Questions tagged [schwartz-space]
For question about the Schwartz space, a vector space of smooth functions stable under the Fourier transform.
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Almost selfadjointness in tempered distributions
Let $\mathbb{F}=\{f^{(s)}\}_{s\in{\mathbb{R}^d}}$ a collection of tempered distributions in $\mathbb{R}^d$ and $F=\{h\in S(\mathbb{R}^d)\,:\,\mbox{the map }s\mapsto f^{(s)}(h)\mbox{ is in }S(\mathbb{R}...
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1answer
37 views
Equivalence of norms on Schwarz space
Consider the following norms on the Schwarz space, for $1\leq q\leq \infty$
$$\lVert f \rVert_{\alpha,\beta, p}=\lVert x^{\alpha}\partial^\beta f\lVert_{L^p}$$
I want to show that the norms $\lVert \...
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1answer
42 views
Fourier transform of $\varphi_m(u)=\int |x|^mu(x)dx$
I'm stuck with the following problem
Let $\varphi_m \in \mathcal{S}'(\mathbb{R}^{n})$, $n \in \mathbb{N}$, $m\in \mathbb{C}$, $0 >\text{Re}(m)>-n$ the distribution defined by
$$\varphi_m(u)...
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0answers
73 views
Definitions of Sobolev Spaces - are they the same?
I have read two definitions of Sobolev spaces.
Definition 1: We let $\lambda$ denote $\lambda^s(\xi)=(1+|\xi|^2)^\frac{s}{2}$ for $s \in \Bbb R$, $\xi \in \Bbb R^n$. We say that $u \in H^s$, if $u ...
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0answers
14 views
Inequality on Schwartz functions
Let $w:\mathbb R^n\to \mathbb R$ be a fixed strictly positive Schwartz funcion with the following property:
$w(x)\geq 1$ on the unit ball $B(0,1)$ in the physical space.
$\widehat {w^{1/2}}$ is ...
3
votes
0answers
28 views
Functions in the Schwartz space $\Bbb S $
I should decide if these functions belong to Schwartz space $\Bbb S $:
$f_1(x) = \frac {1}{1+|x|^4}$
$f_2(x)= e ^{-|x-3|^4}$
$f_3(x)= e ^{-|x|}$
The Schwartz space is a space where the functions ...
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0answers
35 views
Fourier transform on Schwartz Space is a continous automorphism
I am currently trying to understand the Fouriertransform of Schwartz functions. There are two proofs I'm trying to figure out.
$\mathcal{F}$ is a continous operator $\mathcal{S}(\mathbb R) \...
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1answer
30 views
Are radial Schwartz functions tensored harmonic polynomials dense in the Schwartz space?
Let $p \geq 2$ be an integer and denote by $\mathcal{S}(\mathbb{R}^p)$ the space of Schwartz functions on $\mathbb{R}^p$. Let $\mathcal{S}(\mathbb{R}^p)^{O(p)}$ be the subspace of rotation-invariant ...
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0answers
54 views
Is $f(x) = \exp(-|x|)$ in the Schwartz space?
If I have a function given as $f(x) = \exp(-|x|)$, is this function in the Schwartz space?
As a common example of the Schwartz function is considered $ e^{x^2}$. But in my example I got a absolute ...
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2answers
61 views
How to properly estimate a Schwartz function
For every Schwartz function $\varphi \in S(\mathbb{R}^{n})$ there exists a constant $c_{\beta, k}$ such that one can estimate the Schwartz function by $$|\partial^{\beta}\varphi(x)| \leq \frac{c_{\...
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1answer
45 views
On the definition of Schwartz functions
$$
\mathcal{S}(\mathbb{R}^{n})=\Big\{f\in C^{\infty}(\mathbb{R}^{n})\,\Big|\; \sup_{\mathbb{R}^{n}}|x^{\alpha}D^{\beta}f(x)|<\infty\, \forall \alpha,\beta\, \text{multi indexes}\Big\}.
$$
I can't ...
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votes
1answer
40 views
Properties of Schwartz functions
Let $f$ in Schwartz space, and $K, \gamma >0$.
Then I guess the following:
\begin{align}
\int _{ -k }^{ -\gamma }{ \left| f(x) \right| dx } +\int _{ \gamma }^{ k }{ \left| f(x) \right| dx } &=...
