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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5
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1answer
1k views

Proof of Pinsker's inequality.

How to prove the following known (Pinsker's) inequality? For two strictly positive sequences $(p_i)^n_{i=l}$ and $(q_i)^n_{i=l}$ with $\sum_{i=1}^np_i=\sum_{i=1}^nq_i=1$ one has ...
3
votes
2answers
230 views

A question about Lagrange multiplier

Is there any explanation or interpretation of the concept of Lagrange multipliers $\nabla f(x_0)= \delta \nabla g(x_0) $ for some constant $\delta$ and $f$ is a differentiable function and g is the ...
3
votes
4answers
1k views

Moments and weak convergence of probability measures

I was wondering, if you have a sequence of probability measures $(\mu_n)_n$ on $\mathbb R$ and you know that there is a probability measure $\mu$ such that for all $k\in\mathbb N=\{0,1,2,\cdots\}$ $$ ...
4
votes
1answer
248 views

Pointwise convergence counter example.

Can anyone think of a counter example that for $f_n:[a,b] \to \mathbb{R}$ regulated and $f_n \to f$ pointwise but $f$ is not a regulated function? Thanks!
2
votes
0answers
51 views

What happens at “spherical shell” of convergence?

In a Taylor series, the convergence/divergence behavior at the boundary case $|x-x_0|=R$ is not immediately determinable. If I understand correctly, such distribution of convergence and divergence ...
1
vote
1answer
68 views

A simple question about definition of Legendre's transform

Let a function $f:\mathbb{R}^n \rightarrow \mathbb{R}$ be convex and satisfies $$ \lim_{|q| \rightarrow \infty}\frac{f(q)}{|q|}=+\infty.$$ The Legendre's transsformation of $L$ is defined by ...
0
votes
0answers
136 views

Uniqueness of solution of the nonlinear system of equations

Consider the equation system $$ \left\{ \begin{array}{rcl} g(x,y) & = & 1, \\ g(\alpha x, \beta y) & = & 1, \end{array} \right. $$ where $\alpha,\beta > 0$, $a \neq ...
1
vote
2answers
105 views

Equicontinuous at some point and also at all points

how do I go about proving these: Lets F be a family of linear operator and the following are equivalent: 1) F is equicontinuous at some point $v_{0} \in V$ 2) F is equicontinuous at all points of ...
1
vote
1answer
117 views

Infinity norm question

For the infinity norm, for example in the definition of a regulated function on closed interval $A$. Where $\forall \epsilon,\, \, \exists\, \phi \, s.t. \|\phi - f\|_\infty < \epsilon$. The ...
2
votes
2answers
245 views

Proving an asymptotic lower bound for the integral $\int_{0}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^s}$

This is a follow up to the great answer posted to http://math.stackexchange.com/a/125991/7980 Let $ 0 < r < \infty, 0 < s < \infty$ , fix $x > 1$ and consider the integral $$ ...
1
vote
0answers
153 views

Analysis Prelim Question about proving continuity.

Let $(X,d_{disc})$ be a discrete metric space, where $d_{disc}(x,x')=\begin{cases} 1 & \text{if } x\neq x', \\ 0 & \text{if }x=x'.\end{cases}$ Let $(Y,d_Y)$ be an arbitrary metric ...
1
vote
1answer
171 views

Hölder continuity of a function from $[0,1]$ to $[0,1]^2$

I'm trying to prove that if $g: [0,1] \longrightarrow [0,1]^2$ is an $\alpha$-Hölder continuous mapping whose image is the entire square $[0, 1]^2$ then $\alpha \leq 1/2$. I wouldn't know where ...
2
votes
2answers
121 views

Limit involving the laplacian

I'm trying to prove that if $\Omega$ is an open subset of $\mathbb{R}^n$ and $u$ a $C^2$ function then $$\lim_{r\to 0}\frac{2n}{r^2}\left(u(x)-\frac{1}{|\partial B_r(x)|}\int_{\partial ...
8
votes
4answers
341 views

Determining the action of the operator $D\left(z, \frac d{dz}\right)$

This question was motivated by a question by Tobias Kienzler and its wonderful answers. I begin as in the linked question... Using the Taylor expansion $$f(z+a) = \sum_{k=0}^\infty ...
12
votes
1answer
411 views

Proving a complicated inequality involving integers

Let $a,b,c,d$ be integers such that $$\left( \begin{matrix} a & b \\ c & d \end{matrix} \right) = \left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right) \mod 2$$ $$ ad-bc =1$$ ...
23
votes
1answer
730 views

Computing the best constant in classical Hardy's inequality

Classical Hardy's inequality (cfr. Hardy-Littlewood-Polya Inequalities, Theorem 327) If $p>1$, $f(x) \ge 0$ and $F(x)=\int_0^xf(y)\, dy$ then $$\tag{H} \int_0^\infty ...
1
vote
1answer
2k views

What is the infinity norm on a continuous function space?

