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).

learn more… | top users | synonyms (1)

2
votes
1answer
57 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
1
vote
0answers
30 views

Can we identify Fourier transform of continuous compacltly supported functions with finte complex Borel measure?

It is well-known that, $L^{1}(\mathbb R)$ can be embed into $M(\mathbb R)$ (= The space of complex Borel measure on $\mathbb R$); by identifying $f\in L^{1}(\mathbb R)$ with the measure $d\mu= f dm.$ ...
2
votes
2answers
36 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
2
votes
2answers
79 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants $...
1
vote
2answers
361 views

Asymptotic Notations in Limits

Can the asymptotic notations, like Big O, be defined using limits? example: $\lim_\limits{x\to\infty} (f(n)/g(n))=c$ for defining $f(n)=O(g(n))$ If not, why??
2
votes
1answer
222 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
2
votes
2answers
325 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? (...
1
vote
1answer
42 views

How to choose, $\phi \in C^{\infty}_{c}(\mathbb R)$ such that its Fourier transform $\hat{\phi}$ is 1 in some neighbourhood of the given point?

Put $C_{c}^{\infty}(\mathbb R)=$ The space of $C^{\infty}$ functions on $\mathbb R$ whose support is compact. Fix $x_{0}\in \mathbb R.$ My Question is : Can we expect to choose, $\phi \in C^{\...
0
votes
1answer
30 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
2
votes
4answers
184 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
1
vote
0answers
47 views

Di Perna-Lions theory for transport equation

Does someone know if some notes on the topic mentioned in the title are available online? I'm reading the paper "Ordinary differential equations, transport theory and Sobolev spaces" by Di Perna and ...
2
votes
3answers
305 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
2
votes
1answer
79 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge 1$...
2
votes
1answer
62 views

holomorphic function with nonvanishing derivative on unit disk $D$

Let $f$ be a holomorphic function on the unit disk $D$. Suppose for any $z\in D$, $f'(z)\neq 0$. Then does $f$ have to be a conformal map from $D$ to $f(D)$?
1
vote
1answer
105 views

Weighted $L_2$ Hilbert space

this is a question where I am trying to find a reference for a result but I haven't been able to find one at all. Define $L_2(\mathbb R,d\mu) = \{g\in \mathbb R: \int g^2d\mu <\infty\}$. I am ...
1
vote
1answer
50 views

The Lipschitz property of an upper envelope

I'm self studying (for fun) the book "Functional Analysis, Calculus of Variations and Optimal Control" (by Clarke), and I'd appreciate some feedback for my proposed solution for an exercise from the ...
1
vote
1answer
93 views

Question on $x$-section of measurable rectangle in product measure space $X \times Y$

I'm reviewing my analysis notes. We have that $(X, \Sigma, \mu)$ and $(Y, \tau, \nu)$ are complete measure spaces. We are considering the product measure space $(X \times Y, \Sigma(\lambda^{*}), \...
2
votes
0answers
45 views

recurrence relation with trigonometric function

What is the explicit formula for the sequence? $a(n+1) = \sin ( \frac{\pi a(n)}{2} )$ and $a(1) = 0.5$ What should I read to solute such questions?
7
votes
1answer
651 views

Visualisation of the smash product

wedge product, join etc. all of them are no problem for my head, but I am really failing to get a visual idea of what the smash product wants to tell me. For example if I take two spheres, I have no ...
4
votes
2answers
88 views

Borel measure supported on $\mathbb{Q}$

Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?
3
votes
2answers
71 views

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$.

For $1 \leq r < p < \infty$ prove the continuous injection of $L^p([0, 1])$ into $L^r([0, 1])$. I am having a hard time starting. Any suggestions. I tried a straight forward approach. That ...
2
votes
1answer
51 views

Proof of a distance

I have one distance shown as an example in a book but I'm striving to demonstrate that it is effectively a distance. here it is said : let $U=\{z\in\mathbb{C, |z|=1}\}$ we can get a distance on $U$ ...
1
vote
0answers
72 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
3
votes
1answer
38 views

Classify the continuous bilinear functional on $L^p \times L^q$.

