Questions tagged [schwartz-space]
For question about the Schwartz space, a vector space of smooth functions stable under the Fourier transform.
345
questions
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16 views
Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass
It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...
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34 views
Is it true that $\widehat{f(x,0)}(\xi) = \widehat{f}(\xi,0)$?
Let $f \in S(\mathbb{R}^2)$ and $g(x)= f(x,0)$. What is the relationship between $\widehat{g}$ and $\widehat{f}$?
It's clear that $g \in S(\mathbb{R})$ so both $f$ and $g$ have a Fourier transform.
I ...
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0answers
6 views
Find all $K \in S(R)'$
Find all $K \in S(R)'$ such that $T_m : L_1(\mathbb{R}) \rightarrow L_1(\mathbb{R})$ defined by $T_m(f) = K * f$ for all $f \in S(R)$. Where, $K *f$ is convolution of $K$ and $f$.
I am not sure what ...
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0answers
60 views
Is there a metrizable topology on tempered distributions for which sequential convergence is pointwise?
Is there a metrizable topology on $S’$ (the set of linear continuous functionals in Schwarz’s space) whose convergent sequences are the sequences that converge pointwise?
The obvious topology with ...
2
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1answer
83 views
Show that $f(x) = e^x \cos(x)$ on $\mathbb{R}$ is a tempered distribution
As shown in the title.
I know that the anti-derivative of $f(x) = e^x \cos(x)$ is $ \frac{\sin(x)+\cos(x)}{2} e^x$, whose anti-derivative is $\frac{\sin(x)}{2}e^x$...but not sure how to prove it.
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33 views
Can I generalize the classical Bernstein inequality?
I have know the proof of the classical Bernstein inequality as follows.
Given some function $ f\in S(\mathbb{R}^d) $, where $ S(\mathbb{R}^d) $ denotes the Schwartz space of functions. Let $ \hat{f} $ ...
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1answer
42 views
A problem regarding a generalization of Schwartz space
Consider a space of functions $f:\mathbb{R}^n\to \mathbb{C}$ satisfying
$$\sup_{x\in \mathbb{R}^n}|x^\alpha f(x)|<\infty, $$
for any multi index $\alpha$.
This space contains Schwartz space. Does ...
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1answer
45 views
Contraction principle and ODE
I would like to find a bounded $C^{\infty}$ solution of the following equation:
$$- \frac{d^2}{dx^2} u + u -\epsilon u^2 = f,$$
where $f$ is a function in the Schwartz space and $\epsilon >0$ is ...
2
votes
2answers
88 views
Show that $0$ is the only distribution solution of $u''-u=0$ in $\mathcal{S}'(\mathbb{R})$
Show that $0$ is the only distribution in $\mathcal{S}'(\mathbb{R})$ that satisfies $u''-u=0$.
So, I have that for each $\varphi\in\mathcal{S}(\mathbb{R})$ $$0=\langle u''-u,\varphi\rangle = \langle ...
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0answers
18 views
L1 control of subgaussian using decay function and decay fourier
I have a subgaussian with pdf $p(x) \in SG(a,C)$, where $SG(a,C)$ is the set of pdf of all random variables $X \in \mathbb R^d$ s.t. $\mathbb E e^{aX^2} < C,$ i.e. a set of subgaussian pdf.
I wanna ...
4
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0answers
38 views
Generalized form of the Bernstein inequality
I'm using Muscalu and Schlag's book to study harmonic analysis by myself and I encountered the following problem:
Take an arbitrary function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\...
1
vote
2answers
46 views
Bound $L^1$-norm of Schwartz function using fourier transform
I have a function $f \in \mathcal S({\mathbb R^d})$, where $\mathcal S(\cdot)$ denote the Schwartz Space. In this case, the fourier transform of $f$:
$$
\hat f (x) = \int_{\mathbb {R}^d} e^{i \langle ...
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28 views
Regarding a property of Schwartz class.
Let $f\in\mathcal{S}(\mathbb{R}^n),$ the Schwartz class. Is it true that the function $g$ defined on $\mathbb{R}_+=[0,\infty),$ by
$$g(r)=\int_{S^{n-1}} f(rw)d\sigma(w) $$
also in Schwartz class $\...