As I understand it the norm $\|f\|_2$ on the set of continuous functions on $[0,1]$ is defined by $$\|f\|_2 = \sqrt{\int_0^1|f(t)|^2dt}$$ but what is the infinity norm $\|f\|_\infty$? Is it $$\int_0^1 ...
1
vote
2answers
338 views

Example of an infinitely differentiable function with a given property

What is an example of an infinitely differentiable function $f$ such that $f(x) = 1$ on $[-1, 1]$ and $f(x) = 0$ on $(-\infty, -2] \cup [2, \infty)$?
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vote
0answers
87 views

Extremizing an Integral under a cyclic condition

Let $h$ be a nonnegative, smooth and convex function on $[0,1]$ and let $f(x,y):[0,1]\times[0,1]\rightarrow[0,1]$ with $f(x,y)=f(y,x)$ and $f$ continuous. Suppose I fix $r>0$ and demand that $$ ...
1
vote
1answer
364 views

An additivity property of outer measure

By definition, as set $E$ is measurable if for any set $C$ $$\mu^\ast(C) = \mu^\ast(C\cap E) + \mu^\ast(C \cap E^c).$$ Is it true that if either $A$ or $B$ is measurable, then we can't have $$ ...
0
votes
1answer
543 views

Nonincreasing derivative implies nondecreasing function

Suppose $f\colon[0,\infty)\to[0,\infty)$ is differentiable with continuous derivative such that its derivative $f'$ is nonincreasing. Does this imply that $f$ is non-decreasing?.
1
vote
1answer
113 views

Equivalent norms question

Two norms $\| x \|_\alpha$ and $\| x \|_\beta$ are said to be equivalent if there exists positive real numbers $C$ and $D$ such that $$C\|x\|_\alpha\leq\|x\|_\beta\leq D\|x\|_\alpha$$ does this mean ...
3
votes
1answer
237 views

Distance between bounded and compact sets

Let $(X,d)$ be a metric space and define for $B\subset X$ bounded, i.e. $$\operatorname{diam}(B)= \sup \{ d(x,y) \colon x,y\in B \} < \infty,$$ the measure $$\beta(B) = \inf\{r > ...
12
votes
1answer
656 views

Meaning of “kernel”

In analysis, there are at least three kinds of "kernel" concepts: In probability theory, there is a concept called transition probability, also called probability kernel, from one measure space $X$ ...
4
votes
1answer
783 views

Uniform convergence, but no absolute uniform convergence

Can someone give an example of a series of functions $f_k(x)$ for which $\sum_{k=0}^{\infty} f_k(x)$ converges uniformly, and $\sum_{k=0}^{\infty} |f_k(x)|$ converges pointwise, but ...
19
votes
2answers
1k views

Continuity and the Axiom of Choice

In my introductory Analysis course, we learned two definitions of continuity. $(1)$ A function $f:E \to \mathbb{C}$ is continuous at $a$ if every sequence $(z_n) \in E$ such that $z_n \to a$ ...
8
votes
1answer
278 views

Lebesgue measure in $\mathbb{R}^2$ of uncountable union

Suppose we have a collection $(A_r)_{r\in\mathbb{R}}$ of Lebesgue subsets of $\mathbb{R}$, each with Lebesgue measure $0$. Consider the set $$E=\bigcup_{r\in\mathbb{R}}\{r\}\times ...
-1
votes
1answer
252 views

strictly convex space ---> strictly convex function

How would you prove that in a strictly convex normed vector space, the function $f(x) = \| x \|^2$ is strictly convex?? FYI: $E$ is strictly convex iff $\| t x + (1-t) y \| <1$ for all $x,y \in ...
1
vote
3answers
268 views

L'Hopital's rule, why is this incorrect?

As the title says, why is this an incorrect use of L'Hopital's rule? Clearly this isn't the correct limit?$$\lim_{x \to 2} \frac{\sin x}{x^2} =\lim_{x \to 2} \frac{\cos x}{2x} =\lim_{x \to 2} ...
3
votes
1answer
188 views

Help with fixing a proof for an upper bound of $\int_{0}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$

This is a follow up to Help with removing singularities involving $ \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}$ I am getting a little confused at the ...
2
votes
1answer
231 views

Show that $\prod_{i=1}^n a_i- \prod_{j=1}^n b_i =$ $\sum_{t=1}^{n-1}(\prod_{i\leq t-1}a_i)(\prod_{j\geq t+1} b_j)(a_t-b_t)$

Pardon the cryptic notation and possibly trivial question. I believe the following holds. Define $$X_t=(\prod_{i\leq t-1}a_i)(\prod_{j\geq t+1} b_j)(a_t-b_t).$$ Show that ...
0
votes
1answer
109 views

Classification of a point with null hessian

I have to classify the point $(2,-4)$ for the function $f(x,y)=(x-2)^2-(y+4)^4$ which have a null hessian. What kind of point is (max/min)? Thanks
6
votes
1answer
357 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
2
votes
1answer
561 views

Uniformly Continuous Function sending Bounded Set to Unbounded One

Let $(X,d)$ and $(Y,\rho)$ be metric spaces, and let $f: X \to Y$ be a uniformly continuous function. If $A \subset X$ is bounded, must $f(A) \subset Y$ be bounded? It is clear to me that in metric ...
0
votes
1answer
393 views

