Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

learn more… | top users | synonyms (1)

7
votes
3answers
1k views

Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?

Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain $$f(x+a) = ...
0
votes
1answer
94 views

Differentiability for a function made of built-in functions and basic operations

If a function is made up of some standard built-in functions such as $\ln$, $\exp$, $\sin$, $\cos$, $\operatorname{abs}$ and the basic operations $+$, $-$, $\times$, $\div$, is it true that this ...
3
votes
1answer
196 views

Identifying recursive polynomials

I need to evaluate the following function and want to proceed analytically as far as possible: $F(y) =e^{ i \beta \left ( y \frac{d}{d y} \right )^2} y \, e^{-y^2/2}$ My plan is to expand into ...
7
votes
2answers
382 views

Prove that $\left|\int_a^b f(x) dx - (b-a) f(a) \right| \leq \frac{(b-a)^2}{2}$ when $f$ satisfies $|f(u)-f(v)|\leq |u-v|$

this is a question from Apostol's Calculus, Volume I, p. 139 (Exercises 33) which I've gotten completely stuck on. Let $f$ be a continuous function such that $|f(u) - f(v)| \leq |u - v|$ for all ...
21
votes
3answers
1k views

Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.

Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$ Computing the Fourier series of $f$ and ...
3
votes
2answers
91 views

About integrable functions.

Let $f_n\colon [0,1]\rightarrow R$ be Lebesgue mensurable with $\int_{0}^{1} |f_n(t)|^3dm(t)<1$ for all $n$. How we can show that $f_n$ is integrable uniformly i.e for all $\epsilon>0$ there ...
0
votes
1answer
1k views

Fourier transform of the characteristic function

My qustion is about the Fourier transform of the characteristic function $\chi_{[0,1]}$. How can I find what it is? The problem is I got something really messy, so I think I didn't get it right.
6
votes
8answers
555 views

How to show that $\lim_{n \to +\infty} n^{\frac{1}{n}} = 1$?

I've spent the better part of this day trying to show from first principles that this sequence tends to 1. Could anyone give me an idea of how I can approach this problem? $$ \lim_{n \to +\infty} ...
4
votes
1answer
599 views

integral of Laplacian of a positive function

I've encountered the following, rather elementary, problem: $K$ is a compact subset of some 2-dimensional oriented manifold with smooth boundary, $f$ is a positive smooth function on $K$ that ...
2
votes
3answers
122 views

polynomial-torus

I am wondering if it is possible to prove (or come up with an explicit example) that there is a polynomial $f ( x,y,z )$ of degree 8 such that the set $f ( x,y,z )=0$ is a union of two torri? Any ...
2
votes
2answers
217 views

derivative of a map of vector space of matrices

Question: Let $A_{n\times n}$ be the vector space of all real $n\times n$ matrices. If I define a map $$g:A_{n\times n}\rightarrow A_{n\times n}$$ such that: $$g\left ( X \right )=X^{2}$$ In ...
6
votes
2answers
140 views

$C^{2} ( \mathbb{R}^{2}) $ function-proof of inequality

Let $f\in C^{2}( \mathbb{R}^{2} )$. Suppose that $\triangledown f=0 $ on a compact set $A\subseteq \mathbb{R}^{2}$. I want to prove that there is a strictly positive constant $\lambda > 0$ such ...
2
votes
1answer
253 views

example of Diffeomorphism

I am trying to come up with a diffeomorphism of the upper half plane $y> 0$ onto the first quadrant $x> 0 , y> 0$ Can anyone come up with such a diffeomorphism?
4
votes
1answer
143 views

Showing $V_a^b\alpha = \sup \left\{\int_a^bfd\alpha:\|f\|_\infty\leq 1\right\}$

Let $\alpha:[a,b]\to\mathbb{R}$ be of bounded variation and right-continuous. Given $\varepsilon>0$ and a partition $P$ of $[a,b]$, construct $f\in C[a,b]$ with $\|f\|_\infty \leq 1$ such that ...
2
votes
0answers
48 views

When is a subset of {0,1} valued borel functions on a standard borel space (polish space) complete (see *) under the pointwise convergence topology?