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0answers
48 views
Convergence in Schwartz Space of Fourier transforms?
Let $f_n$ a sequence in Schwartz space. Suppose that $\lim _{ n\rightarrow \infty }{ { \left\| { f }_{ n }-f \right\| }_{ \alpha ,0 }=0 } $, where ${ \left\| f \right\| }_{ \alpha ,0 }=\underset { ...
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0answers
33 views
Find all solutions $T$ of $x^{2006} T = 0$ in the space of tempered distributions, $\mathcal{S}'(\mathbb{R})$
I'm modeling my solution after this answer to a similar question. This is as far as I've gotten:
Every $\phi \in \mathcal{S}(\mathbb{R})$ that vanishes at $0$ can be expressed as $\phi(x) = x \psi (x)...
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1answer
65 views
Show that the function $f$ is a tempered distribution
Let $f$ be a locally integrable function on $\mathbb{R}^n$ such that $f$ is of polynomial growth at infinity. Prove that $f$ is a tempered distribution.
Here the distribution is defined as $f(\phi)=\...
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1answer
89 views
For a Schwartz function $f$, if $\int_{\mathbb{R}} f(x) x^n dx = 0$ for all nonnegative integers $n$, is $f$ identically 0?
This is an old exam question I'm practicing with. The associated hint is to use the Fourier transform. I'm pretty lost, but here are my thoughts so far.
First, in this old stack exchange question a ...
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1answer
54 views
How do you show that $e^{-a\sqrt{1+\|x\|^2}}$ is in the Schwartz space
I want to show that $e^{-a\sqrt{1+\|x\|^2}}$ is in $\mathcal{S}(\mathbb{R}^d)$. ($a>0$)
Please tell me proof.
Where,
$$\|x\|^2 = \left(\sum_{j=1}^{d} |x_j|^2\right)$$
$$ f(x) \in \mathcal{S} \...
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1answer
80 views
Equivalent definition of Schwartz space
Please tell me about the equivalent definition of schwartz space.
Definition of Schwartz space is the following.
$$ f(x) \in \mathcal{S} \overset {\mathrm{def}} {\Leftrightarrow} \displaystyle \sup_{...
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1answer
51 views
Differentiating Fourier Transform of a Schwarz function
My question is related to this question Conditions for the differentiation of Fourier Transform of a function. Suppose my function $f(x)$ is from a Schwarz function https://en.wikipedia.org/wiki/...
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0answers
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Constructing a Schwartzfunction
Hi I am working on a problem and as a hint I received the following:
Construct a Schwartz function $\phi \in \mathscr s (\mathbb R^n) $which disappears outside a neighbourhood P but $\phi (x) = 1$ for ...
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0answers
24 views
Inequality in the sense of distributions
What does
$$
f\leq g
$$
mean for two Schwartz distributions/generalized functions $f $ and $g$? I assume it means that the inequality holds when tested with positive functions. The next question ...
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votes
1answer
46 views
Is the Schwartz topologically emebedded in space of tempered distributions?
Let $g\mapsto ( \cdot, g)_2$ denote the map from the Schwartz space $S$ into its dual space $S'$ where $(f,g)_2$ is the inner product in $L^2$. Then is this a linear topological embedding ($S'$ is ...
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1answer
19 views
Wk* continuity and wk* sequential continuity on $S'$
Let $S$ be the Schwartz space and $S'$ be its dual space. If a linear operator $T:S'\rightarrow S'$ is weakly* sequentially continuous, is it also weakly* continuous?
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1answer
93 views
Is translation continouos in Schwartz space?
It is true that translation is continuous in Schwartz Space {S}($\mathbb{R}$) with its topology?, in other words, I'm trying to prove that if $\phi \in {S}\left(\mathbb{R}\right)$ then the function $\...
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0answers
20 views
For which functions/distributions X, $S(\mathbb R) \ast X \subset \ell ^{1}({\mathbb Z})$?
Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz space) and define $f_k(x) =f(x-k) \ (k\in \mathbb Z^d)$ (translation of $f$).
Question: For which kernel $h:\mathbb R^d \to \mathbb C,$ one can ...
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vote
1answer
41 views
How to deduce that the rotation matrix is constant from the following?
