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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Upper bound on L2 norm of derivative of Schwartz function $f \in \mathcal S(\mathbb R)$
I'm preparing for my prelims and here's a practice question related to Schwartz functions.
Let $f \in \mathcal S(\mathbb R)$. Prove that for some $C > 0$:
$$||f'||_{L^2(\mathbb R)} \leq C||f||_{L^2(...
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General conditions for the invertibility of integral mappings
I am wondering when is an integral mapping invertible. More precisely, let $\mathcal{S}(\mathbb{R}^n)$ be the space of Schwartz smooth and rapidly decaying functions, its dual $\mathcal{S}'(\mathbb{R}^...
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Find the fundamental solution in tempered distribution
I am a beginner of distribution theory.
I would like to find the fundamental solution in the sense of tempered distribution $\mathcal{S}'(\mathbf{R}^3).$(the duality of Schwartz space)
\begin{equation}...
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difference between $H_s^0(\mathbb{R}^n)$ and $ H_s(\mathbb{R}^n)$
we have $\mathscr{S}(\mathbb{R}^n)$ schwartz space and $C_{c}^{\infty}(\mathbb{R}^n)$, and for $s\in \mathbb{R}$ the Sobolev norm
$||u||^2_s={\int_{\mathbb{R}^n}^{}}{(1+|\xi|)^{2s}\widehat{u}(\xi)\...
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Analytic continuation of $I(s)=\frac{1}{\Gamma(s)}\int^\infty_0 f(x)x^{s-1}\, dx$
This is Exercise 17 of Chapter 6 from Stein and Shakarchi Complex analysis
Let $f$ be an infinitely differentiable function on $\mathbb{R}$
that has compact support, or more generally, let $f$ ...
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Can Schwartz class functions be nowhere analytic?
By "Schwartz class functions" I will be referring to the functions of the Schwartz space on $\mathbb{R}$, that is, smooth ($\mathcal{C}^\infty$) functions $f : \mathbb{R} \to \mathbb{R}$ ...
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If $\varphi\in\mathcal{S}(\mathbb{R}):\int_\mathbb{R}\varphi dx=1$, does it then follow that $\sum_{k}\hat{\varphi}(k) \leq 2$?
Let $\varphi\in\mathcal{S}(\mathbb{R})$ be a Schwartz test signal such that $\int_{\mathbb{R}}\varphi(x)dx = 1$. Does it then follow that the sum of the Fourier coefficients of $\varphi$ are at most ...
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Exact definition of Fourier transform on L^p and some calculation
I'm interested in the definition of Fourier transform of a function $ f \in L^p(\mathbb{R}^d) $, where $ p \in [1, \infty] $ and $ d \in \mathbb{N} $ is the dimension.
The way I understand is that one ...
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Showing Boundedness of Linear Operator on Schwartz Space
In class, my professor said that for $\mu \in L^p(\mathbb{R}^n), p \in [1,\infty]$, we have a bounded linear operator $\mu':\mathcal{S(\mathbb{R}^n)} \rightarrow \mathbb{R}$ such that $\mu'(\phi):= \...
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Show that $ T_f(\varphi):=\int f \varphi d x, \quad \varphi \in \mathcal{S} $ is a tempered distribution.
Let $f$ be in $ L_{\text {loc }}^1$, and suppose that there exists a polynomial $p$ such that $f / p$ is integrable on $\mathbf{R}$. Show that
$$
T_f(\varphi):=\int f \varphi d x, \quad \varphi \in \...
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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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Estimate on the first term in $(\ast)$
Here, $\mathscr{S}(\mathbb{R})=\{f\in C^{\infty}(\mathbb{R}):\lVert f\rVert_{(N,\alpha)}<\infty\text{ for all non-negative integers }N\text{ and }\alpha\}$ is the Schwartz space ; where $$ \lVert f\...
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Threefold Convolution of a Schwartz function produces the same gaussian
I am wondering if there is a way to find a (or multiple) Schwartz function f such that its threefold convolution produces the same gaussian. Aka
$ f_3\left(x\right) = e^{-π(x-1)^2}$ where $f_3\left(x\...
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Does weak convergence on a dense subset imply weak convergence for boundary values of analytic functions?
Generally, weak convergence of a sequence of operators $T_n$ on a dense subset $\mathcal D \subset \mathcal F$ does not imply weak convergence on the full set,
$$
\lim_{n\to \infty} T_n(f) = T(f) ~~\...
