# 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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### [Fourier Inversion Theorem]: conditions for a function to be "schwartz", or at least square integrable.

Let's consider the function in $(eq. 1)$, i'm not solving explicitely that integral, but since it is the convolution of some other functions, it's easier to express $f(x)$ as the inverse of its ...
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### Is $\{ f \in \mathcal{S}(\mathbb{R}) \mid \text{supp }f \subset [a,b]\}$ another Frechet space?

The question is as in the title. Given a compact interval $[a,b]$ and the Schwartz space $\mathcal{S}(\mathbb{R})$, is the subspace \begin{equation} \{ f \in \mathcal{S}(\mathbb{R}) \mid \text{supp }f ...
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### Establishing estimates between equivalence Schwartz seminorms

On the Schwartz space $\mathcal{S}(\mathbb{R}^m)$, I am considering two types of seminorms: \begin{equation} \rho_{k,n}(f):=\max_{\lvert a \rvert \leq k , \lvert b \rvert \leq n}\sup_{x \in \mathbb{R}^...
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### For a Schwartz function $f(x,y)$, is $f(x+y,y)$ also a Schwartz function?

The question is as in the title. This seems intuitively clear but I cannot see how to prove rigorously. Let me phrase more precisely: For a Schwartz function $f(x,y)$ on $\mathbb{R}^2$, is the ...
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### Function $f$ subordinate to function $g$

I'm trying to study a research paper, during that i come across this word Subordinate, i searched this a lot on google but didn't get any satisfactory answer. I understand the definition they gave in ...
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### Semi-norms generating the usual topology of Schwartz space

Consider the following family of semi-norms on the Schwartz space $$\|f\|_{m,n}=\sup_{x\in \mathbb{R}}|(1+|x|)^m f^{(n)}(x)| \;\;\; m,n\in \{0,1,2,...\}$$ It is well known in the litterature that ...
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### Is there a smooth tempered distribution $u$ such that $\sum_{j=0}^\infty 2^{-j} u(x-j)$ is not smooth?

I found the following claim in page 3 of Weinstein, Alan, A symbol class for some Schrödinger equations on ${\mathbb{R}}^ n$, Am. J. Math. 107, 1-21 (1985). ZBL0574.35023: Operators whose symbols ...
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### function $T^n(e^{-\pi x^2})\neq 0$ for all $n\in\mathbb N$

Let $\mathcal S(\mathbb R)$ be the Schwartz-space. Look at the linear operator $T:\mathcal S(\mathbb R)\rightarrow \mathcal S(\mathbb R)$ $(Tf)(x)=\sqrt{2\pi}xf(x)-\frac{1}{\sqrt{2\pi}}f'(x)$ and ...
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### Quotient of polynomial in Schwartz space.

Let $P(\xi)$ a polynomial in $\mathbb{R}^{n}$ and $\varphi$ a function in Schwartz space. Is true that if $P$ has no roots, then $\dfrac{\varphi}{P}$ is in Schwartz space? Could anyone help me, please?...
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### Limit sequence of Schwartz functions

Let $\left\{f_n\right\}_{n\in\mathbb{N}}\subset \mathcal{S}(\mathbb{R}), f\in L^2(\mathbb{R})$ with $\lim f_n=f$ in $L^2$ and $p(x)$ an polynomial. Since $\mathcal{S}$ is closed under the product of ...
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### Function that is in the Schwartz space but is not a bump function

Wikipedia states that the function $f(x)=x^\alpha e^{-a|x|^2}$ for a multi-index $\alpha$ and a positive real number $a$ is in the Schwartz space. I need to find a function that is Schwartz but not a ...
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### The Schwartz-space is not normed.

I'm supposed to prove that the topology of the Schwartz-Space $\mathcal{S}:=\mathcal{S}(\mathbb{R}^n)$ is not induced by a norm by assuming that there exists such a norm $||\cdot||$. As a first step, ...
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### Product between a Schwartz function and a polynomial with a real exponent.

If $f\in\mathcal{S}$ (Schwartz space) by definition, $(1+|x|)^{N}|\partial^{\alpha}f(x)|\leq C_{\alpha,N}$ for all $N\in\mathbb{N}$ and $\alpha$ multi-index. This definition implies that the product ...
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### Showing that an exponential function is Schwartz function.

I would like to know if I am right with the following reasoning. My goal is to prove that a function is Schwartz. Let $g:\mathbb{R}\to\mathbb{R}$ with $g\in\mathcal{C}^{\infty}([0,\infty))$. Question ...
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### Schwartz function absolute value [duplicate]

Let $\phi$ be an element of $\mathcal{S}(\mathbb{R})$, the space of Schwartz functions on $\mathbb{R}$. Show that there exists $\psi\in\mathcal{S}(\mathbb{R})$ such that $|\phi|\leq\psi$. I am not ...
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### Counstruct a sequence of Schwartz functions

Here is an exercise from Wolff's lectures on harmonic analysis, p26: Using translation and multiplication by characters, construct a sequence of Schwartz functions $\lbrace\phi_n\rbrace$ so that Each ...
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### Spherical Mean of Schwartz Function Decreases Rapidly - Stein, Shakarchi, Fourier Analysis.

I'm studying Stein and Shakarchi's Fourier Analysis book, and I'm stumped on page. 189, where it talks about the spherical mean of a function. The book defines the spherical mean of a complex-valued ...
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### Bound of norm in Schwartz class of functions

I am currently working my way through Stein's Functional Analysis and at one point in the text he claims without proof that for all $\phi \in\mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of ...
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### How can I show a Schwartz function is identical 0? [duplicate]

I have the following problem: If a Schwartz funcetion $f\in\mathcal{S}(\mathbb{R})$ satisfies $f(2\pi n)=0$ and $\hat{f}(n)=0,\forall n\in\mathbb{Z}$, where $\hat{f}(\xi)=\int e^{-ix\xi}f(x)dx$ is the ...
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### Solving Transport Equation with Velocity Switch using Tempered Distributions

Imagine we have the following two transport problems. The first is well known and can be solved using the Fourier transform, but I do not know how to solve the second one. Problem 1: With Constant ...
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### Clarification of the definition of tempered growth.

This is a rather simple question, but I don't have many resources I've been able to consult in this regard (meaning that a Google search hasn't yielded any meaningful results). I just want to clarify, ...
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### Equivalence of topologies on the Schwartz space

I am dealing with the Schwartz space: set of all inifinite differentiable functions on $\mathbb{R}$ such that for all $a, b\geq 0$, $x^{a}f^{(b)}(x)$ is a bounded function. The topology is generated ... 1 vote
I'm trying to show that $x \mapsto e^x$ is not a tempered distribution. For that I want to find a schwartz function $\varphi \in S(\mathbb{R})$ such that the support of $\varphi$ is contained in \$[-1,...