-4
votes
0answers
38 views

Calcul of limit [on hold]

What is the limit of $$\lim_{f \rightarrow 0} \frac{ \nabla {f(x)} }{\sin{(f(x))}}?$$ We can use the Poincare inequality and the famous limits: $$\lim_{x\rightarrow 0} ...
0
votes
1answer
31 views

Inverse of an operator on two functions

I have the following operator, defined for two twice-differentiable functions $f,g$: $X(f,g):=\frac{(g')^3+fg'f''+g'(f')^2-ff'g''}{g'f''-f'g''}$ This operator has the following property: A curve ...
2
votes
1answer
90 views

Does a function that is twice weakly differentiable have a version that is classically differentiable?

I have been wondering about the idea of functions that are weakly differentiable. My intuition tells me that the weak derivative allows one to differentiate functions that either have a removable ...
0
votes
1answer
32 views

$L^\infty(\Omega)$ space

Consider Lebesgue spaces $L^p(\Omega)$, $\Omega$ is a bounded domain. Let $f \in L^p(\Omega)$ for all $p$. Is it true that $f \in L^\infty(\Omega)$?
5
votes
2answers
196 views

Prove that $f$ is constant on $[a,b]$

$\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$ And $f$ is continious on $[a,b]$ and ...
2
votes
1answer
57 views

If $\phi\in \mathcal{S}(\mathbb R) $ then $\phi_{t}(x)=\frac{1}{t} \phi(x/t)\in\mathcal{S}(\mathbb R)$?

Let $\phi:\mathbb R \to \mathbb C$ be a function; and define $$\phi_{t}(x):=\frac{1}{t}\phi (x/t), (t>0).$$ We note that, if $\phi\in L^{1}(\mathbb R),$ then $\phi_{t}\in L^{1}(\mathbb R);$ in ...
3
votes
3answers
42 views

Finding domain of $f\text{ o }g$

I am having a small question, please don't close this before answering, I just want to know whether its a matter of convention or not. If $f(x) = \dfrac{1}{x}$ and $g(x) = \dfrac{1}{x}$ $ $ Then ...
3
votes
1answer
43 views

a function with infinity L^p norm

Let $1\leq p<\infty$, $1/p+1/q=1$. For a function $f$ with $||f||_q=\infty$, can we write $$ ||f||_q=\sup_{g\in L^p(\Omega),||g||_p\neq 0}\frac{\int_\Omega |fg|}{||g||_p}? $$ or $$ ...
1
vote
2answers
70 views

Can all functions be expressed in terms of elementary functions?

After being introduced to the non-elementary function through an attempt to evaluate $\int x \tan (x)$, an interesting question occurred to me: Can the non-elementary functions be decomposed to ...
1
vote
1answer
65 views

Approximation of the Heaviside Function whose derivative has a compact support

I am looking for a smooth approximation $H_\delta$ of the Heaviside function, which has the property that $$ \lim_{\delta\rightarrow 0^+}H_\delta =H $$ in the distribution sense, and $$ ...
3
votes
2answers
53 views

What is $T^nf(t)$? (Question on integrals)

I am supposed to prove the following: For the operator $T$ defined by $$Tf(t)=\int_0^t(t-s)f(s)\,ds,\quad f\in C[0,1]$$ Show that $$T^nf(t)=\int_0^t\frac{(t-s)^{2n-1}}{(2n-1)!}f(s)\, ds$$ I ...
0
votes
0answers
25 views

Can Duhamel's Formula be applied Functional Equations?

Consider an n'th order linear Differential Equation of the form $$y^{[n]} = a_0(x) + a_1(x)y + a_2(x)y' + a_3(x)y'' \ ... \ + a_{n-1}(x)y^{[n-1]}$$ It is well known that to solve this we can create ...
1
vote
1answer
46 views

I'm searching for the formula of the series $ \sum_{n=0}^{\infty}a^{n^l} $

I'm searching for the sum-formula (if exists) of the following power series: $$ \sum_{n=0}^{\infty}a^{n^l} $$ where $l=2,3,....$, and $|a|<1$.
1
vote
1answer
29 views

conditions for Convergence of sequence of functions

Suppose $\{f_n\}$, $n \in \mathbb{N}$ is a sequence of a positive real-valued functions defined on $[0, T]$ and continuous on $(0, T)$. If {$f_n$} satisfies the following conditions : $f_n( iT/2^n ) ...
0
votes
0answers
24 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
0
votes
0answers
24 views

does the limit of the ratio of $p+1$ norm and $p$ norm equal to $\infty$ norm

Suppose that $f\in L^1(\Omega,\Sigma,\mu)\cap L^\infty(\Omega,\Sigma,\mu)$. Then I have proved that for any $1\leq p\leq \infty$, $f\in L^p(\Omega,\Sigma,\mu)$. Moreover, I have proved ...
1
vote
1answer
33 views

Sturm Liouville problem with additional term.