*In other words, what restrictions on a family F of {0,1} valued borel functions will tell us that the pointwise limit of any net in F is borel. I feel like there must be lots known about this but I ...
11
votes
1answer
471 views

About a measurable function in $\mathbb{R}$

Let $h:\mathbb{R}\rightarrow\mathbb{R}$ be a measurable function such that $$\left|\int_I h\right|\leq c \sqrt{|I|}$$ for each interval $I$. Then $h_\epsilon(x)=h(x/\epsilon)$ satisfies ...
1
vote
2answers
112 views

Am I correct with the following converging/diverging series

Have two series, just a quick check of some simple series: $\sum _{1}^{\infty} \frac {1}{\sqrt {2n^{2}-3}}$ Considering $\frac {1}{\sqrt {2n^{2}-3}}$ > $\frac {1}{\sqrt {4n^{2}}}$ = $\frac ...
2
votes
1answer
241 views

How to show that $\lim_{n\to \infty} f_n(x) = 0$.

I hate to admit it, but I don't know how to begin solving this problem: Let $f\in L^1(\mathbb R)$. Let $\displaystyle f_n(x) = \frac{f(nx)}{n},~n\geqslant 1$. Then $\displaystyle \lim_{n\to \infty} ...
5
votes
1answer
142 views

Integral equation and existence: $g(x)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$

I'd like to know how one would go about showing that the following function, $f$, that is almost everywhere positive exists: $$g(x_1,\cdots,x_n)=\int_{-\infty}^{\infty} \prod _{j=1}^nf(u-x_j)du$$ ...
7
votes
2answers
169 views

Showing $ \sum_{k=1}^{n} k^{1/k}\sim n$

I would like to show that: $$ \sum_{k=1}^{n} k^{1/k} \sim n$$ by using integrals. We have: $$ \int_{3}^{n+1} t^{1/t} \mathrm dt +\sqrt{2}+1 \leq \sum_{k=1}^{n} k^{1/k} \leq \int_{3}^{n} t^{1/t} ...
2
votes
1answer
76 views

Question about normed spaces

Let $(X,||\cdot||)$ be a complete normed space. Let $F_1, F_2, F_3,\ldots\subseteq X$ be closed, non-empty subsets of $X$. Assume that $F_1 \supseteq F_2\supseteq F_3\supseteq \cdots$ and ...
2
votes
1answer
73 views

A functional in $C[-1,1]$ that is zero in the even functions marks the norm of an odd function

Given a $g$ an odd function the question is to exhibit a continuous linear functional from $C[-1,1]$ $\phi$ such that $|\phi|=1$ , $\phi(g)=|g|_\infty$ and $\phi$ is zero on the even functions.
4
votes
1answer
116 views

A Hilbert basis for $L^2 ([0,1]\times[0,1])$

Let $\{f_n(x)\}$ and $\{g_n(x)\}$ be two Hilbert basis of $L^2 ([0,1])$ then $\{g_n(x)f_k(y)\}$ is a Hilbert basis for $L^2 ([0,1]\times[0,1])$. Obs: That is Orthogonal, and unitary is I proved with ...
2
votes
2answers
101 views

Cauchy sequence

Show that if $(x_{n})_{n}$ is a Cauchy sequence in X and $\lambda \in \mathbb{R}$, then the sequence $(\lambda x_{n})_{n})$, is also Cauchy in X. We know that for $(x_{n})_{n}$, we have $\forall ...
4
votes
2answers
232 views

Showing $\int_0^{2\pi} \log|1-ae^{i\theta}|d\theta=0$

This is a homework problem for a second course in complex analysis. I've done a good bit of head-bashing and I'm still not sure how to solve it-- so I might just be missing something here. The task is ...
2
votes
2answers
106 views