1
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1answer
39 views
Uniform convergence and Schwartz space
Consider a function $f \in S(\mathbb{R})$ in the Schwartz space and define the sequence of functions $(f_n)_{n\in \mathbb{N}}$ by $f_n(x): = f(\frac{x}{n})$. I would like to show that $f_n$ does not ...
1
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2answers
117 views
Hilbert transform of Schwartz function is in $L^1$ iff $\int_{\mathbb R} f=0$
Suppose $f:\mathbb R \to \mathbb R$ be a Schwartz function. Then the Hilbert transform $Hf\in L^1(\mathbb R)$ iff $\int_{\mathbb R} f(x) \mathrm dx=0$.
I could prove $Hf\in L^1(\mathbb R)$ implies $\...
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0answers
20 views
Does $fT_n \to fT$ for a Schwartz Function and Tempered Distribution?
Let $f \in \mathcal{S}(\mathbb{R})$ and $T_n$ be a sequence of tempered distributions converging to $T$, which is also a tempered distribution. Is it true that $fT_n \to fT$?
My solution to this seems ...
5
votes
1answer
149 views
If $\phi$ is in $\mathcal S$ and $\phi(0)=0$, is $\int_0^1 \nabla \phi(tx)dt$ in $\mathcal S$ too?
Let $\phi\in\mathcal S(\mathbb R^n)$ be a Schwartz function vanishing at $0$. Then the FTC implies
$$ \phi(x) = x\cdot \int_0^1 \nabla \phi (tx)\ dt.$$
Let this integral term be $\psi(x) := \int_0^1 \...
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0answers
8 views
Sufficient and necessary condition for a linnear combination of Dirac derivative distributions to be a tempered distribution
I have to find a necessary and sufficient condition for $(a_n)_{n\in \mathbb{Z}}$ so that $\underset{n\in \mathbb{Z}}{\sum}a_n\delta^{(k)}_n$ is a tempered disctribution where $$\left\langle\underset{...
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0answers
15 views
Show that the finite part of $\frac{1}{x_+}$ is a distribution
I would like to show that the finite part of $\frac{1}{x_+}=\mathbb{1}_{[0,+\infty[}\frac{1}{x}$, noted Fp$\left(\frac{1}{x_+}\right)$ and given by $$\left\langle\text{Fp}\left(\frac{1}{x_+}\right),\...
1
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1answer
32 views
Does a dense subspace of test functions induce the (weak-$\ast$) topology on the space of distributions?
Suppose $E$ is a dense subspace of $\mathcal{S}(\mathbb{R}^n)$. Is it true that the topology on $\mathcal{S}’(\mathbb{R}^n)$ induced by the linear functionals $u\mapsto u(\phi)$, $\phi\in \mathcal{S}(\...
2
votes
1answer
54 views
About convergence in Schwartz space
The Question
Let $f,\varphi\in\mathcal S(\mathbb R^d)$, $\varepsilon>0$ and denote $\varphi_\varepsilon(x)=\frac{1}{\varepsilon^d}\varphi\left(\frac
x\varepsilon\right)$. Prove that $\varphi_\...
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0answers
25 views
Density of Schwartz functions in $L^2$ on a compact domain
I'm trying to prove that the space $AC^2([0,1])$ of absolutely continuous, complex-valued functions defined on $[0,1]$, with the property that their derivatives are in $L^2([0,1])$ is dense in $L^2([0,...
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0answers
6 views
Riesz Interpolation for Bourgain space, and a question about Schwartz space
1)If I have an operator $T$ s.t $||Tv||_{X_{-s,0}} \le c_1 ||v||_{X_{-s,1-b}}$
and $||Tv||_{X_{-s,1-b}} \le c_2 ||v||_{X_{-s,1-b}}$. Where $X_{s,b}$ is the Bourgain space .Then how to apply Riesz ...
-2
votes
1answer
17 views
A function in the Schwartz space composed with a linear transformation.
Let $A$ a $n\times n$ matrix such that $\det(A)\neq0$, $f\in S(\mathbb{R}^n)$ in the Schwartz space, then $f\circ A$ is in the Schwartz space.
4
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1answer
32 views
If $f$ is Schwartz, does $f=\mathcal{O}\left(e^{-a|x|^{\epsilon}}\right)$ for some $a,\epsilon>0$?
