All Questions
Tagged with functional-analysis calculus
1,287 questions
0
votes
1
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35
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Without Banach-Steinhaus, can we prove the continuity of the bilinear form?
Let $B$ be a normed linear space, and let $f : {M}_{n}\left( \mathbb{R}\right) \times B \rightarrow \mathbb{R}$ be a bilinear function, $f(\cdot,y)$ and $f(x,\cdot)$ are both continuous, prove that ...
- 3,713
0
votes
0
answers
50
views
Bound for Sup norm of piecewise continuous function
I’m working for this paper, but I encountered some problem for proving Lemma 2. It stated that if $\psi \in C([a,b])\cap_{i=1}^n C^2([x_{i-1}, x_i])$, where $a=x_0<x_1<\cdots<x_n=b$, one has ...
- 1
5
votes
1
answer
181
views
Approximating a continuous function by translations implies uniform continuity?
Suppose $f\in C^0(\mathbb R)$ is a continuous function on real numbers. If for any $\varepsilon>0$, there exists $\delta>0$ such that $|f(x+\delta)-f(x)|<\varepsilon$ for all $x\in\mathbb R$....
- 274
2
votes
1
answer
69
views
Locally Lipschitz continuity of the Fourier transform of $L^1$ function
Let $f \in L^1(\mathbb{R}^d)$. It is very famous that $\xi\mapsto \mathcal{F}f(\xi)$ is continuous on $\mathbb{R}^d$, where $\mathcal{F}f$ is the Fourier transform of $f$.
My question is the following:...
- 659
1
vote
1
answer
112
views
Determine a function $g(x)$ to maximize a definite integral in the form of $\int^{+\infty}_{t} x g(x)f(x) \mathrm dx$
Suppose I have a random variable $X \sim N(0, \sigma^{2})$ with density function $f(x)$. I have another function $g(x) > 0$ defined on $(t, +\infty)$, where $t > 1$ is a constant, and $\int^{+\...
- 169
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0
answers
25
views
Inequality by induction $ n^2 \leq \left( \sum_{j=1}^n x_j \right) \left( \sum_{j=1}^n \frac{1}{x_j} \right)$ [duplicate]
I am trying to prove the following inequality: for all $n \geq 1$ and all $x_1, \ldots, x_n \in \mathbb{R}^{*}_+$, one has
$$
n^2 \leq \left( \sum_{j=1}^n x_j \right) \left( \sum_{j=1}^n \frac{1}{x_j} ...
- 169
1
vote
0
answers
44
views
Solving $\int_0^\infty{f(x,x_o) \lambda(x) dx} = 0$ for $\lambda(x)$, such that this is satisfied for all $x_o$ for a known $f(x,x_o)$
How would one solve the following functional equation for a strictly nonnegative function $\lambda(x)$:
$$\int_0^\infty{f(x,x_o) \lambda(x) dx} = 0$$
if this condition holds for all $x_o$, given a ...
- 11
0
votes
0
answers
22
views
Homoclinic orbits as critical points of an Hamiltonian
Good morning, I'm studying an article by P.H. Rabinowitz and he states that a critical point of given Hamiltonian is an homoclinic orbit for the associated system.
I don't understand why that is. Does ...
- 61
6
votes
2
answers
186
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Hilbert space question: if $M=\{f\in L^2(0,+\infty)\ |\ \int_0^\infty f^2(x)e^xdx<+\infty\}$, what is $M^\perp$?
This was from a Real Analysis exam:
In the Hilbert space $H=L^2(0,+\infty)$. Consider
$$M=\left\{f\in L^2(0,+\infty)\ \bigg|\ \int_0^\infty f^2(x)e^xdx<+\infty\right\}$$
Find $M^\perp$.
Since the ...
- 3,824
1
vote
1
answer
77
views
Probability density for function to have a certain y-value
An at first sight simple problem, to which I couldn't readily find the answer nonetheless.
Assume a real-valued function $y=f(x)$ of a single real variable $x$, on the domain $D=[a,b]$. We can ...
- 195
-2
votes
1
answer
67
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If the integral of a monotonic f converges, does it mean f approaches 0? [closed]
I have come across this question:
Say $f(x): [0,\infty) \rightarrow \mathbb{R}$ is monotonic non-increasing, and $\int_{0}^{\infty} f(x)dx$ converges.
Does it mean that $\lim_{x\to\infty}{f(x)}=0$? If ...
0
votes
1
answer
77
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Is a Toeplitz operator just a multiplication operator with a bounded function on the unit disk?
