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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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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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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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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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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 ...
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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
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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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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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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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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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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
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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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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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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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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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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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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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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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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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1answer
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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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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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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| ...
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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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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. $\psi \in C^{\infty}(\mathbb{R}^k)$ ...
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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 $\...
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$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 ...
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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}, \...
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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\...