Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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0answers
125 views

Would this be bounded?

Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric positive-semi-definite matrix with exactly $m/2$ positive eigenvalues and every entry of $M$ is less than $1$. Let $I_{r}$ be an $m$ ...
3
votes
3answers
228 views

$L_p$ Spaces and limits of translated functions

If $g\in L^p(\mathbb{R}^n)$ and $1\leq p<\infty$ then show $$\lim_{|t|\to \infty}\lVert g_{(t)}+g\rVert_p=2^{1/p}\lVert g\rVert_p,$$ where $g_{(t)}(x):=g(t+x)$. Any hints? Try to give me only ...
1
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1answer
64 views

simple calculus inquiry

Define $$f(x) = \begin{cases} \frac{\sin x}{x} & \text{if } x \neq 0\\ 1 & \text{if } x = 0\\ \end{cases}$$ Show that $f$ is uniformly continuous (UC) on $\mathbb{R}$. My Approach: ...
0
votes
2answers
120 views

Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$

Let $C_c^\infty$ denotes the set of real valued function with compact support. Show $\exists \{g_n\} \subset C_c^\infty$ such that $(f \ast g_n)(x) \to f(x)$ a.e., when $f \in L^1_\text{loc}$. If ...
2
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9answers
2k views

Limit of square root without L'Hopital's rule.

How might one go about taking the following limit without using L'Hopital's rule? I am stumped: $$\lim_{x \to \infty} \sqrt{x^2 + x} - x$$
1
vote
2answers
90 views

Cauchy Principal Value, some sufficient condition.

This is a qualifying exam problem from Indiana University. Prove or provide a counterexample to the following statement: If $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, then ...
1
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1answer
79 views

Continuity and Subsequences

Suppose that $f$ is continuous on $[a,b]$. Prove that given $\epsilon>0$, there exist points $x_0=a<x_1<...<x_n=b$ such that if $E_k=\{y: f(x)=y\ for\ some\ x \in [x_{k-1}, x_k]\}$, then ...
4
votes
1answer
800 views

Finding a such that x^a*sin(1/x) is uniformly continuous.

Assuming that $\sin x$ is continuous on $\mathbb R$, find all real $\alpha$ such that $x^\alpha\sin (1/x)$ is uniformly continuous on the open interval (0,1). I'm guessing that I need to show that ...
1
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1answer
121 views

Uniform Continuity on $D$ and Boundedness on $D$

I have to prove or find an counterexample of the statement: if $f$ is uniformly continuous on $D$ then $f$ is bounded on $D$. I think this statement is not true since if $f(x)=x$ is uniformly ...
1
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1answer
46 views

Continuity of $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$

I have a question about a proof in my analysis textbook. They show that if $E$is a banach space, then $J: GL_c(E) \rightarrow GL_c(E): T \mapsto T^{-1}$ is continuous by first showing that it is ...
1
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2answers
195 views

Why it is wrong? (differentiation under the integral sign)

Why is it wrong? $$ \frac{d^2}{dx^2}\int_{-1}^1\log|x-t|dt=\int_{-1}^1\frac{\partial^2}{\partial x^2}\log|x-t|dt=\int_{-1}^1\frac{-1}{(x-t)^2}dt. $$
4
votes
3answers
96 views

convergent series, sequences?

I want to construct a sequence of rational numbers whose sum converges to an irrational number and whose sum of absolute values converges to 1. I can find/construct plenty of examples that has one or ...
1
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1answer
176 views

Monotonic Function and the opposite monotonicity

Does there exists $f: \mathbb{R} \to \mathbb{R},g: \mathbb{R} \to \mathbb{R}$, such that $f,g$ are onto function and satisfies: $f(g(x))$ strictly monotonically increasing and $g(f(x))$ strictly ...
2
votes
2answers
914 views

Mean of the terms of convergent sequence [duplicate]

Consider a convergent sequence $a_1,a_2,a_3\cdots a_n$ tending to a limit A. Now consider the sequence $K_1,K_2,K_3 \cdots K_n$ such that $K_n =\cfrac {a_1+a_2+...a_n}n$. Now what I guess is that as ...
0
votes
1answer
95 views

How does a myopic interpret Wiener's Tauberian?

I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made ...
3
votes
1answer
735 views

Space of all Lipschitz-continuous functions and the compact and separable sets

Let $C^{0,1}([a,b])$ be the space of all Lipschitz-continuous functions $x\colon [a,b] \to \mathbb{R}$ with the metric $$ d_{0,1}(x,y) := \sup_{a \le t \le b} |x(t) - y(t)| + \sup_{a \le s,t \le b, ...
8
votes
1answer
585 views

Transforming a distance function to a kernel

Fix a domain $X$: Let $d : X \times X \rightarrow \mathbb{R}$ be a distance function on $X$, with the properties $d(x,y) = 0 \iff x = y$ for all $x,y$ $d(x,y) = d(y,x)$ for all $x,y$ Optionally, ...
0
votes
2answers
266 views

Convergent series 1/e

I want to prove that: $$\sum\limits_{i=0}^\infty \frac{(-1)^i}{i!}=\lim_{i \to \infty}\bigg(1-\frac{1}{i}\bigg)^i$$ First I need to prove series is convergent. But the partial sums of the series ...
0
votes
0answers
101 views

How to observe infinity?

