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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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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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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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 ...
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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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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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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 ...
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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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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 ...
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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}(\...
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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 $\...
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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 ...
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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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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 ...
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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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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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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),\...
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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}(\...
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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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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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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 ...
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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.
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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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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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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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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 ...
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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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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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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 $K>0$ ...
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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 ...
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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 ...
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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 $f$ ...
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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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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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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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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 ...
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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: $$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$ ...
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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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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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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 ...
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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-...
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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 \...
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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\...
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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 ...
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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 ...
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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 ...
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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}^...

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