I have been recently studying Hardy spaces on the unit disk (most specifically, $H^2$) and I have come across the so-called Toeplitz operators.
In every book I have found about the topic they define a ...
0
votes
1
answer
56
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Asymptotics of 2Ai(x)Ai'(x)
I'm looking for the asymptotics of the derivative of Airy squared $2Ai(x)Ai'(x)$ as $x\to \infty$. I know that $A i^2(x) \sim \frac{1}{4} \pi^{-1}(x)^{-1 / 2} e^{-\frac{4}{3}(x)^{3 / 2}}$. Would it ...
- 3,161
1
vote
0
answers
40
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Continuous function supported in $[2-\delta,2]$, satisfying integral of $f(x)\sqrt{4-x^2}$ equals $1$.
Fix $\delta >0$.
I want to show the existence of continuous function $f$, satisfying $supp(f) \subset [2-\delta, 2]$ and $$\frac1{2\pi}\int_{2-\delta}^{2} f(x)\sqrt{4-x^2}\,\mathrm dx=1\;.$$
How do ...
- 135
2
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0
answers
71
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A generalization of integration by parts
Many years ago, I came up with a short generalization of integration by parts that was definitely known, but I could never find a reference for it. I was considering throwing it on arxiv, but before I ...
- 382
0
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0
answers
39
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Is this calculus derivation process correct?
This is part of the economics romer model, finding the Lagrangian value
$$\mathcal{L}=\int_{i=0}^Ap(i)L(i)di-\lambda([\int_{i=0}^AL(i)^\phi di]^{\frac{1}{\phi}}-1)\\
s.t.\int_{i=0}^{A}L(i)^{\phi}di=1
$...
- 101
2
votes
1
answer
86
views
Does there exist a continuous fuction $g(x)$ such that: $\int \frac{g(x)}{x} dx$ is finite at $x=0$ and $g(0)\ne0$?
Does there exist a continuous fuction $g(x)$ such that: $\int \frac{g(x)}{x} dx$ is finite at $x=0$ and $g(0)\ne0$?
Based on my calculations I am finding that for the integral condition to be true $g(...
- 21
2
votes
0
answers
62
views
If $f_n\to f$ a.e. and $\limsup_{n\to +\infty} \|f_n\|\le K,$ does it imply that $\|f\|$ is bounded?
Let $(H, \|\cdot\|)$ denote a Hilbert space which is continuously embedded in $L^2(\mathbb R^n)$. Let $\{f_n\}$ be a sequence such that
$$ f_n\to f \text{ a.e. in } \mathbb R^n,$$
and
$$\limsup_{n\to +...
- 425
0
votes
1
answer
108
views
Is $C(X)$ complete if $X$ is compact? [duplicate]
Let $X$ be a compact space, $C(X)$ be the metric space of continuous functions from $X$ to $\mathbb{R}$ with sup norm.
Is $C(X)$ complete?
I can prove it when $X$ is a metric space.
- 45
0
votes
0
answers
25
views
Relative compactness of sequences and sets
I am currently solving an exercise that is concerned with relative compactness.
The space $C(X)$ is assumed is assumed to be equipped with the supremumnorm. My problem is with e). I concluded that $(...
- 1,277
3
votes
1
answer
183
views
Find self-similar solution for the heat equation
I would like to find solutions to the equation
$$\partial_t \phi = \partial_{rr}\phi + \frac{1}{r}\partial_r \phi - \frac{1}{r^2}\phi$$
of the form $\phi(t, r) = t^{-\gamma} g(r/\sqrt{t})$. This ...
- 4,368
0
votes
0
answers
64
views
Lower Bounds for the Operator norm of Integral Operators with square-integrable kernels
Let $k$ be a measurable function on $E\times E$ where $E \subset \mathbb{R}^2$, define the integral operator $L_k$ by
$$
L_k f(x) = \int_{E} k(x,y) f(y) dy, \qquad x \in E
$$
If $k$ is is square-...
- 6,198
0
votes
1
answer
52
views
Open and dense subsets of continous functions with respect to L1-norm
I am having an issue with an exercise that is concerned with the $L^1$-norm on the space of continous real-valued functions. More, specifically I am in doubt whether my solutions are correct.
I am ...
- 1,277
0
votes
0
answers
54
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Richard Feynman's Definition of Half Derivative [duplicate]
I read from Leonard Mlodinow's Feynman's Rainbow that Feynman has defined the concept of a half-derivative, an operation $g$ on a function $f$ such that
$$g(g(f))=f'.$$
I wonder if it is possible to ...
- 493
3
votes
1
answer
276
views
Is there a closed form for the linear operator $T$ such that $T(x^n) =f(n)x^{n-1}$?