Complete normed spaces

Let $(X, ||\cdot||)$ be a complete normed space. Let $||\cdot||$ be a norm on $X$, and assume that there are constants $c_{1}$, $c_{2} \in (0,\infty)$ such that: $c_{1}||x-y||\le||x-y||_{0}\le ...
2
votes
1answer
551 views

Sum of Cauchy sequences [duplicate]

Possible Duplicate: Sum of Cauchy Sequences Cauchy? Let $(X,||\cdot||)$ be a normed space. Show that if $(x_{n})_{n}$ and $(y_{n})_{n}$ are Cauchy sequences in $X$, then the sequence ...
4
votes
2answers
120 views

Is Completeness intrinsic to a space?

Is completeness an intrinsic property of a space that is independent of metric? For example, since $\mathbb{R}^n$ is complete with the Euclidean metric, is it complete with any other metric? If ...
2
votes
1answer
86 views

Change of Variables Clarification

How can we show that $v(L(C)) = |\det DL|v(C)$ for any open cube $C$ an element of $\mathbb{R}^n$ and any linear transformation $L: \mathbb{R}^n \rightarrow \mathbb{R}^n$, without direct applying the ...
0
votes
2answers
118 views

Problem in normed spaces

Some help with the following would be great. Let $(X,||\cdot||)$ be a normed space. Let $(x_{n})_{n}$ and $(y_{n})_{n}$ be Cauchy sequences in $(X, D)$. Say also that $s_{n} = ||x_{n} + ...
6
votes
1answer
172 views

Smoothness of radially defined functions.

Suppose that for each $v \in \mathbb{R}^2$ with $|v| = 1$, there is a smooth ($C^{\infty}$) function $f_v : [0, 1] \rightarrow \mathbb{R}$ such that $f_v(0) = 0$. Now, let $\bar{D}$ be the closed ...
2
votes
1answer
155 views

Ratio of limits

In arithmetic there is a property that if $\frac{a}{b}=\frac{c}{d}=\alpha$ then $\frac{a-c}{b-d}=\frac{a+c}{b+d}=\alpha$, with the first we assume $b\neq d$. With the limits, for example, if ...
2
votes
1answer
281 views

Existence and uniqueness theorems for ODE. Log-Lipschitz regularity.

Let $\mathbb{X}$ be a linear space with a complete metric $d:\mathbb{X}\times\mathbb{X}\to [0,+\infty)$. Let's $B[x_o,b]$ is a compact ball of radius $b$ centered at $x_o$. THEOREM:If ...
0
votes
0answers
94 views

Solving $ \int_{0}^{\infty} \frac{e^{-t}}{\sqrt{t}}e^{-\alpha^2/4t} \mathrm dt $ [duplicate]

Possible Duplicate: Is My Solution on Integration by Parts Correct? I'm trying to compute the following integral: $$ \int_{0}^{\infty} \frac{e^{-t}}{\sqrt{t}}e^{-\alpha^2/4t} \mathrm dt $$ ...
1
vote
1answer
115 views

Measure (mathematical analysis)

Measure, wikipedia article According to Wikipedia: In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size ...
2
votes
3answers
165 views

“direct” ways in which a non-computable number is used?

I was wondering whether non-computable numbers are ever of "direct" use ? I understand they are immensely useful indirectly, because we need them to do analysis in the real numbers for instance. ...
15
votes
3answers
353 views

Computing $ \int_{0}^{\infty} \frac{1}{(x+1)(x+2)…(x+n)} \mathrm dx $

I would like to compute: $$ \int_{0}^{\infty} \frac{1}{(x+1)(x+2)...(x+n)} \mathrm dx $$ $$ n\geq 2$$ So my question is how can I find the partial fraction expansion of $$ ...
2
votes
4answers
1k views

Computing $ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $

I would like to compute: $$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)} \mathrm dx $$ We have: $$ \int_{0}^{2\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx=2\int_{0}^{\pi}\frac{\sin(nx)}{\sin(x)}\mathrm dx$$ ...
7
votes
1answer
103 views

How can one prove that $\sum_{k=1}^{\infty}\prod_{m=1}^{2k}\text{ctg}\frac{m\pi}{2k+1}=\frac{\pi}{4}-1$?