In my calculus course, there's example stated on the book: Given that $M$ is an ordered set and the sequence $\{a_n\}\subset M$, prove that there's a (weakly) monotonic subsequence of $\{a_n\}$. ...
0
votes
1answer
80 views

The convergence of the improper integrals!

Suppose $f'(x)$ exists on $[0,\infty)$, prove or disprove that: the following two integrals $$\int_{0}^{+\infty}\frac{2dx}{f(x)} \ \ \text{and}\ \int_{0}^{+\infty}\frac{dx}{f(x)+f'(x)}$$ have the ...
4
votes
2answers
517 views

Notation: What's meant by $C^{\infty}_{0}(\mathbb{R}^{+})$?

In Chapter 0 of Iwaniec's Spectral Methods of Automorphic Forms Iwaneic uses the notation $C^{\infty}_{0}(\mathbb{R}^{+})$ without definition. I assume that it's the set of infinitely differentiable ...
1
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1answer
123 views

Diagonalization theorem and convergence

Let $\{f_{n}\}$ be a sequence of pointwise bounded continuous functions on a separable metric space $X$. There is a common diagonalization theorem (see Baby Rudin, Theorem 7.23) which states that if ...
2
votes
1answer
159 views

When can I cancel out a differentiation and integration operation?

Is it correct to state that $\frac{d}{ds} \int_{u=0}^{u=s} f(u)du = f(s)$ if $f(u)$ is continuous? If so, what is the relevant theorem in action? If not, what else would be needed?
1
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1answer
157 views

Ratio test Rudin Example 3.35B

The series in question is:$$\frac{1}{2}+1+\frac{1}{8}+\frac{1}{4}+\frac{1}{32}+\frac{1}{16}+\frac{1}{128}+\frac{1}{64}...$$ where $$\liminf\limits_{n\rightarrow \infty} ...
-1
votes
1answer
65 views

How to find the derivative with respect to the transformed co-ordinates.

I am stuck with something very simple , would be glad to get help . Suppose if i have a transformation matrix J , how do i find the derivative with respect to new co-ordinates , and derivative of ...
0
votes
1answer
86 views

If $X_n$ converges to $X$ almost surely how can we proof that $1/X_n$ will converge to $1/X$ almost surely?

If $X_n$ converges to $X$ almost surely how can we prove that $\cfrac 1{X_n}$ will converge to $\cfrac 1{X}$ almost surely?
0
votes
2answers
61 views

Limit in metric space.

I have , In $X$ $\lim_{n\to \infty} f_n=f_0 $ and similarly $g_n$ converges to $g_0$, $d$ is a metric defined in $X$ . I have to show that $\lim_{n\to \infty} d(f_n, g_n)=(f_0, g_0)$ This is what i ...
6
votes
1answer
316 views

Prove that $\int_0^1[f''(x)]^2dx\ge4.$

Let $f$ be a $C^2$ function on $[0,1]$ such that $f(0)=f(1)=f'(0)=0,f'(1)=1.$ Prove that $\int_0^1[f''(x)]^2dx\ge4.$ Find all $f$ for equality to occur.
1
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2answers
74 views

At which points is this function differentiable

Find all points $(x_1,x_2,\ldots,x_k) = x \in \mathbb{R}^k$ such that $f(x)=|x_1 \cdot x_2 \cdots x_k|$ is a differentiable function. I know that if $\forall i, x_i \neq 0$ then this function is ...
0
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0answers
25 views

$u \in C^{\alpha}(\Omega) \Rightarrow |u(x) - u_{B_R(x)}| \le (2R)^{\alpha} \| u\|_{C^\alpha(\Omega)}.$

Let $u \in C^{\alpha}(\Omega)$ and $B_R(x) \subset \Omega$. How can I see that \begin{equation} |u(x) - u_{B_R(x)}| \le (2R)^{\alpha} \| u\|_{C^\alpha(\Omega)}. \end{equation} where $u_{B_R(x)}= ...
0
votes
1answer
115 views

Rigorous proof that a simple closed curve with a given parametrization in $\mathbb{R}^2$ either has positive or negative orientation

I am asking for a rigorous proof of the following: Theorem: Let $R$ be a region and $C$ be a simple closed curve so that $C=\partial R$. If $\gamma:[a,b]\to \mathbb{R}^2$ is parametrization of $C$ ...
3
votes
2answers
233 views

Is this metric space incomplete? [duplicate]

Possible Duplicate: which of the following metric spaces are complete? I have doubt to this problem., $X=(0,\pi/2)$ and the metric is $d(x,y)=|\tan x-\tan y|$ is it complete metric space? ...
2
votes
1answer
250 views

Is it possible to solve this equation using the Lambert function?