Let $1<p<\infty$, $1/p+1/q=1$ and let $L(\cdot,\cdot)$ be continuous bilinear functional on $L^p(\mathbb{R}) \times L^q(\mathbb{R})$. The continuity means that if $f_{n} \rightarrow f$ in $L^p$ ...
0
votes
1answer
47 views

Extension of $C^k$ functions: lower bounds

Consider $f\in C^k(U)$ for an bounded open set $U\subset\mathbb{R}^d$, such that $0<A\leq|D f|\leq B$ on $U$, and suppose that $U^\prime$ is a smooth bounded open whose closure is within in $U$. ...
5
votes
3answers
160 views

Integral with parameter

Is it possible to express in a closed form the integral $$\int_{0}^{\pi/2}\frac{\sin \left ( ax \right )}{\sin x+\cos x}\, {\rm d}x,\,\,\, a\in \mathbb{N}$$ Well, I find it very difficult. Well, I ...
1
vote
1answer
27 views

can we approximate $f,$ in $L^{p}$-norm, by a function $f+h$ which is constant in a some neighbourhood of the point?

Suppose $f\in L^{p}(\mathbb R), (1<p <\infty), \epsilon > 0, \gamma_{0}\in \mathbb R.$ Then My Question is: Can we expect to find, $h\in L^{p}(\mathbb R)$ such that $\|h\|_{L^{p}(\...
0
votes
2answers
27 views

Minima point is a solution point

Consider $$f:\left[0, \dfrac{\pi}2\right] \to \mathbb R$$ defined as $$f(x)=\sup\{x^2,\cos x\}.$$ It is easy to show that $f$ has an absolute minimum point at $x_o \in I$ , but how to show that $\cos ...
1
vote
1answer
68 views

Formal construction of $\mathbb Q$: interpretation and equality of elements

Formally the rational numbers are defined as $\mathbb Z \times \mathbb Z / \{0\}$, where $(m_1, n_1)$ and $(m_2, n_2)$ being equivalent if $m_1n_2 = m_2 n_1$. This set equipped with $+$ and $\times$ ...
-1
votes
3answers
178 views

Differentiation in Banach spaces

Let $E$ be a Banach space, and $F:=L(E,E)$, with $L(E,E)$ the set of continuous linear funtions in $E$. Prove that the function $\exp: F → F$, defined by $$\exp(A)=\sum\limits_{n=0}^\infty\frac{A^n}{...
2
votes
1answer
100 views

Spherical coordinates and Lebesgue integral

I found a textbook that says that $r \in [0,\infty), \theta \in [0,\pi], \phi \in [-\pi,\pi]$ are the spherical coordinates. Then he goes on with: This corresponds to a unitary map $L^2(\mathbb{R}^3) ...
1
vote
2answers
51 views

$V$ is bounded. Does this imply that $\lim_{x\to \infty}\int_{a}^{x}\left | f \right |$ exists?

I am given that the set $V=\left \{ \int_{a}^{x}\left | f \right |:\ x\geq a \right \}$ is bounded. Does this imply that $\lim_{x\to \infty}\int_{a}^{x}\left | f \right |$ exists? Thanks!
5
votes
1answer
58 views

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f||_1 = 1$.

Let $f_n:\mathbb{R}\rightarrow [0, 1]$ be functions such that $\sup_{x \in \mathbb{R}}f_n(x) = 1/n$ and $||f_n||_1 = 1$. Set $F(x) = \sup_{n \in \mathbb{N}}f_n(x)$. Prove that $\int_\mathbb{R}F(x)dx ...
10
votes
3answers
256 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: $$\frac{1}{\operatorname{vol}(A)}\displaystyle\int_{\...
1
vote
3answers
270 views

Is Inverse Function Composition Commutative?

Given $f: \mathbb{R}\to(-1,1)$ is there a theorem that states $f\left(f^{-1}(x)\right) = f^{-1}\left(f(x)\right)$. In example, is $\tanh{\left(\tanh^{-1}{(x^2)}\right)} =\tanh^{-1}{\left(\tanh(x^2)\...
3
votes
0answers
76 views

$\|\phi_{\lambda}- \phi_{\lambda} \ast f \|_{L^{2}(\mathbb R)}\to 0$ as $\lambda \to \infty$? ($\phi_{\lambda}(x)=\lambda^{-1} \phi(x/\lambda).$)

