let $f_n:[a, b] \rightarrow \mathbb{R}$ be a sequence of continuous functions which is uniformly bounded i.e. $||f_n||_{L^{\infty}} \leq M <\infty$ and satisfies $f_n(a)=A$ for all $n\in \mathbb{N}....
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0 votes
1 answer
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Nonincreasing sequence of functions

Let $f_n:[a,b]\to[0,\infty)$ be a sequence of differentiable functions such that $\lim_{n\to\infty}f_n'(t)\leq 0$. The derivative is taken with respect to $t\in[a,b]$. Then is it possible to conclude ...
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1 vote
1 answer
52 views

Showing that the sequence of functions defined by $f_n(x):=\frac{n^2x+n}{n^2+n+1}$ for all $x \in \Bbb R$ and $n \in \Bbb N$ is converges to $g(x)=x$

Let $(f_n):\Bbb R \to \Bbb R$ be a sequence of functions defined by $f_n(x):=\frac{n^2x+n}{n^2+n+1}$ for all $x \in \Bbb R$ and $n \in \Bbb N$. Let $g:\Bbb R \to \Bbb R$ be a function defined by $g(x):...
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  • 1,951
1 vote
1 answer
33 views

Showing that the given sequence of functions converges to a given function on the given set as follows

Let $(f_n): \Bbb R \to \Bbb R$ be a sequence of functions defined by $f_n(x) := x^n$ for $x \in \Bbb R$ and $n \in \Bbb N$. Let $f:(-1,1] \to \Bbb R$ be a function defined by \begin{align*} f(x):= \...
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  • 1,951
2 votes
0 answers
26 views

Convergence of Sequence of functions with nested radicals.

I am reading Lascota and Mackey's "Chaos, Fractals, and Noise" (which is great). In an early chapter, they define the following operator (really this is a special case of the Frobenius-...
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0 votes
1 answer
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Calculate $\lim_{n\to\infty} \frac{n\sin(x/n)}{x(x^2+1)}$ in $\mathbb{R}^+$

For all $n\in\mathbb{N}$, let $f_n:(0,\infty)\to\mathbb{R}$ be the function defined by $f_n(x)=\frac{n\sin(x/n)}{x(x^2+1)}$. Find the pointwise limit of $(f_n)$. I know that for all $x>0$ we have ...
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3 votes
1 answer
33 views

Showing that this particular sequence of functions defined by recursion is point-wise bounded

Let $(s_n)_n$ be a summable sequence. Consider the following sequence of functions $(z_n)_{n\geq 0}$, $z_n:[0,+\infty)\rightarrow\mathbb{R}$, defined recursively as \begin{align*} z_0(x)&\equiv 0\\...
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4 votes
1 answer
97 views

Riemann integral of product of sequence of function with a continuous function at a point.

Here is the setup to a questions I've been looking at. I mainly just need a hint on how to proceed from where I'm at. Or suggest a theorem/corollaries that would help. The problem: Let $\{g_n\}$ be a ...
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0 votes
2 answers
97 views

Existence of closed form formulas for integer sequences

Suppose that z x y ———————— 0 1 0 1 2 0 2 2 1 3 3 0 4 3 1 5 3 2 6 4 0 7 4 1 8 4 2 9 4 3 . . . . . . . . . n xn yn the z column ...
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0 votes
0 answers
70 views

Uniform convergence of $f_n(x)=nx^n(1-x^n)$ on $[0,1]$

I want to check the uniform convergence of the sequence of function $f_n(x)=nx^n(1-x^n)$ on $[0,1]$. My approach: We can see $f_n$ converges pointwise to $f(x)=0$ Now If we consider the sequence $x_k=(...
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2 votes
2 answers
37 views

Give an example of a sequence of functions $f_n$ which is uniformly convergent on $(-1,1)$ and for which $\{f'_n(0)\}$ is unbounded.

Give an example of a sequence of functions $f_n$ which is uniformly convergent on $(-1,1)$ and for which $\{f'_n(0)\}$ is unbounded. I thought of making the $f'_n$ part first. If I take, $f'_n(x) = \...
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  • 1,340
-3 votes
2 answers
63 views

Sum of inverse functions [closed]

Given a sequence $a_1,\ldots,a_n$, we can take the average $\bar{a}=\frac{\sum_{i=1}^n a_i}{n}$. For some real valued function $f$ it seems to me intuitively that $$\sum_{i=1}^n \frac{1}{f(a_i)}=\frac{...
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0 votes
1 answer
34 views

Uniform convergence of the sequence, $f_n:[a \geq 0,\infty) \to \mathbb R$ defined by $f_n(x)=\frac{n^2x}{1+n^3x^2},~n \in \mathbb N,$

Let $f_n:[a \geq 0,\infty) \to \mathbb R$ be a sequence of functions defined by $$f_n(x)=\frac{n^2x}{1+n^3x^2},~n \in \mathbb N,$$ I found that, each $f_n$ has critical point $x_c=\frac{1}{n^{3/2}}$ ...
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0 votes
0 answers
39 views

Prove polynomic sequence with each $P_i$ containing one critical point and all intersecting at know $x$ define a convergent sequence of arc lengths

Have a sequence of polynomials, the initial $p_0$ is a quadratic, with degree of $p_i$ greater than $p_{i-1}$. Sturm analysis of each $p$'s derivative suggests one critical point. They are defined to ...
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1 vote
0 answers
37 views

Left side derivative [duplicate]

Let $(a,b)$ be an open interval in $\mathbb{R}$. $f:(a,b)\rightarrow\mathbb{R}$ $z\in (a,b)$ and let $f$ be continuous in $z$. Let $f$ be differentiable over $(a,b)\setminus\{z\}$. My question is ...
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