Imagine you want to solve an ODE on $[a,b] \subset \mathbb{R}$ $f''(x) + (A(x) + B(x))f(x) = \lambda_n f(x)$, where $A,B$ are some smooth functions and $\lambda_n$ the n-th eigenvalue. Furthermore, ...
0
votes
0answers
26 views

Question about function and primitive

I have a function$f$ such that: $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ is continuous , there exist $C>0$ and $\theta>2$ such that $|f(t,u)|\leq C(1+|t|^{\theta-1}~ a.e t\in ...
1
vote
1answer
35 views

Monotone property of $L^p$ norm

My question is: Is there monotone property of $\|f\|_p$ when $p$ is increasing, where $\|f\|_p=(\int_a^b f(x)^pdx)^{1/p}$ is the classical $L^p$ norm and $f\in L^p$? . This proposition is ...
0
votes
1answer
14 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
2
votes
2answers
50 views

Is this operator bounded ??

Let $X$ be the Banach space $X:=\{ f\in C(\mathbb{R},\mathbb{R}),\sup_{t\in \mathbb{R}}|e^{-s^2}f(s)|<+\infty \}$ equipped with the norm $$|f|_X=\sup_{t\in \mathbb{R}}|e^{-s^2}f(s)|$$ I want to ...
1
vote
1answer
40 views

continuous extension and smooth extension of a function

Let $X$ be a metric space. Let $E$ be a subset of $X$. (1). any continuous function $f:E\longrightarrow \mathbb{R}$ can be extended to a continuous function $g: X\longrightarrow \mathbb{R}$ such ...
1
vote
1answer
46 views

Fréchet derivative, is this true?

I was just wondering whether the following statement is true: Let $H_1,H_2$ be Hilbert spaces and $\{e_n\}_{n\geq 0}$ be an orthonormal basis of $H_2$. Let $f:H_1\rightarrow H_2$ be an operator (not ...
1
vote
0answers
49 views

bilinear form, anti symmetric part

$\mathcal{H}$ : real Hilbert space with inner product $(\,,\,)$ and norm $||\,||:=(\,,\,)^{1/2}$ Let $D$ be a linear subspace of $\,\mathcal{H}$ and $\mathcal{E}$ : $D\times D\to \mathbb{R}$ a ...
3
votes
1answer
76 views

UNBEATABLE recurrence relation

Hi I don't know where to start to solve this reccurence relation: $g(1)=2$ $ g(2n)=3g(n)+1$ $ g(2n+1)=3g(n)-2$ of coures I can make it: $ g(1)=2$ $ g(n)=g(2n)/3-1/3$ $ ...
0
votes
3answers
57 views

Matrix norm in Banach space

How can I calculate the following matrix norm in a Banach Space: $$ A=\begin{pmatrix} 5 & -2 \\ 1 & -1 \\ \end{pmatrix} ?$$ I have tried ...
0
votes
0answers
11 views

Rearranging 2 discriminant function to solve for 1 parameter (to derive a decision boundary)

I have a task where I want to classify patterns from 2 classes where the samples are drawn from a bivariate Gaussian distribution. I use the 2 discriminant functions ($g_1$ and $g_2$) to classify the ...
2
votes
1answer
25 views

What assumptions are needed to get two integrals close to each other?

I have functions $A,B,C$, where $\int_{\mathbb{R}} |A\cdot B - C| <\varepsilon$, and want to be able to say that $\int_{\mathbb{R}} A \approx \int_{\mathbb{R}} \frac{C}{B}$. What extra assumptions ...
0
votes
1answer
19 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0
votes
0answers
23 views

What is the contour lines of $\frac{2x+y}{2x-y}$?

What is the contour lines of $$f(x)=\frac{2x+y}{2x-y}$$? I need help to describe them... Id like to get help... Thank you!
0
votes
3answers
88 views

a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$ [duplicate]

Hi need some help with this problem: Assume $f : \mathbb{R} → \mathbb{R}$ is a continuous function, satisfying $f(α) = f(β) +f(α −β)$ for any $α, β ∈ \mathbb{R}$, and $f(0) = 0$. Then $f(α) = α ...
0
votes
0answers
50 views

Mean value of a function over the n-sphere's surface.

We know that we can use the bloch sphere to represent an unitary vectors $v$ in $\mathbb{C}^{2}$, due to the fact $su(2) \approx so(3)$. Then, if we have the function $f:\mathbb{C}^{2} \rightarrow ...
1
vote
1answer
35 views

prove $\lim_{n\to\infty}\sup\{|f(x) - f_n(x) | : x \in S \} =0$

I understand how the theorem works but how would you prove that a sequence $f_n$ of functions on set $S \subset \mathbb{R}$ converges uniformly iff $$\lim_{n\to\infty} \sup\{|f(x) - f_n(x) | : x \in ...
0
votes
1answer
61 views

