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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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Interaction of pullback and the Fourier transform

I'd like to understand how the spectra of functions on a given domain are affected by (different kinds of) maps of that domain. Specifically, consider the Schwartz space $S(\mathbb R^n)$ of test ...
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Given $f \in L^p_{\text{loc}}$ and $\phi \in S$ (Schwartz class), do we have that $f \ast \phi \in C^\infty$? What if $f,\phi$ have compact support?

Context. Consider the usual Lebesgue measure on $\mathbb R^n$ and let $L^p_{\operatorname{loc}}$ denote the space of measurable functions that are locally $p$-integrable on $\mathbb R^n$, with $1 \...
xyz's user avatar
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Why is it stated that polynomials are embedding inside a Schwartz space? [closed]

From Wiki, Schwartz space is a vector space of very nice functions $\mathcal{S}(\mathbb{R}_n)$ on $\mathbb{R}_n$. So any polynomial $\phi(\boldsymbol{x}^{\boldsymbol{\alpha}})$ where $$\boldsymbol{x}^{...
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Schwartz function realized as the divergence of a schwarz vector field has null integral

I am learning some fourier analysis on my own and I encountered the following problem for which I would greatly appreciate if anyone could guide me through my attempt. Assume that $\psi$ is a Schwartz ...
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Schwartz function dense in weighted Sobolev Space

Given a function $u$ on the Gevrey space with norm defined by \begin{equation} ||u||_{G^{\sigma,s}}=||e^{\sigma|\xi|}(1+|\xi|)^s\hat{u}(\xi)||_{L^2} \end{equation} for $\sigma\geq 0$ and $s\in \mathbb{...
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What is the meaning of this definition?

Let $\mathscr{S}(\mathbb{R}^{d})$ the Schwartz space of rapidly decreasing functions and $\mathscr{S}'(\mathbb{R}^{d})$ the space of tempered distributions. Let $f$ be such that $f^{\gamma} \in C^{\...
InMathweTrust's user avatar
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Any closed subspace of the Schwartz space is dense in its dual?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space and $\mathcal{S}'(\mathbb{R}^n)$ be the continuous dual space. Then, it is well-known that for any $T \in \mathcal{S}'(\mathbb{R}^n)$, we can ...
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Is any Schwartz function a Fourier multiplier?

If $m$ is a Schwartz function then $m=\widehat{n}$ for another $n$ Schwartz function because, fourier transform is an isomorphism between the Schwartz space. Now, the operator $\mathcal{F}^{-1}(m(\xi) ...
eraldcoil's user avatar
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Example of an even nonnegative bandlimited Schwartz function with nonnegative Fourier transform

I am looking for an example of an even Schwartz function $f$ such that $f \geq 0$, $\widehat{f} \geq 0$ and $\text{supp}(\widehat{f}) \subset [-1, 1]$, i.e., $\widehat{f}(\xi) = 0$ for $|\xi| >1$. ...
Epsilon-Delta's user avatar
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Convolution between $f$ and $g$, with $g$ being in the Schwartz class. Does it follow that $f \ast g \in C^\infty$?

Usually, the convolution between two functions $f,g$ defined on $\mathbb R^n$ is given by $$ (f \ast g)(x) = \int_{\mathbb R^n} f(x-y)g(y) \, dy. $$ Right now I am wondering about a specific property ...
Temirbek Alikhadzhiyev's user avatar
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Nuclear spaces: the Schwartz class

Good morning, I'm studying quantum mechanics as a mathematician. I read that the Schwartz class $$\begin{equation*} \mathcal{S}(\mathbb{R}) := \{ \varphi \in \mathcal{C}^\infty (\mathbb{R}, \...
Marco Lugarà's user avatar
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analytically continue functions on independent planes in $\Bbb R^3$

I know techniques to analytically continue some functions off the positive real line. I am less sure given I have 2 independent euclidean planes sitting in $\Bbb R^3$ with some function $f$ defined on ...
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Showing that $((1+\xi^2)u,\phi)=0$ implies that $u=0$.

