Tagged Questions
5
votes
1answer
62 views
Taylor series and tempered distributions
Suppose we have a function $\psi$ in $\mathbb{R}$ other than a polynomial which is equal to its Taylor Series for every point in $\mathbb{R}$. Why is the following statement valid?
When we interpret ...
1
vote
0answers
31 views
Can we say approximation by smooth function is an equivalent form of “Weierstrass approximation” theorem in Sobolove?
As I came to know most of properties characterized by approximation by smooth functions in Sobolev space looks equivalent to that of Weierstrass approximation theorem in the space of continuous ...
1
vote
1answer
61 views
Inverse fourier transform 3 dimensions
Hoi, I want to show that for $n=3$ that $$\mathcal{F}^{-1}\left(\frac{1}{1+|s|^2}\right) = \frac{1}{4\pi |x|}e^{-|x|} $$
As a hint I've been given: Its the unique solution to the equation ...
1
vote
1answer
34 views
Multiplication of distributions by smooth functions
Let $u\in D'(\mathbb{R})$ and $f\in C^{\infty}$. I'm trying to figure which of the following statements is true:
I. If $f\restriction_{supp(u)}=1$ then $f\cdot u=u$.
II. If ...
1
vote
1answer
60 views
Calculation of the Laplacian of a function in $\mathbb{R}^3$.
I have to calculate the Laplacian distributional sense) of the following function
$$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$
with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
2
votes
1answer
49 views
Extending a distribution continuously to $C_c^N (\Omega)$
Let $\Omega \subseteq \mathbb{R}^n$ be a domain and let $u\in D'(\Omega)$ be a distribution of order $\leq N$.
How can we show that $u$ can be continuously extended to $C_c^N(\Omega)$?
By ...
0
votes
1answer
84 views
Is this integral 0?
Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$
Is there a way we can conclude ...
0
votes
2answers
110 views
example of a function with compact support
Can you give an example of a function which is $C^\infty (\mathbb{R})$ having support on (-1,1) such that $ \int_{-1}^1 f(x)\,dx$=1 and $ \int_{-1}^1 xf(x)\,dx$=0.
Thank you.
1
vote
0answers
19 views
Function of the incremental ratio tends weakly to a distribution
Let $g:\mathbb{R}^3\to\mathbb{R^2}$ be a continuous function. Suppose that there exists $\Omega$ a neighborhood of $0$ where $Xg, Yg \in L^\infty(\Omega)$, with
...
2
votes
1answer
45 views
A particular product of distributions
Suppose you have two continuous functions $f,g: \mathbb{R}\to\mathbb{R}$; is the product $f'g$ as a distribution, at least locally? I am interested in a local result, actually, so you can as well ...
1
vote
1answer
64 views
The Dirac impulse and Fourier transform
Here wikipedia it is said that the Dirac delta could be thought of as
$$
\delta(x) = \left\{ \begin{array}{ll} \infty &, x = 0 \\ 0 &, x \ne 0 \end{array}\right.
$$
and here that the ...
2
votes
0answers
109 views
Proof that limit goes to zero without Riemann-Lebesgue lemma
Let $\varphi$ be a test function ($\varphi$ is smooth and has compact support - is zero outside some bounded interval).
I know that the following
$$ \lim_{\epsilon \to 0_+} ...
1
vote
0answers
45 views
Lebesgue-Stieltjes integral as a generalized function
Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral
$$
\langle F, \varphi \rangle ...
3
votes
1answer
112 views
Second derivative of convex function
Let $f(x)$, $x>0$ be a convex function. Then it's distributional second derivative is defined by the rule
$$
\langle f''(x),\varphi(x)\rangle = \langle f(x), \varphi''(x)\rangle
$$
for any ...
1
vote
0answers
19 views
Solve an equality for distribution
I have an equality that holds for any $\lambda > 0$
$$
\int\limits_{0}^{\infty}{e^{-\lambda t^{\alpha}}} T(t)dt = \int\limits_{0}^{\infty}e^{-\lambda t^{\alpha}} dg(t),
$$
where $\alpha > 0$ ...
3
votes
1answer
111 views
Show the usual Schwartz semi-norm is a norm on the Schwartz space
Let $f \in C^\infty(\mathbb R)$. Define the semi-norm
$$
\|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)|
$$
where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$.
Show ...
4
votes
1answer
82 views
How to prove that $x \rightarrow e^{1/x}$ is not a restriction of any real distribution to $ \mathbb {R}_+$?
This is an excercise 2.2 from Hormander, vol. I:
Does there exist a distribution $u$ on $\mathbb{R}$ with the restriction
$x \rightarrow e^{1/x}$ to $\mathbb{R}_+$?
The answer, provided in the book, ...
