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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2
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1answer
52 views

Rademacher complexity of regularized linear function class: does it depend on dimension or not?

I am going through some lecture notes on Learning Theory here: http://ttic.uchicago.edu/~tewari/LT_SP2008.html trying to learn about Rademacher complexities. I'm getting confused about the Rademacher ...
1
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0answers
45 views

Hausdorff dimension of closed intervals is not changed under $f(x)$

Let $f(x)=x^2$. Prove that for any $E\subseteq\mathbb{R}$ the dimension of image is not changed i.e $$\dim_HE=\dim_H(f(E))$$ Any set in $\mathbb{R}$ can be represnted as a countable union of ...
0
votes
1answer
118 views

What does it mean for an integral to “vanish”?

I had a question; What does it mean for an integral to vanish in complex analysis? There is supposedly something, which says if the integral "vanishes," the sum of the residues is 0. But what does ...
2
votes
2answers
135 views

Infinitely many times differentiable function with unbounded derivatives?

Let $f$ be an infinitely many times continuously differentiable function on the compact interval $[0,1]$. We denote by $f^{(k)}$ the $k$-th derivative with respect to $x$. Then we know: $\sup_{x \in ...
1
vote
1answer
34 views

Restriction of a finite measure to a set on unbounded function

So I have a measure $(X,\mathscr{F},\mu)$, possibly finite, or $\sigma$-finite, or a completely general finite measure. $B\in \mathscr{F}$ is a set of finite measure. For every measure set $A\in ...
3
votes
3answers
69 views

How to prove that $ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 $?

I would like to prove that $$ \lim_{u \downarrow 1} (u-1) \zeta(u) =1 \quad .$$ However, I am not sure which form of the Riemann-zeta function I ought to pick in order to compute this limit. I ...
2
votes
1answer
143 views

Baby Rudin Problem 2.29

Here's is Prob. 29 in the Exercises following Chap. 2 in PRINCIPLES OF MATHEMATICAL ANALYSIS by Walter Rudin, 3rd edition: Prove that every open set in $\mathbb{R}^1$ is the union of an at most ...
1
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1answer
43 views

How prove this usefull identities with equations

let $f(x):R\to R$ be $C^{k+1}$, show that ...
1
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1answer
21 views

an injective map can not take several intersecting arcs onto line segment

I read a result in the theory of harmonic mappings, and i think it might be true in general setting as well. But i am unable to get a proof of this. Can anyone help me with proving it. The statement ...
0
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0answers
33 views

Checking for a function to be homeomorphism

Let $h=(h_1,h_2):U\to \mathbb{R}^2$ be one-one and continuous in a neighborhood $U$ of the origin in $\mathbb{R}^2$ with $h(0)=0$ and $h_1$ harmonic. Then $h:U\to h(U)$ is bijective and continuous. If ...
2
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0answers
179 views

How to apply Fubini's theorem in proof of Osgood's lemma

In the proof of Osgood's lemma for seperate holomorphicity, at one step we get that $$f(z)=\frac{1}{(2\pi \iota)^n}\int_{|w_i-\zeta_i|=r_i}\sum_{v_1,v_2,\ldots,v_n}\frac{f(\zeta)z_1^{v_1}\ldots ...
3
votes
1answer
120 views

Closed Subspaces of Hilbert Spaces

I read the following statements. But I do not know how to show it or any example to support it. Could anyone provide some explanation and examples, please? Thank you! The subspace $C^\infty$ ...
1
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1answer
27 views

Showing that a measure is lower continuous.