When I first learnt calculus I was so surprised to learn that there is a meaningful mathematical operator $D$ that
$$
D(x^n)= n x^{n-1}.
$$
It seemed to be a very random thing to multiply the exponent ...
- 6,781
2
votes
1
answer
81
views
Does a function having the property that for every $a_n\rightarrow a$ $f(a_n)$ diverges exist?
Could you help me solve the following tricky exam problem? Let $f: [
-1, 1]\rightarrow \mathbb R$. Let $a\in [-1, 1]$. Suppose that for every not ultimately constant sequence {$a_n$}, $a_n\in [-1, 1]$,...
- 303
2
votes
0
answers
62
views
Computing the distributional derivative of a particular function
Let us consider the function
$$
f(x) =
\begin{cases}
1 & \text{if } x \in \mathbb Q \\
-1 & \text{if } x \in \mathbb R \setminus \mathbb Q
\end{cases}
$$
I have a main question about the ...
- 21
57
votes
3
answers
5k
views
A very odd resolution to an integral equation
Here is something I've found on the internet
$$\begin{aligned}
f-\int f&=1\\
\left(1-\int\right)f&=1\\
f&=\left(\frac1{1-\int}\right)1\\
&=\left(1+\int+\int\int+\dots\right)1\\
&=1+...
- 5,643
1
vote
1
answer
145
views
If a sequence is bounded, does its weak limit happen to be bounded?
Let $(H, \|\cdot\|)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$, i.e. there exists $C>0$ such that
$$\|f_n\|\le C \quad\forall n.$$
Let $f$ denote the weak limit of $f_n$.
...
- 425
1
vote
2
answers
238
views
Relationship between $L^2$-distance and cosine similarity
Given a vector space $X$, the cosine-similarity can be defined as:
$$c(x,y)= \frac{\langle x, y\rangle}{\| x \| \| y \|} $$
and distance is:
$$d(x,y) = \| x - y\| $$
First, I expect to estimate some &...
- 21
0
votes
0
answers
19
views
Functions of bounded variation - inequality
Let $u: \mathbb{R} \to \mathbb{R}$ such that $u \in C^1 (\mathbb{R}) \cap \mathrm{BV} (\mathbb{R})$. Prove that
$$\frac{1}{\varepsilon} \int_{-\infty}^{\infty} |u(x+\varepsilon) - u (x)| dx \leq TV (u)...
- 31
0
votes
1
answer
96
views
Prove that the following limit exists and is finite.
Let $f: (0, \infty)\to\mathbb{R}$ be a function that admits the antiderivative $F$ which has the propery that the limit of $\lim\limits_{x\rightarrow \infty} F(x) $ exists and is finite.
a. Give an ...
- 65
1
vote
1
answer
74
views
Is the function $ f $ here Lipschitz?
Assume that $ f:\mathbb{R}^n\to\mathbb{R} $ is a continuous function and $ E\subset\mathbb{R}^n $ is a closed set with zero Lebesgue measure, i.e. $ m(E)=0 $. Suppose that $ f\in C^1(\mathbb{R}^n\...
- 1,082
5
votes
1
answer
98
views
Inequality related to function $f(x)=x\ln x-\frac{x^2}e+ax$
I'm trying to solve a hard question about function and here is the question below:
For the given function $f(x)=x\ln x-\frac{x^2}e+ax$ (where $a$ is a parameter), we have such property:
$$\exists\;...
- 555
-1
votes
1
answer
119
views
What is the natural logarithm of d/dx? [closed]
What is $\ln\left(\frac{d}{dx}\right)f(x)$?
If $e^{\frac{d}{dx}}f(x)=f(x+1)$ is the shift operator, and we can conclude that using the taylor expansion of the exponential function, how would we ...
- 11
0
votes
2
answers
75
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Space bounded from norm real analysis [duplicate]
Question:
\begin{equation}
\text{Let} \: \: \: T : (C[0, 1], \|\cdot\|_{\infty}) \rightarrow \mathbb{R} \: \: \: \text{with} \: \: \: T(f) = \int_0^1 f(t)\,dt.
\end{equation}
Prove that $T$...
- 11
1
vote
1
answer
96
views
Necessary assumptions for direct method of calculus of variations
In "Calculus of Variations" by F. Rindler I learned about the following result (direct method of calculus of variations):
If $X$ is a topological space, $f: X \to \mathbb{R}$ lower ...