Hoelder functions with power $>1$

Do there exist in Banach space any nonconstant functions satisfying Hoelder condition with power $>1$ defined on open connected set with values in $\mathbb{R}$? Thanks.
0
votes
0answers
50 views

smooth functions with values in sequences

Let $S$ be the space of all real sequences with direct product topology (i.e. $S=\prod_{n=0}^\infty \mathbb{R}$) and let $U$ be an open set in $\mathbb{R}^n$. Is there a way to define smooth function ...
2
votes
3answers
321 views

Continuous and uniformly continuous proof

How would you show that if a continuous function $f:[0,1) \to \mathbb{R}$ satisfies $f(x) \to 0$ as $x \to 1$ then it is uniformly continuous?
0
votes
2answers
259 views

A uniform convergence counter example?

Can anyone think of a sequence of functions $f_n:[0,\infty) \to \mathbb{R}$ such that $f_n \to f$ uniformly but $\int_0^\infty f_n \nrightarrow \int_0^\infty f$ ?
1
vote
1answer
83 views

A question on pointwise convergence.

The function $f_n(x):[-1,1] \to \mathbb{R}, \, \, \,f_n(x) = x^{2n-1}$ tends pointwise to the function $$f(x) = \left\{\begin{array}{l l}1&\textrm{if} \quad x=1\\0&\textrm{if} \quad ...
2
votes
2answers
850 views

Simple example of a continuous onto function mapping $[0,1]$ to $\mathbb R$

There should exist such a function, but I cannot think of any example. Onto continuous functions mapping $(0,1)$ to $\mathbb R$ are easy to find. Edit: Sorry - I mentioned Tietze extension theorem ...
2
votes
1answer
108 views

A condition for a function to be constant

I need to proof this result: Let $\alpha >1$ and $c\in\mathbb{R}$. If $f:U\subset\mathbb{R}^m\rightarrow\mathbb{R}^n$, U open, satisfies $|f(x)-f(y)|\leq c|x-y|^\alpha$ for every $x$, $y$ $\in U$, ...
2
votes
1answer
2k views

Limit at infinity of a uniformly continuous integrable function [duplicate]

Possible Duplicate: $f$ uniformly continuous and $\int_a^\infty f(x)\,dx$ converges imply $\lim_{x \to \infty} f(x) = 0$ This is an exercise from Berkeley preliminary exams, Fall 1983 Let ...
2
votes
2answers
146 views

what's the difference between “convergent” and “reconstruct-able”?

I am reading this book: http://www.abdn.ac.uk/~mth192/html/maths-music.html There is a sentence on page 54: "However, the question of convergence of the Fourier series is not the same as the question ...
3
votes
1answer
129 views

Is the ball measure of non-compactness a Lipschitz map?

Let $(M,d)$ be a metric space and let $H(M)$ denote the set of closed and bounded subset in $M$. Then $(H(M),d_H)$ is a metric space where $d_H$ denotes the Hausdorff distance. Let $\chi$ be the ...
1
vote
1answer
312 views

What's an admissible parametrization?

I was browsing a book in complex analysis which said 'the contour is defined by admissible parametrization'.Unfortunately,no definition followed. I presumed it was just z(t) defined on a closed ...
0
votes
0answers
683 views

Asymptotic equivalence?

Let there be two functions $f(x)$ and $g(x)$. If we consider $\lim_{x \rightarrow x_{0}} \frac{f(x)}{g(x)} = k$, we say that $k=1$, then $f(x)\sim g(x)$, $f(x)$ is equivalent to $g(x)$ as $x ...
2
votes
1answer
256 views

Positive sequence of integrable functions

The question was: Given $\mu$ a positive measure in $(X, \Sigma)$ and $f_n, f:X\rightarrow [0,\infty)$ $\mu$-summable then show that if $\liminf f_n\geq f$ almost everywhere and $$\limsup_n \int_X ...
4
votes
5answers
1k views

Analysis Problem: Prove $f$ is bounded on $I$

Let $I=[a,b]$ and let $f:I\to {\mathbb R}$ be a (not necessarily continuous) function with the property that for every $x∈I$, the function $f$ is bounded on a neighborhood $V_{d_x}(x)$ of $x$. Prove ...
1
vote
1answer
93 views

Derivation of Borel measures w.r.t a “bigger” measure?

Given two Borel measures $\mu_1$ and $\mu_2$ on $\mathbb R$, is there always a Borel measure $\mu$ on $\mathbb R$ such that $$ d\mu_1=w_1 d\mu,\qquad d\mu_2=w_2 d\mu, $$ for some functions $w_1$ ...
0
votes
1answer
170 views

Disproving a function is a step function.

The function $f:[0,1] \to \mathbb{R}$ where $f(x) := 0$ if$ x \notin \mathbb{Q}$ and $f(p/q) : = \frac{1}{q} , q > 0, p,q$ coprime. How would you show that this is not a step function? Thanks!