A question from some Russian book, about different summations an integrals, by Prudnikhov, Brichkhov and Marichev. Page 746, 20, they write: $$ ...
3
votes
3answers
198 views

Convergence of $ u_{n}=\sqrt [n]{\frac{(a+1)(a+2)…(a+n)}{n!}} $

I would like study the convergence of the following sequence: $$ u_{n}=\sqrt [n]{\frac{(a+1)(a+2)...(a+n)}{n!}} $$ where $a>0$ We have: $$ \ln(u_{n})=\frac{1}{n}\sum_{k=1}^n ...
2
votes
1answer
88 views

A nonempty compact convex subset $A\subset \mathbb{R}^n$ has an extreme point.

A nonempty compact convex subset $A\subset \mathbb{R}^n$ has an extreme point. How do you prove this result? Can you give me sketch? Thanks
1
vote
0answers
118 views

Incrementally compute the conditional entropy

Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase ...
13
votes
1answer
263 views

Convexity of $\theta(q)$

Define Jacobi's (fourth) theta function with argument zero and nome $q$: $$\theta(q) = 1+2\sum_{n=1}^\infty (-1)^n q^{n^2}$$ plot of the function via Wolfram|Alpha plot of the function via Sage I ...
1
vote
1answer
88 views

Linear Isomorphism And Completeness

I am just a beginner. So please be patient with me. Consider $X=\{\frac 1 n : n \in \mathbb N\} \cup \{0\}$. Prove that the vector space of continuous functions on $X$ is linearly isomorphic ...
1
vote
1answer
111 views

The set of continuities of a function is measurable.

Let $f$ be a real valued function then the set where $f$ is continuous is measurable. Note the statement does not even requires that $f$ is measurable.
2
votes
1answer
394 views

How to prove this subadditivity?

Let be $X$ a positive random variable. I would like to prove that the function $\varphi:\mathbb{N}\to [0,+\infty]$ defined by $\varphi(n)=-\log \mathbb{E}[\exp(-nX)]$ satisfies $\varphi(m+n)\leq ...
5
votes
2answers
591 views

Isometric Immersion of a separable Banach Space into $\ell^{\infty}$

The problem is: Let $X$ be a separable Banach space then there is an isometric immersion from $X$ to $\ell^{\infty}$. My efforts: I showed that there is an isometry from $X^*$ (topological dual) to ...
10
votes
3answers
386 views

Converging series question, Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then does $\sum_{n=1}^{\infty} \frac {a_n}{n}$

Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then does $\sum_{n=1}^{\infty} \frac {a_n}{n}$ For this I have shown the case for when $ a_n^{2} \le\frac {|a_n|}{n}$ $\Rightarrow$ $ ...
6
votes
3answers
247 views

Integral of a function defined in the set of Surreal Numbers

Given ${\{C}\}\ $ the set of all the $Surreal\ numbers$, is it possible to define the integral: $$\int_a^b{dxf(x)}$$where $$a\in{\{C}\},b\in{\{C}\},x\in{\{C}\}$$ Thanks
9
votes
3answers
379 views

Showing that $\gamma = -\int_0^{\infty} e^{-t} \log t \,dt$, where $\gamma$ is the Euler-Mascheroni constant.

I'm trying to show that $$\lim_{n \to \infty} \left[\sum_{k=1}^{n} \frac{1}{k} - \log n\right] = -\int_0^{\infty} e^{-t} \log t \,dt.$$ In other words, I'm trying to show that the above definitions ...