If $u$ is a tempered distribution and $((1+\xi^2)u,\phi)=0$ for all $\phi$ Schwartz function. Then $u$ is the zero tempered distribution? Actualization. I think that I proves this. My attempt: $\left\{...
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Find a $m>0$ such that $(1+|x|)^{-m} f(x) \in L^1\left(\mathbb{R}^n\right) $

Consider $f \in L_{l o c}^1\left(\mathbb{R}^n\right)$ and $T_f$ is the regular distribution associated to $f$. Assume now $f \in L_{l o c}^1\left(\mathbb{R}^n\right)$, nonnegative satisfying $T_f \in \...
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Can be use $u$-substitution for calculating the adjoint of an operator in Schwartz space?

I only have seen that for calculating the adjoint of an operator in $\mathcal S$, it used integration by parts, but I was thinking that if one can use substitution to find the adjoint. For exmple, for ...
Daniel Muñoz's user avatar
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If $f_n(x) \to f(x)$ in the Schwartz space, $f_n(Ax) \to f(Ax)$ for any invertible matrix $A$?

Let $f_n$ be a sequence of Schwartz functions on $\mathbb{R}^N$ converging to another Schwartz function in the Schwartz space $\mathcal{S}(\mathbb{R}^N)$. Now, let $A : \mathbb{R}^N \to \mathbb{R}^N$ ...
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For Schwartz $f$ on $\mathbb{R}^2$ with $f(x,x)=0$, is the integral $\int_{\mathbb{R}^2} \frac{f(x,y)}{\lvert x-y \rvert^{n}}dxdy$ finite?

Let $f$ be a Schwartz function on $\mathbb{R}^2$ such that $f(x,x)=0$ for all $x \in \mathbb{R}$. Then, I wonder if the integral \begin{equation} \int_{\mathbb{R}^2} \frac{f(x,y)}{\lvert x-y \rvert^{n}...
Keith's user avatar
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Convolution: tempered distribution

Let $u$ be a tempered distribution and $\psi$ a Schwartz function. I have seen two definitions for $\psi*u:$ $\psi*u:$ defined as a function of $\mathbb{R}^d,$ that is, for $x \in \mathbb{R}^d,\psi*u(...
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Is it really true that $\mathcal{S}(\mathbb{R}^n)$ is identified with smooth functions on $S^n$ vanishing at a fixed point?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space on $\mathbb{R}^n$ and $C^\infty(S^n)$ be the space of smooth functions on $n$-sphere. Now fix a point $x \in S^n$ and define \begin{equation} C^\...
Keith's user avatar
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Formal Demonstration of an Integral Inequality Involving Nested Sets

I am delving into an analysis involving a specific integral inequality that arises within the context of Fourier analysis, particularly focusing on expressions involving a series of integrations over ...
Sara Testori's user avatar
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$\sqrt{ f(x) + \alpha e^{-\lVert x \rVert^2}}$ is a Schwartz function for any nonnegative Schwartz $f$ and $\alpha>0$?

This question has been motivated by original and comments therein. Let $f$ be a nonnegative Schwartz function on $\mathbb{R}^n$ and let $\alpha \in (0,\infty)$ be fixed. Then, I wonder if \begin{...
Keith's user avatar
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Is it possible to approximate any nonnegative Schwartz function by square of Schwartz functions?

Let $f$ be a nonnegative Schwartz function on $n$-dimensional Euclidean space. Then I wonder if it is possible to find a sequence of complex-valued Schwartz functions $f_n$ such that $\lvert f_n \...
Keith's user avatar
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Convolution of Schwartz and $C^\infty$ function with bounded increasing [duplicate]

I have the following problem: Let $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in C^\infty(\mathbb{R}^n)$ such that exists $\alpha>0$ $$ \left|g(x)\right|\leq \frac{1}{1+|x|^\alpha} \quad \forall x\in\...
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Question regarding the indexes $\alpha$ and $\beta$ in the definition of Schwartz space

I was asked to prove that the Fourier transform of any Schwartz function belongs to the Schwartz class, i.e: $$||\xi^{\alpha}\frac{d^{|\beta|}\hat{f}}{d\xi^{\beta}}||<\infty$$ For $\alpha$ and $\...
Pedro Ye's user avatar
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Is the Schwartz space "sequentially dense" in the space of tempered dsitributions with respect to some topology?

Let us consider the Euclidean space $\mathbb{R}^n$ of any $n \in \mathbb{N}$. Then, I somehow vaguely recall that the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ is dense in the space of tempered ...
Keith's user avatar
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For $f \in H^s$, then $\exists g \in C_c$ such that $f=g$ a.e.