As it is clear for everyone the deformation gradient $\mathbf{F}$ can be decomposed into,
$\mathbf{F}=\mathbf{Q} \cdot \mathbf{U}$
where $\mathbf{Q}$ is the rotation matrix.
Now if I consider $\...
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0answers
64 views
On linear operators on Tempered Distributions
Consider $\mathscr{S}$ the Schwartz space of rapidly decreasing complex smooth functions over $\mathbb{R}^{d} $, equipped with its usual metric topology, and S′ its topological dual, the space of ...
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0answers
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On reflexivity properties of the Schwartz space
Consider $\mathscr{S}$ the Schwartz space of rapidly decreasing complex smooth functions over $\mathbb{R}^{d}$, equipped with its usual metric topology, and $\mathscr{S}'$ its topological dual (the ...
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1answer
33 views
Given $a>0$, $\frac{1}{x^2+a^2}$ is not a Schwarz function.
Given $a>0$, $f(x) = \frac{1}{x^2+a^2}$ is not a Schwarz function.
Please verify if this is correct:
Although Poisson Sumation formula is working for this function $f$, I think it is not Schwarz,...
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votes
1answer
226 views
The convolution of functions in the Schwartz Space lies in the Schwartz Space
I'm reading Fourier Analysis by Rami Shakarchi and Elia.M.Stein. When I started reading the chapter about Fourier Transform on R I came across very difficult inequalities. One of them take me a long ...
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0answers
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Fourier transformation on $\{T_\theta\}$ is surjective
Let $S$ Schwartz space on $\mathbb{R}^2$. For $f\in S$, let $T(f):=\int _{-\infty}^{\infty} f(0,y)dy$ and $T_{\theta} (f):=T(f(x \cos{\theta} -y \sin{\theta}, x\sin{\theta} +y\cos{\theta}))$ and we ...
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0answers
99 views
Applying the generalised Leibniz rule multi-index form
The multi-index Leibniz rule states $\partial^{\alpha}(fg) = \sum_{\beta \leq \alpha}{\alpha \choose \beta} (\partial^{\beta}f)(\partial^{\alpha - \beta}g)$
Where $ \alpha, \beta$ are multi-indices.
...
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0answers
103 views
Understanding Schwartz Space / How to show a function is or is not in Schwartz Space
I'm confused about Schwartz Space. In particular I'm finding it difficult to show a function is / isn't in Schwartz space.
Definition:The Schwartz Class $\mathcal{S}(\mathbb{R}^{n})$ is the set of ...
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1answer
41 views
Use Dominated Convergence theorem in Schwartz convergent sequence
Let $\{f_k\}_{n \in \mathbb N} \subset S(\mathbb R^n)$ converging to $f \in S(\mathbb R^n)$ in the $S(\mathbb R^n)$ topology.
Is it true that $\int_{\mathbb R^n} f_k$ converges to $\int_{\mathbb R^n} ...
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votes
0answers
50 views
Structure of tempered distribution
I was reading a note on distributions. The author left the following Schwartz representation theorem as an exercise:
I'm trying to prove the theorem. According to the hints, I've done all the ...
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0answers
52 views
Translation invariance of distributions
Let $\mathcal{S}(\mathbb{R}^4)$ denote the Schwartz space on $\mathbb{R}^4$, $\mathcal{S}^{\otimes n}(\mathbb{R}^4)$ the tensor product of $n$ of these spaces and $\mathcal{S}(\mathbb{R}^{4n})$ the ...
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0answers
80 views
Prove that $ \Vert (\sum_{j \in \mathbb{Z}} |\phi_{2^j} * f|^2)^{1/2}\ \Vert_p \le C\Vert f\Vert_p $
I was trying to prove the following inequality:
Given $\phi$ a Schwartz function in $\mathbb{R}^d$ such that $\int\phi=0$. Let $\phi_t(x)=t^{-d}\phi(xt^{-1})$.
I need to prove that:
$$ \Vert (\sum_{...
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votes
3answers
105 views
Abuse of notation with distributions
A distribution is an element of the continuous dual space of some function space. Let us take the Schwartz space $\mathcal{S} := \mathcal{S}(\mathbb{R}^n)$ just as an example. A distribution $\phi \in ...
