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
43 views
+100

Boundedness of singular integral operator on normalized bump functions implies boundedness on Schwartz functions controlled by suitable seminorms

In the book "Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory integrals" by E. Stein, the author considers on page 294 a singular integral operator $T:\mathcal{S}(\...
user avatar
  • 36
1 vote
0 answers
25 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 ...
user avatar
0 votes
0 answers
10 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}...
user avatar
2 votes
0 answers
49 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 ...
user avatar
  • 2,129
0 votes
0 answers
11 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{\...
user avatar
  • 2,509
0 votes
1 answer
48 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(\...
user avatar
1 vote
1 answer
63 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 ...
user avatar
1 vote
0 answers
44 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 ...
user avatar
  • 31
2 votes
2 answers
106 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 ...
user avatar
0 votes
0 answers
8 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, ...
user avatar
0 votes
0 answers
16 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
41 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,...
user avatar
  • 337
1 vote
0 answers
49 views

Does this integral inequality hold?

Take the Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ with family of norms: $$\|f\|_{\alpha,\beta} = \sup_{x\in \mathbb{R}^{n}}|(\partial^{\alpha}f)(x)x^{\beta}|.$$ Now consider the following mapping: ...
user avatar
0 votes
0 answers
25 views

Question regarding rapidly decreasing in complex analysis

I have encountered some problem when trying to prove this theorm, where the definition for rapidly decreasing is that We say that φ is rapidly decreasing if the function φ(x)p(x) is bounded for all ...
user avatar
  • 29
0 votes
0 answers
28 views

An identity regarding Dirac delta function

From the Wolfram Mathworld website, I have it that: $$ \int f(x)\delta^{(n)}(x) dx \equiv - \int \frac{\partial f}{\partial x} \delta^{(n-1)}(x)dx $$ Now, taking $f(x)=x^2, n=1$ on the LHS, and ...
user avatar
1 vote
2 answers
50 views

Multi-variable Schwartz functions can be controlled by one variable Schwartz functions

I met the following concrete problem in my study, which I have no idea to write down a precise proof. So let $f(x_1,...,x_n)$ be a Schwartz function defined on $\mathbb{R}^n$. I want to show that ...
user avatar
0 votes
0 answers
37 views

Convolution in Schwarz Space

Suppose $f,g \in \mathcal{S}(\mathbb{R})$, where $\mathcal{S}(\mathbb{R})$ denotes the Schwarz space on $\mathbb{R}$. Then we need to show that $$ x (f *g) = (xf)*g \;\;\;\; \mbox{ a.e. }$$ So, ...
user avatar
0 votes
1 answer
43 views

On a property of Schwartz functions in $\mathbb{R}^2$ [closed]

Let $f\in \mathcal{S}(\mathbb{R}^2)$, space of Schwartz functions on $\mathbb{R}^2$ and define $$g(x)=\int_\mathbb{R} f(x,y) h(y) dy,$$ where $h$ is a compactly supported continuous function on $\...
user avatar
4 votes
1 answer
87 views

Fourier transform of $2\pi$-periodic function

I want to describe the image of $2\pi$-periodic tempered distributions under the Fourier transform $$F: \mathcal S'(\mathbb R) \to \mathcal S'(\mathbb R), f \mapsto \left(\varphi \mapsto \langle F[f](\...
user avatar
  • 460
0 votes
0 answers
20 views

Question that regards multiindex; if I have $x \in \mathbb{R}^n$ how may I express $|x^\alpha| = |x|$ for some $\alpha$?

I mean, the notation $x^{\alpha} = x_1^{\alpha_1}x_2^{\alpha_2}...x_n^{\alpha_n}$ for $\alpha = (\alpha_1,..., \alpha_n)$ and $|x|$ is the sum of all (?) So I don't see how that would be possible (if ...
user avatar
2 votes
0 answers
35 views

Periodic sum smooth iff it comes from Schwartz function?

Let's start with a continuous function $f : \mathbb R \to \mathbb R$ with enough decay to make $$F(x) := \sum_{k \in \mathbb Z} f(x+k)$$ converge absolutely and uniformly (side question: does in this ...
user avatar
0 votes
1 answer
40 views

Fourier coefficients of smooth functions behave like Schwartz functions? [duplicate]

Let $f : \mathbb R \to \mathbb R$ be a $1$-periodic $L^1([0,1])$ function. Hence, the Fourier coefficients $$a_n = \int_0^1 f(x)\exp(-2\pi i nx) dx$$ are well definied. Do we now have the equivalence $...
user avatar
1 vote
0 answers
16 views

How to apply the Differential Operator $D^\beta$ to a map of Schwartz functions $F:\mathcal S^k \to \mathcal S$

I'm currently trying to prove, that we can use any function $F \in C^\infty (\mathbb{C}^k , \mathbb C)$ with $F=0$ to map Schwartz functions $f_j \in \mathcal S(\mathbb R ^d)$ to themselves. So $$F \...
user avatar
  • 171
1 vote
1 answer
43 views

For which $s \in \mathbb{R}$ is the surface measure an element of $H^{s}(\mathbb{R}^{m})$

I'm working on Q17 of sheet 2 of the course Distribution Theory and applications of part iii of the mathematical Tripos. I've left the link for the question out due to copyright reasons but you may ...
user avatar
  • 301
1 vote
1 answer
41 views

Product of a Schwartz function and a function with polynomial bonuded derivatives is schwartz?

