Questions tagged [schwartz-space]

For question about the Schwartz space, a vector space of smooth functions stable under the Fourier transform.

Filter by
Sorted by
Tagged with
1 vote
0 answers
21 views

[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 ...
  • 177
1 vote
1 answer
43 views

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(...
  • 425
0 votes
0 answers
11 views

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}^...
2 votes
0 answers
41 views

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}...
1 vote
0 answers
39 views

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)\...
  • 386
2 votes
1 answer
74 views

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 $f$ ...
  • 95
0 votes
1 answer
54 views

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 $f : \mathbb{R} \to \mathbb{R}$ ...
  • 1,337
1 vote
1 answer
41 views

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 ...
0 votes
1 answer
36 views

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 ...
  • 73
0 votes
0 answers
21 views

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):= \...
  • 415
0 votes
1 answer
41 views

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 \...
  • 163
0 votes
0 answers
20 views

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 ...
  • 5,822
0 votes
0 answers
9 views

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}^...
  • 5,822
1 vote
0 answers
33 views

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 ...
  • 5,822
0 votes
0 answers
52 views

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 ...
  • 1,052
1 vote
1 answer
41 views

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\...
0 votes
0 answers
23 views

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\...
1 vote
0 answers
14 views

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) ~~\...
  • 382
3 votes
2 answers
76 views

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_{...
1 vote
1 answer
48 views

Properties of the Schwartz space

I uploaded a photo of my problem because latex isn't working very well here.
0 votes
2 answers
78 views

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 ...
  • 61
0 votes
0 answers
39 views

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}^{-|\...
  • 2,880
-1 votes
1 answer
30 views

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.
  • 197
3 votes
1 answer
52 views

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 \...
0 votes
0 answers
62 views

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 ...
1 vote
1 answer
113 views

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, \...
1 vote
0 answers
33 views

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 ...
0 votes
1 answer
41 views

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 $\...
  • 442
1 vote
1 answer
66 views

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\...
  • 13
5 votes
1 answer
83 views

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 ...
  • 32.9k
2 votes
0 answers
55 views

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 ...
  • 77
0 votes
0 answers
12 views

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?...
0 votes
1 answer
24 views

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 ...
  • 2,880
0 votes
0 answers
64 views

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 ...
0 votes
0 answers
43 views

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, ...
  • 41
0 votes
0 answers
17 views

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 ...
  • 2,880
0 votes
0 answers
35 views

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 ...
  • 2,880
1 vote
0 answers
34 views

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 ...
  • 1,143
3 votes
1 answer
56 views

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 ...
  • 71
1 vote
0 answers
51 views

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 ...
0 votes
0 answers
85 views

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}...
2 votes
0 answers
55 views

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 ...
  • 2,273
0 votes
0 answers
14 views

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{\...
  • 2,880
0 votes
1 answer
67 views

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(\...
1 vote
1 answer
91 views

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 ...
1 vote
0 answers
52 views

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 ...
  • 71
2 votes
2 answers
248 views

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 ...
  • 117
0 votes
0 answers
27 views

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, ...
0 votes
0 answers
32 views

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 ...
user avatar
1 vote
1 answer
158 views

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,...
  • 163

1
2 3 4 5
9