Here is the problem and my attempt. Let $\mu$ be a measure defined on sets $E_1, E_2, \ldots$ such that $E_{n+1} \subset E_n$ for all $n$. Additionally, let $\mu(E_1) \lt \infty$. Show that ...
9
votes
1answer
121 views

What are the “right” spaces for the Laplace transform

There are for example several canonical spaces to define the Fourier transform (i.e. Schwartz's space). Is there also a particularly suitable space to define the Laplace transform, so that the Laplace ...
1
vote
1answer
137 views

An analytic function with minimum and maximum at the boundary

Suppose there is a complex valued function analytic on some open connected set U and continuous on the boundary of that set. Then the maximum of $|f|$ is attained at some point on the boundary. Say ...
1
vote
1answer
41 views

The concept of integrating along a square?

I had a question; What is the idea (in complex analysis) of integrating along a square? Take a look at @M.N.C.E.'s method on Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$ I am not quite sure what ...
3
votes
1answer
47 views

Why does the whole integral converge but not part of it? (Dilogs)

$\newcommand{\Li}{\operatorname{Li}}$Consider the integral: $$\int_0^1 \frac{(-\Li_2(x) - \Li_3(x) - x^2/8 + 3x - x\log(1-x) + \log(1-x))}{x^2} \, dx$$ This integral converges to $\sim 0.01$ But ...
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0answers
55 views

Convergence of Fourier Sine Series for Gerneral Continuous Function

This is my question: How do I should that, for $f \in C[0,\pi]$ with $f(0) = f(\pi) = 0$, the Fourier sine series $$\tilde f_n = \sum_{r=0}^n b_r \sin(r s)$$ converges uniformly to $f$ on ...
2
votes
0answers
82 views

Image of $f(A)$ when $A$ is compact and $f$ is analytic in the interior of $A$ and continuous on the boundary of $A$

Suppose there is a non constant complex function $f$ and a compact set $A$ such that $f$ is analytic in the interior of $A$ and continuous on the boundary of $A$. Then clearly $f(A)$ is compact. Now ...
0
votes
1answer
545 views

By applying term-wise differentiation and integration find the sum of the series $\sum_{k=1}^{\infty}\frac{x^k}{k}$

I need to find the sum of the following series: $$\sum_{k=1}^{\infty}\frac{x^k}{k}$$ on the interval $x\in[a,b], -1<a<0<b<1$ using term-wise differentiation and integration. Can anyone ...
0
votes
2answers
38 views

Find the values of N such that for n>N |$a_n$-3|< $\epsilon$

I have the sequence $a_n$ = $\frac{9n^2 - 1}{3n^2 -2}$ I need to find the values of N such that for n>N |$a_n$-3|< $\epsilon$ So far I have the following |$\frac{9n^2 - 1}{3n^2 -2}$ - 3| < ...
2
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1answer
120 views

there doesn't exist any sequence of polynomials which converge to $\sin x$ uniformly on $\mathbb{R}$

Show that there doesn't exist any sequence of polynomials which converge to $\sin x$ uniformly on $\mathbb{R}$. Suppose there is a sequence of polynomials $\{p_n\}$ which converges to $\sin x$ ...
1
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1answer
37 views

Continuity of $\sup_{x\in\Omega}\varphi(x,\cdot)$

Let $\Omega\subset\mathbb{R}^n$ be open,bounded and (I don't know if this matter) of class $C^{1+\alpha}$. Let $\varphi:\bar\Omega\times\partial\Omega\to\mathbb{R}$ such that $\varphi(x,\cdot)$ is ...
0
votes
2answers
87 views

Show a bound sequence with a cluster point is indeed convergent

I have been wondering for a while now: How do I show that $(i) \space (a_{n})_{n \in \mathbb{N}}$ is convergent. $(ii)\space (a_{n})_{n \in \mathbb{N}}$ is bounded and has a cluster point. ...
0
votes
1answer
70 views

Eigenvalue of Compact Operators

To prove that the set of eigenvectors of a compact linear operator on a normed space $X$ is countable, I read "it suffices to show that for every real $k > 0$ the set of all eigenvalues whose ...
2
votes
1answer
2k views

Prove sin(1/x) is discontinuous at 0 using epsilon delta definition of continuity

Let $$f(x) = \begin{cases} 0 &\text{ if $x=0$}\\ \sin(1/x) &\text{ otherwise} \end{cases} $$ Prove that $f$ is discontinuous at $0$ using the $\epsilon \delta$ definition of continuity. I ...
2
votes
1answer
49 views

All holomorphic functions such that $|f(z)| \leq |z|^k$ for some non-negative integer $k$?