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0answers
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Schwartz seminorm of Fourier transform of $\phi$ bounded above by a Schwarz seminorm of $\phi$
The inequality I am trying to prove is the following:
For all $k,m \in \mathbb{N}$, there exists constants $C > 0$, $k', m' \in \mathbb{N}$ such that
\begin{align} q_{k,m}(\hat{\phi}) \leq C q_{k',...
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1answer
214 views
Homogeneous Sobolev space and distributions
The standard way to define the homogeneous Sobolev space $\dot{H}^m(\mathbb R^n)$ for $m\in \mathbb N$ is via the condition $f\in \dot{H}^m(\mathbb R^n)$ if and only if $f \in \mathscr S' (\mathbb R^n)...
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0answers
42 views
Smooth tempered distribution
We have $ f \in \mathcal{C}^{\infty}( \mathbb{R}) $ such that
$ \forall n \geq 0 \:, \exists C_n, \: \: |f^{(n)}(x)| \leq C_n (1 + |x|)^{2-n} $
Show that the distribution $ T_f $ is ...
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vote
1answer
20 views
Trying to show $\mathcal S(\Bbb R)\subset L^p(\Bbb R)\ \forall 1\leq p\leq\infty$
$$\mathcal S(\Bbb R):=\big\{f:\Bbb R\mapsto\Bbb R\ :\ \forall k,l\in\Bbb N \ \sup\limits_{x\in\Bbb R}\{|x^k|\cdot|f^{(l)}(x)|\}<\infty\big\}$$
I am trying to show that $\mathcal S(\Bbb R)\subset ...
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vote
1answer
31 views
A function on $\mathcal{S}(\mathbb{R}^n)$ written as a combinations of Schwartz functions
I am having a hard time trying to figure out how to do this exercise:
Let $\psi \in \mathcal{S}(\mathbb{R}^n)$ be such that $\psi(0)=0$. Prove that there are functions $\rho_j \in \mathcal{S}(\mathbb{...
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1answer
96 views
convergence for tempered distributions
$\mathcal S(\mathbb R^n) := \{f \in C^{\infty}(\mathbb R^n): \|\varphi\|_{k,l} < \infty \}$ where $\|\varphi\|_{k,l} := \operatorname{sup}_{x \in \mathbb R^n} (1 + |x|^2)^{k \over 2} \sum_{|\alpha| ...
2
votes
1answer
47 views
Proving existence of a Brownian functional
Let $W$ be a standard Brownian Motion indexed by $\mathbb{R}$, i.e., $W\restriction_{\mathopen{[}0,\infty\mathclose{[}}$ is a standard Brownian Motion, and $W\restriction_{\mathopen{]}-\infty,0\...
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1answer
112 views
Definition of Schwartz space
We have the following definition of Schwartz space $S(\mathbb{R}^k)$: we tell that the function $\psi: \mathbb{R}^k \to \mathbb{C}$ is in $S(\mathbb{R}^k)$ iff
1. $\psi \in C^{\infty}(\mathbb{R}^k)$ ...
2
votes
0answers
111 views
A “Schwartz function” of Schwartz function?
Let $f_1:\mathbb R\to \mathbb R$ be a Schwartz function, and $\mathcal S(\mathbb R)$ be the space of Schwartz functions (Schwartz space). $\mathcal B(\mathcal S)$ is the Borel $\sigma$-field of $\...
3
votes
2answers
60 views
$L^p(\mathbb R^n) \subset \mathcal S'(\mathbb R^n), 1 \le p < \infty$
For $p$ = 1, it's clear because for $f \in L^1(\mathbb R^n)$, one can define a tempered distribution by $T_{f}(\varphi) := \int_{\mathbb R^n} f(x) \varphi(x) dx$.
Apparently, one can make the ...
1
vote
1answer
270 views
Equivalent definitions of Schwartz Space
I'm trying to show the following but have no idea how to begin. I'm quite new to analysis and multi-index notation.
$$ f \in \mathcal{S} \quad \Longleftrightarrow \quad \forall N \in \mathbb{N}, \...
4
votes
1answer
153 views
If $\hat{f}\in L^2(\mathbb{R})$ then $\hat{f}$ is rapidly decreasing.
This is problem 2.3.6 from the book of Mckean, "Fourier Series and Integrals".
Problem:
Check that if $\hat{f}\in L^2(\mathbb{R}^1)$ then $\hat{f}$ is rapidly decreasing in the sense of $\gamma^n\...
