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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119 views

An integral inequality related to maximal function

Suppose that $f\in C^1([0,\infty),\mathbb{R})$,and $F(x)=\max_{x\leq y\leq 2x}|f(y)|$,then show that $$ \int_{0}^{\infty}F(x)dx\leq \int_{0}^{\infty}|f(x)|dx+\int_{0}^{\infty}x|f'(x)|dx $$ EDIT ...
2
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0answers
29 views

variation of a final state due to changes in period (where the period is a parameter)

I have a simple ordinary differential equation $\frac{dx}{dt}=f(x,t,p,T)$ $x(0) = x_0$, $x(T) = x_T$ where $p$ and $T$ are constant parameters. How do I compute $\frac{dx_T}{dT}$ ? Thanks! NOTE: I ...
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1answer
248 views

Why Euler's formula is true? [duplicate]

Possible Duplicate: How to prove Euler’s formula: $\exp(i t)=\cos(t)+i\sin(t)$? I need to know why Euler's formula is true? I mean why is the following true: $$ e^{ix} = \cos(x) + i\sin(x) ...
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0answers
89 views

A variation of fundamental lemma of variation of calculus .

I have a question on a variation of the fundamental lemma . If $\int_\Omega f(x) g(x)=0$ and $f, g $ are $C^0\Omega$ functions and $\int_\Omega g(x)=0 $ then is it possible that there exist some ...
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4answers
584 views

Evaluate $\lim_{n\to\infty}n\int_0^1x^nf(x)dx$

Let $f:[0,1]\mapsto\mathbb{R}$ be a continuous function. Evaluate $\lim_{n\to\infty}n\int_0^1x^nf(x)dx$
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1answer
93 views

Show that $U \subset V \Leftrightarrow V^\bot \subset U^\bot$ for $U,V$ subspaces in a Hilbertspace

Let $(\mathcal H, \langle\cdot,\cdot\rangle)$ be a Hilbertspace, $U,V \subset \mathcal H$ are closed subspaces. I want to show $$U \subset V \Leftrightarrow V^\bot \subset U^\bot$$ $\Rightarrow$ is ...
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3answers
214 views

Showing that a set is not compact (not using the usual metric)

Consider $\mathbb{R}^2$ with the metric: $$d((x_1,y_1),(x_2,y_2))= \begin{cases} |y_1-y_2| \text{ if } x_1=x_2 \\ 1+|y_1-y_2| \text{ if } x_1 \neq x_2 \end{cases}$$ Show that $E= \{(x,0) : -1 \leq x ...
4
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3answers
75 views

For what values of $a$ will the following sequence converge?

$a \in \mathbb R$ has the decimal expansion $a = a_0.a_1a_2a_3 \ldots a_n \ldots$ Find all values for $a$ for which the sequence $\{a_n\}_{n=1}^{\infty}$ converges. I rule out irrationals first, ...
1
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1answer
99 views

If $B$ is a continuous bilinear function such that $B(h,k) = o(\lVert(h,k)\rVert^2)$, then $B=0$.

Suppose that $B: H \times K \Rightarrow F$ is a continuous bilinear function, where $H,K$ and $F$ are real normed spaces. I have to prove (not as homework) that if $B(h,k) = o(\lVert(h,k)\rVert^2)$, ...
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1answer
99 views

Prove the limit of a sequence

Prove that if $a_n$ is a nonnegative sequance and: $$\lim_{n\to \infty} a_n=a$$ then $$\lim_{n\to \infty} \sqrt[5]{a_n}=\sqrt[5]{a}$$ I tried to do this using the definition of the limit of a ...
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1answer
44 views

Prove the lower bound

Knowing That $ A \land B \subset \mathbb{R}$ and they both have a lower bound. Prove that (most likely using the definition of a bound): $$\inf (A \div B)=\min\{ \inf A,\ \inf B \}, $$ where $A \div ...
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1answer
92 views

Proof $a + \infty = \infty$

Given a convergent sequence $(a_n)$ with limit $a \in \mathbb{R}$ and a divergent sequence $(b_n)$ tending to infinity. I want to prove now using the boundedness of $(a_n)$: $\exists C \in ...
2
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0answers
141 views

$f$ mapping open sets to open sets

I know that the definition of a continuous mapping between two topologies is defined as: For $\mathcal{X}$ and $\mathcal{Y}$ and $f$ such that $f:\mathcal{X}\rightarrow \mathcal{Y}$ if $f^{-1}$ maps ...
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1answer
46 views

Does such a statement follow from compactness or connectedness of $[a,b]$

Assume that $I$ is a compact interval in $\mathbb R$. Does the following statement (I hope true) follows from compactness or from connectedness of $I$? For arbitrary family of open in $\mathbb R$ ...
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1answer
35 views

Problem with definition of $R$-analytic function

Let $f:[a,b]\rightarrow \mathbb R$. We say that $f$ is $\mathbb R$-analytic if for each $x_0 \in [a,b]$ there is $R(x_0)>0$ and power series $\sum_{k=0}^\infty c_k(x_0)(x-x_0)^k$ convergent for ...
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1answer
75 views

Question on Gamma Function

In Gelfand and Shilov Vol I (of Generalized Function), on page 257, they write down the following equation that I don't know how to arrive at: $$\int_{0}^{1} (1-t)^{-\frac{n}{2}} t^{\frac{q-2}{2}}dt ...
1
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1answer
66 views

A question involving Poincaré inequality

I am struggling to prove the following. Let $\Omega$ be bounded set in $\bf{R}^n$ and $u$ a $C^2$ function on $\Omega$, such that $u=0$ in $\partial \Omega$. Prove that there is a constant $C$ ...
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1answer
46 views

Just would like a quick explanation regarding Lower Darboux Sums

On Example 1 in this following PDF: http://home.iitk.ac.in/~psraj/mth101/lecture_notes/lecture15-16.pdf Consider the function $f : [0, 1] \to \mathbb R$ defined by $f(1/2) = 1$ and $f(x) = 0$ ...
1
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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
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3answers
226 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 ...
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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: ...
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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 ...
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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: $$\lim_{x \to \infty} \sqrt{x^2 + x} - x$$
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2answers
89 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
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1answer
799 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 ...
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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
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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
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2answers
909 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 ...
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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
733 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
582 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, ...
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votes
2answers
264 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 ...
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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 $\{a_n\}$. ...
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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 ...
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votes
2answers
516 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 ...
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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
158 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?
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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} ...
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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 ...
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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?
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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
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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
votes
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? ...