If $f$ is Schwartz, i.e. $f\in\mathcal{S}\left(\mathbb{R}\right)$, is it true that$f=\mathcal{O}\left(e^{-a|x|^{\epsilon}}\right)$ for some $a,\epsilon>0$ as $|x|\to\infty$?
Functions in the ...
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1answer
46 views
Exchanging limit and integral
If $\sup_{x \in \mathbb{R}^n} |f_m(x)|\rightarrow 0$ for $f$ a Schwartz function (i.e. $f$ is smooth and decays rapidly), is it true that $\|f_m\|_{L^1}=\int_{\mathbb{R}^n} |f_m(x)| dx \rightarrow 0$ ...
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2answers
56 views
Fourier Transform of a compactly supported distribution : how to write the definition in a “better way”?
I am really troubled by the following Theorem :
*Theorem : If $u\in \mathcal{E}'(\mathbb R^d)$ (space of compactly supported distributions), then its Fourier transform is a function, and :
$\hat{u}(\...
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0answers
21 views
Definition Schwarz Function : Which sense to give to a polynomial of R^d ? and relation with multi-indexes generalization
In most of definitions , but we can take the example of Wikipedia EN of Schwartz space, people use the terminology of "monomial" or "polynomial" (depending of if we take a sum or ...
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1answer
54 views
An application of taylor formula
Let $\phi\in \mathcal{S}(\mathbb{R}^n)$, i.e. $\phi$ is a Schwartz function, is true that:
$$|2\phi(x)-\phi(x+y)-\phi(x-y)|\leq |D^2\phi(x)||y|^2,\quad\forall y\in\mathbb{R}^n???$$
I have no idea on ...
0
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1answer
24 views
Fourier transform wrt spatial variable of a distribution
Let $T\in\mathcal{S}'(\mathbb{R}^{n+1})$ be a tempered distribution, i think that:
$$ \mathbb{R}^{n+1}=\{(x,y):x\in\mathbb{R}^n,y\in\mathbb{R}\}.$$
How i can define the Fourier transform of $T$ wrt $x$...
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0answers
15 views
Fourier transform of $f(x)=\frac{1}{|x|^{n-\alpha}}$
What is the Fourier transform of $f(x)=\frac{1}{|x|^{\alpha}}$ s.t $0<\alpha<n$ using polar coordinates for $n \ge 3$ , where $\mathcal{F}f(x)=\int_{\mathbb{R^n}} f(x)e^{-2i\pi x\xi}dx$
I ...
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0answers
27 views
What does $|x|^2$ mean in this proof that the Schwartz space is a subspace of $L^p$?
The following is supposed to be a proof that the Schwartz space is contained in $L^p(\mathbb{R}^n)$ for $1 \leq p <+\infty$:
Given $f \in \mathcal{S}(\mathbb{R}^n),$ there is a constant $K>0$ ...
1
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1answer
28 views
Schwartz functions after periodization has period 1
We define periodization of a function $f$ as
$$
P_1f(x) = \sum_{n\in N} f(x+n) = \lim_{N\rightarrow \infty} \sum_{|n|\leq N} f(x+n).
$$
To show that $P_1f$ has period 1, my idea was to do a change of ...
3
votes
2answers
66 views
Approximating compactly supported $L^2$ functions with Schwartz functions “from within”?
Crossposted from MathOverflow.
It is well known that the class of Schwartz functions $\mathcal{S}$ in dense in all $L^p$ spaces therefore for each $f \in L^2$ there exists a sequence of Schwartz ...
1
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1answer
82 views
Approximate a positive Schwartz function
Context: let $f \in \mathcal{S}(\mathbb{R}^+) $ be a function of the Schwartz space (all functions whose derivatives are rapidly decreasing) on $\mathbb{R}^+$.
We already know that a generic such $f$ ...
0
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1answer
38 views
limit and integral in the Lp space with Schwartz class.
Can we actually interchange limit and integral for $f \in S(\mathbb{R})$. Or it's just the limit and integral happen both equal to 0?
An exercise:
Show that we can interchange the limit and integral:
$...