2
votes
1answer
127 views
Schwartz space: semi norm estimate on translation
the following family of semi norms is commonly used to introduce the space of Schwartz functions $\mathcal{S}(\mathbb{R}^n)$:
$$
\|\phi\|_N := \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha|\,,|\beta| ...
0
votes
1answer
73 views
How does a myopic interpret Wiener's Tauberian?
I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made ...
2
votes
1answer
45 views
Laplacian in $\Bbb R^2$ acting on compact test-function
I am trying to follow an argument in Strichartz's "A Guide to Distribution Theory and Fourier Transforms"
We consider $\langle \Delta u, \rho \rangle$ where $\Delta u$ is the two dimensional ...
2
votes
1answer
96 views
generalized functions (Distributions) elementary question
I am working with Strichartz's "A Guide to Distribution Theory and Fourier Transforms" (self-study -> not a homework question). He says none of the distributions that correspond to $1/|x|$ are ...
-1
votes
1answer
119 views
Approximating an integral
Might be simple, but i don't get it. Why is the integral in the last line approximately equal to $n(\varphi(\frac{-1}{2n}) - \varphi(\frac{1}{2n}))$?
4
votes
1answer
135 views
Questions concerning a proof that $\mathcal{D}$ is dense in $\mathcal{S}$.
I am currently working through
this lecture notes
and on page 164, there it is said
The space of $\mathcal{D}(\mathbb{R}^n)$ of smooth complex-valued functions with compact support is contained ...
1
vote
1answer
36 views
connection between the support and the representation of a distribution
I want to show, that for $u' \in \mathcal{D}'(\mathbb{R}^n)$
supp $u$ = $\{ 0 \}$ iff there exist numbers $m \in \mathbb{N}, c_{\alpha} \in \mathbb{K}$ such that $u = \sum_{|\alpha| \le m} c_{\alpha} ...
0
votes
2answers
65 views
why is $\frac{\phi(x)-\phi(-x)}{x}$ for smooth $\phi$ bounded at $x=0$
Why is $\frac{\phi(x)-\phi(-x)}{x}$ for smooth $\phi$ bounded at $x=0$? If i set $\phi(x) = \sqrt{|x|}$, it definitely not bounded. I saw this on page 293 of
...
2
votes
2answers
224 views
principal value as distribution, written as integral over singularity
Let $C_0^\infty(\mathbb{R})$ be the set of smooth functions with compact support on the real line $\mathbb{R}.$ Then, the map
$$\operatorname{p.\!v.}\left(\frac{1}{x}\right)\,: ...
2
votes
1answer
74 views
what means a integral exists in the distributional sense?
what exactly means if an integral exists just in the distributional sense, for example the fourier-transform of $x^2 e^{-\lambda x}$ or of $H(R-|x|)$ where $R > 0$ and $H$ is the ...
0
votes
2answers
201 views
distributional derivative
I want to calculate the first and second distributional derivate of the $2\pi$-periodic function $f(t) = \frac{\pi}{4} |t|$, it is
$$
\langle f', \phi \rangle = - \langle f, \phi' \rangle = ...
1
vote
1answer
64 views
How can an ordinary function be a distribution?
I think distributions are linear and continuous functionals on the set of testfunctions. In a textbook I saw this question:
Let $f$ be a $2\pi$-periodic function with $f(t) = \frac{\pi}{4}|t|$
...
1
vote
1answer
102 views
Homogeneous and rotational invariant distribution
If $u \in \mathcal D'(\mathbb R^n)$, $u$ is homogeneous of degree $0$ and rotational invariant, it is necessarily that $u$ is a constant? (Since if $u \in C^\infty$, the conclusion obviously hold.)
3
votes
2answers
190 views
A sufficient condition for a function to be of class $C^2$ in the weak sense.
Let $f\colon\mathbb{R}\to\mathbb{R}$ be a continuous function with weak derivative (i.e. the derivative in the sense of distribution) in $C^1(\mathbb{R})$. Does this condition imply that $f$ is two ...
4
votes
3answers
181 views
Borel Measure such that integrating a polynomial yields the derivative at a point
Does there exist a signed regular Borel measure such that
$$ \int_0^1 p(x) d\mu(x) = p'(0) $$
for all polynomials of at most degree $N$ for some fixed $N$. This seems similar to a Dirac measure ...
6
votes
1answer
288 views
Tensor products of functions generate dense subspace?
Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
4
votes
2answers
360 views
On distributions over $\mathbb R$ whose derivatives vanishes
Let $I \subset \mathbb R$ be open, $u \in \mathcal D'(I)$ be a distribution whose distributional derivatives vanishes (i.e. is zero for all test functions, which we may assume to be complex valued ...