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Proving that the Fourier transform of the identity is the Dirac delta using test functions.
For every finite Schwartz function $f$ and every $t \in \mathbb{R}$, define the Dirac delta as
$$\int_{-\infty}^\infty dt f(t) \delta(t-t') := f(t').$$
How do I proof with this definition that
$$\int_{...
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Properties of the Schwartz space
I uploaded a photo of my problem because latex isn't working very well here.
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Integral exists if integrand decays faster than any polynomial.
I am in the process of proving the Fourier transform is closed on the Schwartz class. And I have trouble showing Schwartz functions are $L^1(\mathbb{R}^n)$. So the simple question is suppose $f$ is ...
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Showing that the heat kernel is a Schwartz function.
In the heat's equation, the heat kernel is the function $N_t(x):=\int_{\mathbb{R}}\mathrm{e}^{ix\cdot \xi}\mathrm{e}^{-t (1+|\xi|^2)^{s/2}}\,d\xi$ with $s>0$.
It is well known that $\mathrm{e}^{-|\...
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Inner product of two Schwartz fucntions.
Say I have two Schwartz fucntions $f(x)$ and $g(x)$.
Does $\int f(x)g(x)dx \approx 0$ imply that $f(x)$ $g(x)$ have virtually non-overlapping support ?
Thank you for your time.
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Continuity of translation from $\mathbb{R}$ to Schwartz space
Given $f \in \mathcal{S}(\mathbb{R})$ where $\mathcal{S}(\mathbb{R})$ is the Schwartz space, the space of $C^\infty(\mathbb{R})$ such that: $\forall i, j\in \mathbb{N}$
$$
\rho_{i,j}(f) = \sup_{x\in \...
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Inner Product on Schwartz Space and $L^2$
For $f, g \in \mathcal{S}$: the inner product is usually given by
$$
\langle f,g \rangle = \int f(x) g(x) dx
$$
whereas for $f, g \in L^2$, the inner product is usually given by
$$
\langle f,g \rangle ...
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Convergence in Schwartz Space
The seminorms are on $\mathcal{S(\mathbb{R}^k)}$ are given by $p_{\alpha, \beta}(\phi) = \sup \vert x^\alpha \partial^\beta\phi \vert$, and the metric is given by
$$
d(\phi, \psi) = \sum_{\alpha, \...
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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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continuous/smooth Fourier transform
Let $f$ be a function with
a) $f\in L^1(\mathbb R^d)$
b) $f\in \mathcal S(\mathbb R^d)$ (The Schwartz-space)
c) $f\in L^2(\mathbb R^d)$
and given the following statements:\
(i) $f$ continuous $\...
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Connection between probability distribution and generalized function
I want to understand the connection between probability density function and generalized function.
There are 3 common used generalized function classes: $\mathscr{E}'\subset\mathscr{S}'\subset\...
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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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Schwartz Function Convergence
I am a little bit confused about how convergence in the Schwartz space is defined. I am aware that the topology on the Schwartz space $S(\mathbf{R}^N)$ is metrizable by
$$
d(f, g) := \sup_n \frac{1}{n}...
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If $\psi(t)=\displaystyle\int\limits_{-\infty}^{t}\varphi(x)dx$ then $\psi\in\mathbb{S}(\mathbb{R}) $
Exercices :
Let $\varphi\in\mathbb{S}(\mathbb{R}) $ such that : $\displaystyle\int\limits_\mathbb{R}\varphi (x)dx=0 $
Let defined :
$$\psi(t)=\displaystyle\int\limits_{-\infty}^{t}\varphi(x)dx$$
Then ...
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Proving or disproving that if $0<\epsilon<1$ then $\frac{g(x)}{g(x/\epsilon)}$ is in Schwartz space when $\frac{1}{g(x)}$ is a Schwartz function.
I have a question about functions in Schwartz space.
$f(x)=\frac{1}{e^{x^2}}$ is in $\mathcal{S}(\mathbb{R})$ (Schwartz space)
If $0<\epsilon<1$, then $\frac{f(x)}{f(x/\epsilon)}=e^{x^2(\frac{\...
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Shifted Schwartz Functions are Schwartz?
I am asking about the same question as this post: Translation of a Schwartz function is a Schwartz function?
Namely: If $f\in S(\mathbf{R}^n)$ and for all $y\in\mathbf{R}^n$ then $\tau_{y}f\in S(\...
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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 ...
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Show that exp is not a tempered distribution
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,...
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