I'm trying to reduce a well-known result by finding all holomorphic functions $f(z)$ (on the whole of $\mathbb{C}$) such that $|f(z)| \leq |z|^k$ for some non-negative integer $k$ and for all $z\in ...
0
votes
1answer
109 views

Find $f$ such that the divergence of $(f(\underline{a}\cdot \underline{x}))\underline{x}=1$

Let $\underline{a}\in\mathbb R^n$ be a fixed vector. Find all functions $f:\mathbb R \to \mathbb R $ such that the divergence of the vector field $(f(\underline{a}\cdot \underline{x}))\underline{x}$, ...
1
vote
1answer
74 views

A Question on Compact Operators on Hilbert Space

I read this question which I have no idea how to start. Could anyone provide me with some detailed answer, please? Thanks. Suppose that a linear operator $F$ from a Hilbert space $\mathcal H$ to ...
1
vote
1answer
57 views

Prove that the lower integral is $\geq 0$ if $f:[a,b]\to \mathbb R$ satisfies $f(x) >0$ for all $x \in [a,b]$.

So here's my proof. I hope I am heading in the right direction but if not please correct: Given $f:[a,b]\to\mathbb{R}$ we have that $f(x)$ are all positive. The lower integral of $f(x)$ from $a$ to ...
4
votes
1answer
385 views

Every closed set in a separable metric space is the union of a perfect set and a set which is at most countable [duplicate]

Prove that every closed set in a separable metric space is the union of a (possibly empty) perfect set and a set which is at most countable. (Rudin's Principles of Mathematical Analysis, 3rd ...
0
votes
2answers
36 views

Find the sum with induction

Find the sum $\sum_{j=1}^n 2^{-j}=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{2^n}$ for all $n=3,4,...$. I need to use induction. I don't know how to start, can you show me?
2
votes
1answer
368 views

Positive operator has a positive spectrum?

Let $T : \operatorname{dom}(T) \rightarrow H $ be a positive self-adjoint operator, is it then true that $\sigma(T) \subset [0,\infty)$? This is something that sounds natural and I guess that it is ...
1
vote
1answer
97 views

Schauder estimates of weak solutions of a elliptic PDEs of 2nd order

Let $\Omega$ be a bounded domain in $\mathbb{R}^d$, $0<\sigma<1$, $L$ be a 2nd order strictly elliptic linear differential operator of divergence form: \begin{eqnarray} ...
0
votes
1answer
53 views

Does $d(x,y) = \lvert N(x) - N(y)\rvert$ satisfy the triangular inequality?

Let $N(x)$ be the norm of the vector $X$ and efine $$d(x,y) = |N(x) - N(y)|$$ I want to prove that $d(x,y)$ satisfies the triangular inequality. Here is my attempt: $$|N(x) - N(y)| \leq |N(x)| + ...
2
votes
1answer
41 views

Intergral involving arcus sinus and square root

While computing certain triple integral, I met the following one which I was unable to compute. I would be greatful for any suggestions: $$\int x\arcsin{\frac{\sqrt{2-x^2}}{x}} \,dx$$
0
votes
2answers
279 views

Primitive of an analytic function - Proof verification

Check out the proof for the following corollary. This is only the first part of the proof and I have a issue with this. $F(z)$ is defined by integrating $f$ along a line segment. But isn't it ...
1
vote
1answer
31 views

Another type of derivative, another type of differential equation

Let $\mathbf{v}:(a,b)\to\mathbb{R}^2$ be a continuous function, such that $||\mathbf{v}(t)||=1,\ \forall t\in (a,b)$. Is it possible to find a continuous function $\mathbf{r}:(a,b)\to\mathbb{R}^2$ so ...
0
votes
1answer
50 views