Is true the following?: If $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in\mathcal{C}^{\infty}(\mathbb{R})$ with $|g^{k}(|x|)|\leq p_k(x)$ all $k$ where $p_k(x)$ is a polynomial then $f(x)g(|x|)$ is in $\...
user avatar
  • 2,509
0 votes
0 answers
35 views

Prove smooth functions vanishing at infinity belong to Schwartz space

I believe the definition of $C^{\infty}_0(\mathbb R^n)$ is $$\{f(x):\forall\alpha\in\mathbb N^n,D^{\alpha}f(x)\in C_0 \cap C^{\infty}\}$$ note that $\alpha$ is a multi-index with nonnegative integer ...
user avatar
0 votes
1 answer
52 views

Confirm if this is correct: Proof of convergence with respect to the Schwartz metric implies convergence in Schwartz space

This is the definition of convergence in Schwartz space: We say that $f_n \rightarrow f$ in $S$ if $\lim_{n \to \infty} ||f-f_n||_{ \alpha , \beta} \rightarrow 0$ for all $ \alpha , \beta \in \mathbb{...
user avatar
  • 171
0 votes
1 answer
21 views

When can the Fourier transform change order in the inner product of $L^2? $

In $L^2(\mathbb{R}^n)$. Let $(f,g):=\int fg$. If $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in L^2(\mathbb{R}^n)$. When $(\mathcal{F}^{-1}(f),g)=(f,\mathcal{F}(g))$? This always holds in this case? Thanks....
user avatar
  • 2,509
1 vote
0 answers
40 views

Does this integral operator preserve the Schwartz space?

Let $\varphi_\mu$ denote the characteristic function of an infinitely divisible probability measure $\mu$ on $\mathcal B(\mathbb R^d)$ and $\mu^{\ast t}$ denote the convolution power for $t\ge0$. We ...
user avatar
  • 12.8k
1 vote
0 answers
68 views

It is true that $L^p(\mathbb{R}^n)\subset \mathscr{S}'(\mathbb{R}^n)$, $1\leq p\leq\infty$?

Remember that $\mathscr{S}'$ is the space of tempered distributions. In a certain text they suggest that this statement is true, and that he uses Holder's inequality to prove it. The question is, how ...
user avatar
0 votes
0 answers
15 views

Leibniz Rule in $\mathscr{S}'(\mathbb{R}^n)$

I am interested in proving Leibniz's rule in $\mathscr{S}'(\mathbb{R}^n)$: $$\partial_{j}^m(fu)=\sum_{k=0}^m\displaystyle{m\choose k}\left(\partial_{j}^k f\right)\left(\partial_{j}^{m-k} u\right),\...
user avatar
1 vote
0 answers
38 views

prove that $C_{c}^\infty\subset \mathcal{S}$.

In this exercice $C_{c}^\infty$ denotes the set of all functions defined in $\mathbb{R}^n$ that are infinitely differentiable and compactly supported and let us remember how the space of Schwartz. A ...
user avatar
0 votes
0 answers
50 views

If $f\in\mathcal{S}$, then the functions $\frac{\partial f}{\partial x_j}$ and $x_jf$ are in $\mathcal{S}$

Let us remember how the space of Schwartz. A multi-index $\alpha=(\alpha_1,...,\alpha_d)$ is an element of $\mathbb{N}_{0}^d$, where $\mathbb{N}_0=\mathbb{N}\cup \{0\}$, and the order of the multi-...
user avatar
0 votes
0 answers
16 views

If $u$ is a Schwartz function then so is $\exp(-\partial_{x}^2)u$?

If $u\in \mathcal{S}:=\mathcal{S}(\mathbb{R})$ (Schwartz space) then $\exp(-\partial_{x}^2)u\in \mathcal{S}$? My attempt is the following $\exp(-\partial_{x}^2)u=\sum_{k=0}^{\infty}\frac{(-\partial_{...
user avatar
  • 2,509
2 votes
1 answer
65 views

A Schwartz function is measurable

prove in details that if $f \in \mathcal{S}$ (a Schwartz function) where$$ \mathcal{S}\left(\mathbb{R}^{n}\right):=\left\{f \in C^{\infty}\left(\mathbb{R}^{n}\right): \forall \alpha \in \mathbb{N}_{0}...
user avatar
  • 183
1 vote
0 answers
46 views

Approximate family of tempered distributions by Schwartz functions

I write $S(\mathbb R^d)$ the space of Schwartz functions and $S'(\mathbb R^d)$ the space of tempered distributions. Assume that I have a family $(u(t))_{t=0}^T \subset S'(\mathbb R^d)$ of tempered ...
user avatar
  • 1,714
2 votes
0 answers
38 views

Estimate $N_p$ norm of $\psi : x \longmapsto \int_{-\infty}^x \phi(t)\ \mathrm dt$

Context and Assumptions : we consider the case of a very standard exercice in Schwartz functions theory: Let's suppose that $\phi$ is a $S(\mathbb{R})$ function, and let's suppose additionally that $\...
user avatar
4 votes
1 answer
178 views

Is the Hilbert space of quantum physics the space of tempered distributions?