I am trying to solve the following equation, where $0 \leq a < b \leq 1$ are constants and $x \in (a,b)$; $$\frac{x-a}{b-a} = e^{-2\log(2)/(x+1)}$$ and stumbled across the Lambert W-function which ...
0
votes
1answer
38 views

Two compactness questions

I'm stuck on how to show one of these is compact, and I want to verify my method for the other. This one I am stuck on: Proposition Let $(\mathbb{Q},d)$ be a metric space with $d(a,b)=|a-b|$. ...
2
votes
4answers
563 views

Is $x^n$ Cauchy in $(C[0, 1], ||\cdot||_{\infty})$?

Consider the sequence of functions \begin{equation} f_n(x) = x^n, \quad x \in [0, 1]. \end{equation} Is this sequence Cauchy in $(C[0, 1], ||\cdot||_{\infty})$? The pointwise limit is not ...
1
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1answer
41 views

Describing open neighborhoods

Let $d$ be a metric on $\mathbb{R}^2$ defined as $$d((x_1,y_1),(x_2,y_2))=\begin{cases} |y_1-y_2| \mbox{ if } x_1=x_2 \\ 1+|y_1-y_2| \mbox{ if } x_1 \neq x_2 \end{cases}$$. Let $N((x,y),\epsilon)$ be ...
2
votes
1answer
387 views

Showing a metric space is bounded.

This is from a review packet: Let $d:\mathbb{R} \to \mathbb{R}$ be defined as $$d(x,y)=\frac{|x-y|}{1+|x-y|}.$$ i) Show that $(\mathbb{R},d)$ is a bounded metric space. ii) Show that $A=[a,\infty)$ ...
0
votes
1answer
65 views

Sufficient condition for isometry

They could give me some suggestions on how to use the data on the derivative, in: If $f:\mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ class is $C^{1}$ such that $\Vert f^{\prime}(x)v\Vert =\Vert ...
1
vote
1answer
96 views

Show that the Initial Value Problem does not cross its equilibrium points.

Let $x'=(x-1)(x-4)(x+5)$ and $x(0)=3$ be an Initial Value Problem. Prove the existence and uniqueness of the solution and that the solution will be trapped in between $x=1$ and $x=4$ lines and that ...
1
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2answers
68 views

Boundedness of sequences

In order to prove that the sequence $a_n$=($1^k+2^k+3^k+..n^k$)$/n!$ is bounded(k is fixed natural number).Is it enough to say that $a_n<nn^k/n!$.What bothers me is that following the definition of ...
2
votes
1answer
55 views

Rotation-invariant homogeneous distribution

Can you tell me for every $\alpha \in \mathbb{R}$, whether there is a non-zero homogeneous and rotation-invariant distribution on $\mathbb{R}^n$ with degree of homogeneity $\alpha$?
1
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1answer
99 views

about fixed point set?

let $K$ be a closed convex subset of a normed space $V$. For any $f: K \to K$ define the fixed-point set of $f$ as follows: $fix(f)=\{x$ belongs to $K$ $|f(x)=x \}$. I have to show that a nonempty ...
2
votes
3answers
129 views

Find an exponential function with given condition

How can I have an example of an exponential function defined in the X range 1 - infinity, with values starting at 40 and converging to 1?
1
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0answers
201 views

Monotonic subsequence

Consider a bounded sequence. By Bolzanno Weierstrass theorem we can conclude that it contains a sub sequence which is convergent . The proof of Bolzanno uses Cantor's completeness principle. Let us ...
0
votes
2answers
113 views

Does there exist a variation of Minkowski's inequality with differences instead of sum.

I seek something of the form $\|f-g\|_p \leq |\|f\|_p-\|g\|_p|$.
5
votes
1answer
290 views

Showing $\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$ using complex integration

Recently I had to use the fact that the Dirichlet integral evaluates as $$\int_{0}^{\infty} \frac{\sin{x}}{x} \ dx = \frac{\pi}{2}$$ a couple of times. There already is a question that specifically ...
2
votes
0answers
54 views

If $F:\mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ if of class $C^{1}$ …

I'm studying differentiation and came across this question: If $F:\mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$ is of class $C^{1}$ such that $\Vert f^{\prime}(x)v\Vert \geqslant 2\Vert v\Vert$. Then F ...
2
votes
1answer
92 views

rectifiable continuous function

could you please help me with this question: if $F:[a,b] \rightarrow \mathbb{R}^{m}$ is continuous and rectifiable, then $ F ([a, b]) $ has $ m $ - measure zero. Any suggestions are welcome.
1
vote
1answer
731 views

Dirichlet Integral via complex integration: problem with last step.

I am trying to understand the derivation of the Dirichlet Integral via complex integration (as outlined on wikipedia) but I have a problem with the last steps. We consider $$f(z) = \frac{e^{iz}}{z}$$ ...
0
votes
2answers
242 views

Finding the min of an integral

So I have to find the following $$\min_{a,b,c\in\mathbb{R}}\int_{-1}^{1} |x^3-a-bx-cx^2|^2dx$$ I have a hint at a solution which says to consider $X=\{\mbox{polynomials of degree} \leq 2\}$. So then ...