How to define an explicit bijection from P(N) to 2^N [closed]

How do I define an explicit bijection between the power set of N and $2^N$ with $2^N =\{f|f:N\to\{0,1\} \text{ is a function} \}$?
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votes
0answers
98 views

Prove that these two definitions are equivalent

While answering this question I have used that \begin{equation}\sin x=\displaystyle\sum_{n=0}^\infty\dfrac{(-1)^nx^{2n+1}}{(2n+1)!}\end{equation} Nwe my question is that how can it be shown that the ...
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0answers
45 views

elementry properties of closure

Definition: Let X⊂R and let x'∈R, we say that x' is an adherent point of X iff ∀ε>0 ∃x∈X s.t.d(x′,x)≤ε. the closure of X is denoted as \overline(X) and is defined to be the set of all the adherent ...
0
votes
1answer
12 views

Characterization of this criteria

I'm proving the statement of some limit which has a form of $$\lim_{\|\mathbf{m}\| \to \infty} f(m_{1},m_{2},\cdots,m_{k}) = S$$ where $\mathbf{m} = (m_{1}, \cdots, m_{k}) \in \mathbb{R}^{k}$. I've ...
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0answers
79 views

Calculate nasty integrals

In physics I ran into some nasty integrals involving characteristic functions $\chi$. The ones are given by $$\int_{\mathbb{R}^2} \left(E - \frac{p^2}{2m}-\frac{q^2 \omega^2}{2} \right)^{n-1} ...
2
votes
2answers
138 views

How to integrate $I(k)=\frac{2\pi}{i k^2}\int_0^{\infty}\left(e^{-ir }-e^{ir }\right)dr$

I heard that you can integrate $$\begin{align}I(k)=\frac{2\pi}{i k^2}\int_0^{\infty}\left(e^{-ir }-e^{ir }\right)dr \end{align}$$ in the sense of tempered distribution. Unfortunately, I am only ...
19
votes
3answers
724 views

Prove that a positive polynomial function can be written as the squares of two polynomial functions

Let $f(x)$ be a polynomial function with real coefficients such that $f(x)\geq 0 \;\forall x\in\Bbb R$. Prove that there exist polynomials $A(x),B(x)$ with real coeficients such that ...
2
votes
1answer
110 views

Weak convergence in $\mathcal{l}_p$ and coordinatewise convergence

Let $x^n=(x^n_1, x^n_2,...)$ be a bounded sequence in $\mathcal{l}_p$ for $1<p<\infty$ and such that $x^n_i$ converges to $x_i$ for all $i\in\mathbb{N}$. I'm trying to prove that ...
1
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0answers
52 views

Hausdorff measures and densities

I've been stuck on this one for a while now. It's problem 2.4 from Falconer's "The geometry of fractals" Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with ...
2
votes
1answer
231 views

Coordinate wise convergence of bounded sequences

Show that if $(x^{(n)})$ is a bounded sequence in $l^\infty$, then there exist a subsequence $x^{(n_k)}$ that converge coordinate wise. Is this some generalization of Bolzano-Weierstrass Theorem?
2
votes
2answers
155 views

How can we proof that $2^{\sqrt 2}$ is irrational?

How can we proof that $2^{\sqrt 2}$ is irrational? I was trying with contradiction by taking $\frac xy=2^{\sqrt 2}$ but can't solve it by taking $log$ also.
1
vote
1answer
37 views

Convergence of Power Series $\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$ with $\alpha ,\beta \ge0$

For $\alpha, \beta \geq 0 \in \mathbb{R}$, find the radius of convergence for the series: $$\sum_{n=0}^{\infty}\frac{1+\alpha^{n}}{1+\beta^{n}}z^{n}$$ Ok, so if $\alpha$ and $\beta$ are $\leq ...