Consider the space $H^s(\mathbb R^d)$ ($f \in L^2$ not in Schwartz class), $s \in \mathbb R$. Apply Riemann-Lebesgue Lemma to $\hat{f}$ to show that for some $s>s_0$ then there is a continuous ...
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Paley-Wiener theorem conversion proof

Let $v:\mathbb{C}\rightarrow \mathbb{C}$ be entire function with an estimate $|v(z)| \leq C_N(1+|z|)^{-N}e^{R|Imz|}$ for all $z \in \mathbb{C}$ then there is $u \in S(\mathbb{R})$ such that $supp(u) \...
voroshilov's user avatar
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How to estimate schwartz space functions?

I'm currently learning about Schwartz spaces and I'm having difficulties when I'm trying to estimate the derivatives of the functions. For example when you have a function $e^{-ln(1+x^2)^N}$ and you ...
voroshilov's user avatar
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Confusion with identities in different function spaces

Let $\mathcal{S}(\mathbb{R}^3,\mathbb{R}^3)$ be the space of vector fields on $\mathbb{R}^3$ with Schwartz function components. Next, let $\mathcal{S}_{div}(\mathbb{R}^3,\mathbb{R}^3)$ be the space of ...
Keith's user avatar
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Proving that $\frac{x}{x^2+\varepsilon^2}\to \mathsf{p.v.}\left(\frac1x\right)$ in $\mathscr{S}'$

Let $\mathscr{S}'$ be the space of tempered distributions. I've to prove that: $$\frac{x}{x^2+\varepsilon^2}\to \mathsf{p.v.}\left(\frac1x\right) \ \ \text{ in $\mathscr{S}'$}$$ My book says that it'...
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Proving that $\rho_\varepsilon\ast U\to U$ as $\varepsilon\to 0^+$

I'm trying to understand the proof of the following fact about tempered distributions. Let $U\in \mathscr{S}'$ be a tempered distribution, and let $\{\rho_\varepsilon\}$ be the family of standard ...
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Schwartz kernel theorem and musical isomorphisms

I am reading about the Schwartz kernel/nuclear theorem which states for any bilinear form $B$ on $\mathcal{S}(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^n)$ (where $\mathcal{S}(\mathbb{R}^n)$ is ...
CBBAM's user avatar
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8 votes
3 answers
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How can tempered distributions be identified with functions?

In Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, he defines tempered distribution ($\mathscr S'$) as continuous linear functionals from the Schwartz class. ...
confusedTurtle's user avatar
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Tempered distribution locally integrable function

Let $f \in L^1_{\text{loc}}\left(\mathbb{R}\right)$ be a nonnegative function. Show that if $\Lambda_f$ is a tempered distribution, then there exist $C > 0, N \in \mathbb{N}_0$ satisfying $$\forall ...
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Given a Schwartz function $f(x,y)$ on $\mathbb{R}^{2n}$, is $f(x,x)$ necessarily a Schwartz functoin on $\mathbb{R}^n$?

I am suddenly a bit confused by the question described in the title. Let $f(x,y)$ be a Schwartz function on $\mathbb{R}^{2n}$ with $x,y \in \mathbb{R}^n$. Then, is it necessarily true that $x \to f(x,...
Keith's user avatar
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Product of a Schwartz function and a tempered distribution

Let $\mathscr{S}(\mathbb{R})$ denote Schwartz space, $T \in \mathscr{S}'(\mathbb{R})$, and $f \in \mathscr{S}(\mathbb{R})$. How can we make sense of the product $fT$? In the second volume of Reed &...
CBBAM's user avatar
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3 votes
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Is $1/x$ a tempered distribution?

If $f$ is a integrable function, then, $f$ is a tempered distribution by $$T_f(\varphi)=\int f\varphi,\quad \varphi \text{ Schwartz function }$$ However, I read that the function $f(x)=1/x,\, x\in\...
eraldcoil's user avatar
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Show that the convolution $u\ast\psi$ between compactly supported tempered distribution $u$ and Schwartz function $\psi$ is a test function

I am trying to show that $u\ast\psi=v$ for some $v\in\mathcal{C}_c^{\infty}$ in the following way: Step 1: Show that one can define $\langle u\ast\psi,\phi\rangle$ for $\phi\in\mathcal{S}'$ Step 2: ...
5Somebody's user avatar
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is it an equivalent definition of the Schwartz Space? all real smooth function that $ f^{(k)}(\infty)=0=f^{(k)}(-\infty)$

Background: I have only taken half semester Analysis course It's in a german script from a linear algebra course on scalar products and self-adjoint endomorphisms, and so on. It mentions such a ...
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Is the convolution of a tempered distribution and a Schwartz function also a function?