For $f\in L^{1}(\mathbb R),$ we define its Fourier transform as follows: $\hat{f}(t)=\int_{\mathbb R} f(x) e^{-ix\cdot t} dx ,(t\in \mathbb R).$ Suppose that $f\in L^{1}(\mathbb R)$ with $\hat{f}(0)=...
3
votes
0answers
150 views

Image of Cantor set under Cantor-Lebesgue function

Let $m^{\ast}$ be the Lebesgue outer measure and $m$ the Lebesgue measure. Let $\phi$ be the Cantor Lebesgue function and let $\psi(x) := x + \phi(x)$. Let $C$ be the standard Cantor set, why does $m^{...
6
votes
1answer
72 views

Series in a space which is not complete

Let $X$ be a normed vector space and $\left\lbrace x_n \right\rbrace_{n \in \mathbb{N}} \in X^{\mathbb{N}}$ with $$\sum_{n=1}^{\infty} \|x_n\| < \infty \wedge \sum_{n=1}^{\infty} x_n \notin X,$$ ...
0
votes
2answers
141 views

Intepolate from linear to step function, and one application for shading colors

I'm running after a particular function $f_\sigma : [−1,+1] \rightarrow [-1,+1]$ that could take three different forms depending on the value of its parameter $\sigma$. Could anyone help me finding/...
1
vote
1answer
42 views

Norms and convergence in $\mathcal{C}^{\infty}(O)$

Let $O \in \mathbb{R}^d$ be open, $K \subseteq O$ compact and $n \in \mathbb{N}$. For $f \in \mathcal{C}^{\infty}(O)=\mathcal{E}(O)$ we define $$\|f\|_{n,K}^{(1)}:=\sup_{|\alpha| \leq n} \|D^{\alpha} ...
1
vote
1answer
133 views

Approximation functions in $L^{1}$ by indicator functions of dyadic cubes

Let $\mu$ be a finite positive regular Borel measure on $\mathbb{R}^{d}$ and let $S$ be the family of finite unions of squares of the form $\{a_{1}2^{n} \leq x_{1} \leq (a_{1} + 1)2^{n}, \ldots, a_{d}...
0
votes
1answer
29 views

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$?

Is $H^{1}(R;H^1(R^n))=H^1(R^{n+1})$, where $H^1(R;H^1(R^n))=\{f\colon R\to H^1(R^n)\colon \int \|f(t,\cdot)\|_{H^1}^2\,dt <\infty \quad \text{and} \int \|f'(t,\cdot)\|_{H^1}^2\,dt <\infty \}$ ...
3
votes
1answer
351 views

The Dirac Delta Distribution $\delta_0 : D \to \mathbb R$ is not regular

I have a question on the following proof, that $\delta_0$ is not a regular distribution. We define $\delta_0$ as the linear function on test function with $$ \delta_0(\varphi) = \varphi(0) $$ for ...
3
votes
2answers
165 views

Composition of functions is constant in $\mathbb{R^2}$.

Let $\hspace{0.05cm}f:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ and $\hspace{0.05cm}$$g:\mathbb{R^2}\to\mathbb{R^2}$ $\hspace{0.05cm}$ be such that$\hspace{0.05cm}$ $g\circ f$$\hspace{0.05cm}$ ...
5
votes
3answers
144 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
0
votes
1answer
56 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
8
votes
3answers
594 views

ODE with singular coefficients

I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - q(x) y(x) = \lambda y(x).$ Then I got a more sophisticated differential equation ( second one) and is given by $$-(1-x^2)y''(x) +x y'...
1
vote
1answer
79 views

The Landau symbol $\mathcal{o}$ as in Königsberger Analysis I

I am currently working on Chapter 14 - local approximations of function and Taylor polynomials - in Königsberger Analysis 1 Background: Königsberger introduced the Taylor Polynomial of order (...
0
votes
0answers
126 views

A continuous function which does not converge to its Fourier series. [duplicate]

Where can I find an example (or the theorem) for a continuous function which does not converge pointwise to its Fourier series, as well as its explanation? I would prefer a web page or a free site or ...
1
vote
3answers
54 views

Characterization of $\mathscr{S}(\mathbb R^n)$?

Consider the vector space $$\displaystyle\mathscr{S}(\mathbb R^n)=\{f\in C^\infty(\mathbb R^n): \lim_{|x|\to \infty} |x^\alpha \partial^\beta \phi(x)|=0, \forall \alpha, \beta\in\mathbb N_0^n\}.$$ I'...