I am currently studying quantum mechanics and I was introduced to Dirac notation (bra-ket notation). I have read multiple times that quantum states are represented by vectors called kets ($|\psi\...
user avatar
  • 41
7 votes
0 answers
165 views

If $\langle u(t), \phi \rangle \in L^1\ \forall \phi \in S(\Bbb R^d)$, does $\langle u(t), \phi(t) \rangle \in L^1\ \forall \phi \in S(\Bbb R^{1+d})$?

Let $S(\mathbb R^d)$ be the space of Schwartz functions and $S'(\mathbb R^d)$ be the space of tempered distributions (the continuous linear functionals from $S(\mathbb R^d)$ to $\mathbb R$) endowed ...
user avatar
  • 1,714
0 votes
0 answers
25 views

Symbol of the conjugate of a partial differential operator

Let $P=\sum_{\alpha}p_{\alpha}(x)D_x^{\alpha}$ be a partial differential opperator on $\mathbb{R}^m$ where ${\alpha}\equiv(\alpha_1,...,\alpha_m)$ is a multi-index and $D_x^{\alpha}=(-i)^{|\alpha|}\...
user avatar
0 votes
1 answer
151 views

Grafakos, Modern Fourier Analysis, Third Edition, Exercise 2.1.4. Bounded Tempered Distributions

In Grafakos book, Modern Fourier Analysis, the exercise 2.1.4 is as follows: Let $P$ be the Poisson Kernel. Show that for any bounded tempered distribution $f$ we have $P_t \ast f \to f$ in $\mathcal{...
user avatar
  • 58
2 votes
0 answers
100 views

Fourier transform in Schwartz space

Reading Rudin, I've got several questions related to the Fourier transform in Schwartz space (some not clear consequences or omitted proofs). $ $ We define $\mathcal{S}\left(\mathbf{R}^{2}\right)$ the ...
user avatar
1 vote
0 answers
97 views

Convolution between a Bounded Tempered Distribution and a $L^1$ function is a distribution.

In this question we can see the definition of a convolution between a bounded tempered distribution and a $L^1$ function. In the comments, there are two possible ways to show that this is in fact a ...
user avatar
  • 58
0 votes
0 answers
46 views

Is it true that $\widehat{f(x,0)}(\xi) = \widehat{f}(\xi,0)$?

Let $f \in S(\mathbb{R}^2)$ and $g(x)= f(x,0)$. What is the relationship between $\widehat{g}$ and $\widehat{f}$? It's clear that $g \in S(\mathbb{R})$ so both $f$ and $g$ have a Fourier transform. I ...
user avatar
0 votes
0 answers
10 views

Find all $K \in S(R)'$

Find all $K \in S(R)'$ such that $T_m : L_1(\mathbb{R}) \rightarrow L_1(\mathbb{R})$ defined by $T_m(f) = K * f$ for all $f \in S(R)$. Where, $K *f$ is convolution of $K$ and $f$. I am not sure what ...
user avatar
  • 1,821
3 votes
0 answers
106 views

Is there a metrizable topology on tempered distributions for which sequential convergence is pointwise?

Is there a metrizable topology on $S’$ (the set of linear continuous functionals in Schwarz’s space) whose convergent sequences are the sequences that converge pointwise? The obvious topology with ...
user avatar
  • 41
2 votes
1 answer
148 views

Show that $f(x) = e^x \cos(x)$ on $\mathbb{R}$ is a tempered distribution

As shown in the title. I know that the anti-derivative of $f(x) = e^x \cos(x)$ is $ \frac{\sin(x)+\cos(x)}{2} e^x$, whose anti-derivative is $\frac{\sin(x)}{2}e^x$...but not sure how to prove it.
user avatar
  • 35
1 vote
0 answers
59 views

Can I generalize the classical Bernstein inequality?

I have know the proof of the classical Bernstein inequality as follows. Given some function $ f\in S(\mathbb{R}^d) $, where $ S(\mathbb{R}^d) $ denotes the Schwartz space of functions. Let $ \hat{f} $ ...
user avatar
-1 votes
1 answer
49 views

A problem regarding a generalization of Schwartz space [closed]

Consider a space of functions $f:\mathbb{R}^n\to \mathbb{C}$ satisfying $$\sup_{x\in \mathbb{R}^n}|x^\alpha f(x)|<\infty, $$ for any multi index $\alpha$. This space contains Schwartz space. Does ...
user avatar
0 votes
1 answer
62 views

Contraction principle and ODE

I would like to find a bounded $C^{\infty}$ solution of the following equation: $$- \frac{d^2}{dx^2} u + u -\epsilon u^2 = f,$$ where $f$ is a function in the Schwartz space and $\epsilon >0$ is ...
user avatar
  • 786

1
2 3 4 5
8