Is this construction already a contradiction

Let's say we have $\ell^p$ for $p>2$ and $\ell^2 \subsetneq \ell^p$. Let $U$ be a closed subspace of $\ell^2$. Is it possible that $U$ is isomorphic to $\ell^p$ as a Banach space?- I hardly think ...
2
votes
2answers
73 views

$C^\infty$ approximations of $f(r) = |r|^{m-1}r$

Consider $f(r) = |r|^{m-1}r$ where $m \geq 1$. Is it possible to find $C^\infty$ functions $f_n$, such that $f_n \to f$ uniformly on compact subsets of $\mathbb{R}$, $f_n' \to f'$ uniformly on ...
1
vote
0answers
53 views

Transfer vector space properties to dual space

I am curious about this here (Actually, I don't know if my assumptions are true or not) a) Let $X$ be a Banach space that is isomorphic to $Y$, then $X^*$ is also isomorphic to $Y^*$. I sketched a ...
2
votes
2answers
59 views

How to find maximum/minimum of $y=\frac{x(x^2-x+2)}{x^2-9}$?

How can I find minimum and maximum of $y=\frac{x(x^2-x+2)}{x^2-9}$? In other words, points, where $y'=0$. My current steps are: $$y=\frac{x(x^2-x+2)}{x^2-9}=x-1+\frac{11x-9}{x^2-9}$$ ...
0
votes
2answers
50 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
1
vote
1answer
35 views

Is $l^p$ closed in $L^p$?

Let's assume we have a subspace $X$ of $L^p$ and we know that $X \cong l^p$(this should just mean isomorph no isometry is assumed here). Can we infer from this that $X$ is closed? I have just read a ...
1
vote
1answer
55 views

Basic sequence- what is so special about it?

Let $(x_n)$ be a Schauder basis of a vector space $X$. This means that the $span(x_n)$ is dense in $X$, right? Then wikipedia introduces the notion of a $\textbf{basic sequence} $ when $(x_n)$ is a ...
2
votes
2answers
133 views

Biorthogonal functions in $L^p$

I asked one question that is already answered: 1.) I have a question about Lemma 9.5 on page 93/94 reference. It's about the part of the proof where the sequence of $(g_n^*)$ are introduced. I don't ...
5
votes
1answer
101 views

Perturbation theory PDEs

I have the solution of a PDE of the form: $$ \Delta \Psi(r,\theta, \phi) = k \Psi(r,\theta,\phi)$$ on a set $\mathbb{R}^3 \backslash B(0,R)$. Hence, the actual solution is known there! Regarding ...
0
votes
2answers
63 views

old prelim exam question

In studying for my prelims, I can't quite come up with a solution to the following problem, from a few years ago. It has the feel of a "write the correct thing down, and it's 3 lines" problem. ...
3
votes
0answers
50 views

Riesz Fischer theorem?

I was wondering about the following: I read that Fischer-Riesz says that $L^2([0,1])$ is isomorphic to $l^2(\mathbb{N})$. Now it is obvious, that this should not depent on the fact which compact ...
0
votes
0answers
43 views

When integration by parts applies?

I think integration by parts is really useful, but I don't know when integration by parts applies. Take the following proposition as an example: Let $\Omega \subset \mathbb{R}^n$ be an open set, ...
3
votes
1answer
39 views

Limit of a hyperpower function

i have a question regarding this class of equations: Let $\gamma(x)=x^x$ Let $\Psi_n(x)=\underbrace{\gamma(x)\circ\gamma(x)\circ\gamma(x)}_n$, such that $\Psi_1(x)=\gamma(x)$ and ...
4
votes
1answer
106 views

A inequality of calculus [duplicate]

Let $f \in C^2[a,b]$ and $f(a) = f(b) = 0$, $f'(a) = 1$,$f'(b) = 0$, prove that $$\int_a^b|f''(x)|^2\,dx \geq \frac{4}{b-a}$$ Remark: This question is in the book functional analysis of Peking ...
5
votes
1answer
79 views

Spectral theorem in Quantum Mechanics

In Quantum Mechanics one often looks at self-adjoint(unbounded and closed) linear operators $A,B$ that are defined dense on $L^2$. My question is: Is it true that if we have $[A,B]=0$, where $[,]$ is ...
0
votes
1answer
23 views

question about the continuity of a function

I need to show that: If $f$ is continuous at $x_0$ iff $f^*(x_0)=f_*(x_0)$ where: $f^*(x_0)=\lim_{x \to x_0} \sup f(y)=\inf_{\epsilon > 0} \sup_{|y-x_0|<\epsilon}f(y)$ and $f_*(x_0)=\lim_{x ...
0
votes
0answers
29 views

Differentiating the sum ($\sum_{j=1}^\infty f_j(t)v_j(x,t)$) term by term

Let $f(x,t) = \sum_{j=1}^\infty f_j(t)v_j(x,t)$ where $f_j \in C^1(0,T)$ and $v_j(\cdot,t) \in C^\infty(\Omega)$ where $\Omega \subset \mathbb{R}^n$ is a domain. The sum makes sense in the sense that ...