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0answers
20 views
Convergence of fractional laplacian in $\mathcal{S}_s$
Let $s\in(0,1)$, let:
$$ \mathcal{S}_s=\biggl\{ f\in C^\infty(\mathbb{R}^n): \sup_{x\in\mathbb{R}^n}(1+|x|^{n+2s})|\partial^\alpha f(x)|<\infty,\,\forall \alpha\in\mathbb{N}_0^n\biggr\}.$$
The ...
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0answers
27 views
Understanding the identity $f(x) \delta'(x-\xi) = - f'(\xi) \delta(x - \xi) + f(\xi) \delta'(x-\xi)$ in Fourier space
Consider the distributional identity (see Chapter 2.6 of Kanwal's Generalized Functions, for example)
$$
f(x) \delta'(x-\xi) = - f'(\xi) \delta(x - \xi) + f(\xi) \delta'(x-\xi) \tag{1}
$$
I can see ...
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0answers
52 views
Proving convolution of two Schwartz functions is again a Schwartz function on $\mathbb{R}^n$
This is something I have encountered in my analysis class. So suppose we take the space of Schwartz functions $S(\mathbb{R}^n)$ which are the rapidly decaying functions, or more formally the functions ...
0
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1answer
35 views
Integral Inequality for $u\in\mathcal{S}(\mathbb{R}^n)$
Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that:
$$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$
...
0
votes
1answer
56 views
Extension of Fourier transform to $L^2$ by density of Schwartz functions
The Fourier transform is usually extended to the $L^2(\mathbb{R})$ space by invoking an argument that relies on the density of Schwartz functions in $L^2$.
Often, this extension is explicitly written ...
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0answers
21 views
On the boundary conditions of black body problem
The black body problem tries to find the spectral energy density of electromagnetic field per unit volume in a isolated cavity with electromagnetic free radiation (i.e., with no sources)
The free ...
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0answers
41 views
Uniform estimate on Schwartz functions away from support of Fourier transform
This question is a follow-up to this post, but hopefully it's a better attempt at formulating the same idea. Roughly speaking, I would like to obtain a uniform estimate on the size of a certain class ...
1
vote
1answer
74 views
Uniform estimate on Schwartz functions with compactly supported Fourier transform
Let $\mathcal{C}$ be the class of all even Schwartz functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the following conditions:
The Fourier transform $\hat{f}$ is compactly supported;
$f$ is non-...
1
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0answers
55 views
Convergence of the derivative in Schwartz space
Reading Grafakos' book "Classical Fourier Analysis", I got stuck in the Exercise $2.3.5.$ (a) which states that if $f ∈ \mathscr{S}(\mathbb{R}^n)$ then
$$(\tau^{−he_j} f − f )/h\rightarrow \...
0
votes
1answer
59 views
Lp Convergence of Fourier Integrals Using Hilbert Transform
I am reading about Hilbert transform and its application on Fourier analyisis, and I am triying to prove a statement given by Terence Tao in his notes on Fourier Analysis. He says that if $\varphi\in\...
0
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0answers
54 views
Equivalence of family of seminorms on Schwartz space
On the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ define
$$
|\varphi|_{r,s,p}=||\cdot^r\partial^s\varphi||_{L^p}
$$
for $p\in\{2,\infty\}$ and multiindices $r,s\in\mathbb{N}^n$. Consider the two ...
1
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0answers
32 views
Proving that a $C^\infty$ regularizing contraction semigroup leave a subspace invariant.
In Villani's Hypocoercity, I am faced with proving a statement which is indicated by the title.
Consider a $C^\infty$ function $V: \mathbb{R}^n \to \mathbb{R}$, convergeing to $+\infty$ fast enough ...
2
votes
1answer
89 views
A dense subset for each of two Banach sets respectively
Let $A$ and $B$ be Banach spaces with their own (possibly different) norms. Also, there is a non-empty subset $S \subset A \cap B$ such that $S$ is dense in $A$ and $B$ respectively.
Then, for $x \in ...
2
votes
0answers
83 views
Fubini's theorem for tempered distributions
Let's say I have a distribution $u\in\mathcal{S}'(\mathbb{R}^n)$ and two function $f\in\mathcal{S}(\mathbb{R}^{n+m})$, $g\in\mathcal{S}(\mathbb{R}^m)$.
Then
\begin{align}
\tilde{f}(y)=\int_{\mathbb{R}^...