Let $T \in \mathscr{S}'(\mathbb{R}^n)$ and $f \in \mathscr{S}(\mathbb{R}^n)$. We may define their convolution as $$(T * f)(\varphi) = T(\tilde{f}*\varphi)$$ where $\tilde{f}(x) = f(-x)$. The above ...
CBBAM's user avatar
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Relationship between Schwartz space and fractional Sobolev space

Let $u$ be an element of the Schwartz space $\mathcal{S}(\mathbb{R}^N)$. Given the fractional Laplacian operator $(-\Delta)^s:\mathcal{S}(\mathbb{R}^N) \to L^2(\Omega)$, with $s \in (0,1)$ , defined ...
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Understanding the support of a differential operator thought of as a tempered distribution from perspective of Schwartz kernel

Consider the differential operator $$ A:=\sum_{|\alpha|\leq m}a_{\alpha}(x)D^{\alpha}, $$ where $\alpha$ is a mutiindex on $\mathbb{R^n}$. It can be checked that the Schwartz kernel of $A$ is $$ K(x,y)...
5Somebody's user avatar
4 votes
1 answer
102 views

Dividing a tempered distribution by a polynomial

Let $p=p(x_1,...,x_N)$ be a non-zero polynomial in $N$ variables (real coefficients). Let $\mathscr{S}$ be the Schwartz space on $\mathbb{R}^N$ and let $\mathscr{S}'$ be its topological dual (i.e. the ...
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When $\mathrm{e}^{-f(x)}$ is a Schwartz function?

The gaussian function $\mathrm{e}^{-|\xi|^2}$ is a schwartz function. In general, Are there some critery on the function $f(\xi)$ for which the function $\mathrm{e}^{-f(\xi)}$ is a Schwartz funtion? ...
eraldcoil's user avatar
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2 votes
1 answer
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Inequality between $L^p$ norm and Schwartz seminorm

Let $u\in \mathscr{S}$ (Schwartz space). I can prove easily that $u$ is in $L^p$ for every $p$. But I cannot prove the following. Let $\{p_k\}$ be the family of seminorms that defines the topology of ...
Kandinskij's user avatar
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Obtaining the correct Gamma factor for this Schwartz function

In this question I overlooked that the class $f_s(x)=e^{sT(x)}$ for $T(x)=\frac{1}{\log x},$ is Schwartz, on $x \in (0,1).$ Therefore we can obtain a functional equation, see this answer. However, I ...
zeta space's user avatar
1 vote
2 answers
130 views

Test functions are dense in Schwartz functions (with respect to $L^\infty$-norm)

Given a Schwartz function $\varphi:\mathbb{R}^n\to \mathbb{R}$, I want to construct this way a sequence of test functions that converge to $\varphi$ with respect to $L^\infty$-norm: $$\varphi_n(x):=\...
Kandinskij's user avatar
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2 votes
1 answer
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Equivalent definitions of the Schwarz space

In different places I've seen Schwarz functions defined by $$\sup_{|\alpha|\le N}\sup_{x\in\mathbb{R}^n}\left(1+|x|\right)^N|D^\alpha \psi(x)| < \infty$$ and by $$\sup_{|\alpha|\le N}\sup_{x\in\...
Sam's user avatar
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The representation of p-norm of $f$ using Schwartz class.

For $1<p<\infty$ and the conjugate $p'$, show that the $p$- norm of $f$ is written as $$\| f \|_p=\sup\left\{\left|\int_{\mathbb R^n}fg\right| : g \in \mathcal S(\mathbb R^n), \|g\|_{p'}=1\right\}\...
daㅤ's user avatar
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2 votes
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Search of a Schwartz Function with specific properties.

I want to find a Schwartz function $\varphi$ with the following properties \begin{equation} \varphi > 0 \quad \text{and} \quad \mathrm{supp}(\widehat{\varphi})\subset B_{R}, \end{equation} where $...
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