- 27
3
votes
1
answer
112
views
Convergence of summation operator on $L^2(\mathbb R)$
I am trying to determine whether or not the following translation operator sends $L^2(\mathbb{R})\to L^2(\mathbb{R})$ Let:
$$ Tf(x): f\to\sum_{n\in\mathbb{N}}f(x-\log n)n^{-1/2}$$
Here is my work so ...
- 545
1
vote
1
answer
74
views
Equivalence of norms. How do I continue?
I have been working trying to solve a problem. It is about equivalence of norms. I am very close to finishing the problem. I have obtained a lower bound and I still need to obtain the upper bound. I ...
0
votes
0
answers
33
views
Complex interpolation and distributive law
Let $X, Y, Z$ be separable complex Banach spaces.
I'm wondering whether the following holds or not:
$$
[X\cap Y,Z]_\theta =[X, Z]_\theta \cap [Y,Z]_\theta,
$$
where $\theta \in [0,1]$ and $[A,B]_\...
- 659
0
votes
1
answer
65
views
Find closure to a space of essentially supported $L^2$ functions
Let $\Omega \subset \mathbb{R}^N$, be an open and bounded domain and let:
$$L^2_* = \{ q \in L^2(\Omega)\, |\, \text{Vol(ess sup(}q)) \in [0, 0.9*\text{Vol(}\Omega\text{)}] \}$$
I want to understand ...
- 103
0
votes
1
answer
73
views
When $f_n \to f$, possible to show that also $\text{ess supp}(f_n) \to \text{ess supp}(f)$?
Let $\Omega \subset \mathbb{R}^N$ be an open and bounded domain. Consider $f, f_n \in L^2(\Omega)$, where $f_n$ is a convergent sequence in the sense:
$$\|f - f_n\|_{L^2(\Omega)} \leq \varepsilon.$$
...
- 103
0
votes
2
answers
101
views
Is the function globally Lipschitz?
Let $f : \mathbb{R}^d \rightarrow \mathbb{R}$ be continuous and vanishing at infinity. It is then known that $f$ is uniformly continuous on $\mathbb{R}^d$. Now, if $f$ is Lipschitz on compact subsets ...
- 349
1
vote
0
answers
48
views
Solution verification of continuity of a function in $L^p(\mathbb R)$
We are asked to prove the following:
Let $f\in L^p(\mathbb R)$, with $1\leq p<\infty$. Show that
$$g(x)=\int_{x}^{x+1}f(t)\,dt$$
is continuous. We did the following:
From Hölder's inequality, for $...
2
votes
0
answers
86
views
Exercise 3.19 in Brezis' Functional Analysis
Exercise 3.19:
Let $E = \ell^p$ and $F = \ell^q$ with $1 < p < \infty$ and $1 < q < \infty.$ Let $a:\mathbb{R}\to\mathbb{R}$ be a continuous function such that
$$|a(t)| \leq C|t|^{\frac{p}...
- 79
0
votes
0
answers
53
views
Do other resources exist on Rate-Independent Systems?
Are Rate Independent System new in the world of mathematics? I can't tell because I certainly am new to mathematics.
I couldn't find any more resources than this book:
Rate-Independent Systems: Theory ...
0
votes
1
answer
110
views
Does each operator $T: L^2 \to L^2$ with zero nullspace, have a continuous inverse? [closed]
Consider an operator $T: L^2 (\Omega) \to L^2(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is an open and bounded domain.
Let $T$ have a zero nullspace.
Questions. Does that imply that:
$T$ ...
- 103
1
vote
0
answers
40
views
"Isometrically isomorphic" Hilbert spaces: norm equivalence?
The quotient space $L^2/M$ is isometrically isomorphic to $M^\perp$.
Let $M=\mathbb{R}$ and $[x] = x + \mathbb{R}$ for some $x \in M^\perp$.
Q1: Does the norm equivalence:
$$ C_1\|x\|_{L^2} \leq \|[x]...
- 103
-1
votes
1
answer
77
views
Is $f \in H_0^1(\Omega)$ compactly supported? [closed]
Let $\Omega \subset \mathbb{R}^N$ be an open and bounded domain. Furthermore let:
$$H_0^1(\Omega) = \{f \in H^1(\Omega)\, |\, f|_{\partial\Omega}=0 \}.$$
$f \in H_0^1(\Omega)$ is said to be compactly ...
- 103
0
votes
0
answers
53
views
Do $L^2$ functions with essential support have a finite average?
Let $\Omega\subset \mathbb{R}^N$ be an open and bounded domain, and $f\in L^2(\Omega)$, where $f$ is further chosen such that it is essentially supported on $\Omega$.
Question. Is the average of such $...
- 103