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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Smooth functions into infinite dimensional Hilbert spaces

I am currently working with infinite-dimensional (separable) Hilbert spaces $\mathcal{H}$ and functions of the type \begin{align} f:\mathbb{R}^n\rightarrow\mathcal{H}. \end{align} What does it mean ...
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20 views

What are the elements of $G_0(\Omega)$ and of $G_1(\Omega)$?

Let $d$ be a number $\geq 0$, and $\Omega$ an open subset of $\mathbb{R}^n$. One calls d$^{\text{th}}$ Gevrey class in $\Omega$, and denotes by $G_d(\Omega)$, the space of $C^{\infty}$ functions $f$ ...
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43 views

Why does the metric on a Schwartz space generate the same topology as the family of seminorms?

I am reading Rauch's "Partial Differential Equations", and he makes a jump I don't understand. He defines the Schwarz space as the space of $C^\infty$ functions that decrease faster than any ...
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continuous spectrum of the self-adjoint Schrödinger Operator

I have following problem. Take $A= - f'' + uf $ an unbounded operator with $u \in \mathcal{S}(\mathbb{R},\mathbb{R})$ being the Schwartz class. The domain of $A$ is $H^2(\mathbb{R})$. (i) Show that A ...
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52 views

Eigenvalues of the one-dimensional Schrödinger operator

I am stuck with the following problem: Let $Af = -f'' + uf$ be an unbounded operator, with $u \in \mathcal{S}(\mathbb{R},\mathbb{R})$ in the Schwartz space. The domain of $A$ is the linear subspace $H^...
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1answer
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Equivalence of two different definitions of $s'$

I was reading Treves book and, on page 527, defined the following objects. A sequence $\tau = (\tau_{p})_{p\in \mathbb{Z}^{d}}$ is of slowly growing if there exists an integer $k \ge 0$ and a constant ...
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27 views

When is the product of a Schwartz function and a smooth function guaranteed to be a Schwartz function?

How can we classify functions $g \in C^\infty \left(\mathbb{R}^d\right)$ such that for all Schwartz functions $f$ on $\mathbb{R}^d$, $f g$ is again Schwartz? I believe this question could be answered ...
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1answer
105 views

Poission Summation Formula ProofWiki question

Suppose that $f(x)$ is a Schwarz function, which is defined here to be a function that satisfies for every $c \in \mathbb{R}, n \in \mathbb{N}, |f^{n}(x)| = o(|x|^c)$, then in a proof of the Poisson ...
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78 views

Problem of convergence in Schwartz class

Can someone help me to show the following: Let $\phi \in \mathcal{S}(\mathbb{R}^n)$ satisfying $$\partial^\alpha \phi(0) = 0,\;\;\; \forall |\alpha|<k,$$ for some integer $k>0$. Considere $f \...
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Schwarz space $S(\mathbb{R})$ is dense inside $L_p(\mathbb{R})$-spaces

I was wondering why the schwarz functions $S(\mathbb{R})$ are dense inside the $L_p(\mathbb{R})$ spaces and I was reading this answer, but I don't understand why the $g_t$ are in $S(\mathbb{R})$. ...
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What is the Schwartz-type space for Mellin transform?

It is well known that for $f\in S(\mathbb R)$, the Schwartz space, one can assert that $f^{(a)}$, $Ff^{(a)}$ (the Fourier transform of $f^{(a)}$) are also in $S(\mathbb R)$ for any $a=0,1,2,\ldots$. ...
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34 views

Cartesian product of Fréchet

Let $F_{1},...,F_{n}$ be Fréchet spaces. Is the cartesian product $F_{1}\times\cdots \times F_{n}$ again a Fréchet space? In particular, is the cartesian product $\mathcal{S}(\mathbb{R}^{d})\times\...
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23 views

Radial Schwarz functions are dense in radial tempered distributions

I am trying to show that any radial tempered distribution can be approximated by radial Schwarz functions, where $T\in{S}'(\mathbb{R}^n)$ being radial means $\langle{T},\phi\circ{R^T}\rangle=\langle{T,...
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Convolution operator on $L_\infty(\ell_q)$

Let $\mathcal{S}(\mathbb{R}^{n})$ be the Schwartz space of rapidly decreasing functions on $\mathbb{R}^{n}$ and let $\varphi \in \mathcal{S}(\mathbb{R}^{n})$. I would like to know whether the ...
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23 views

Vanishing Fourier transform and bounded functions

Let us consider $f$ a function in $C(\mathbb{R})\cap L^{\infty}$ admitting a Fourier transform (say, either in the $L^1$ sense or tempered distribution sense if $f \notin L^1$). We assume that $supp(\...
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20 views

Show that $f$ is a Schwartz function if and only if $|\partial^\alpha \phi(x)| \leq C_{\alpha ,N}(1+|x|)^{-N} $

Show that a smooth function $f$ is in Schwartz space if only if for all $\alpha\geq 0$ and $N\geq 0$ there is a constant $C_{\alpha,N}$ such that $$|\partial^\alpha \phi(x)| \leq C_{\alpha ,N}(1+|x|)^{...
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29 views

Distribution theory and Shannon sampling theorem

Let $C$ denote the Dirac comb distribution and let $\mathcal{F}$ denote the Fourier transform for tempered distributions. Let $x$ be any function in the Schwartz class $\mathcal{S}$ with $X = \mathcal{...
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Proving the continuity of Fourier Transform between Schwartz spaces via sequences

Prove that if $f_k \rightarrow f$ in the Schwartz space $\mathcal{S}(\mathbb{R}^n)$, then $\hat{f_k} \rightarrow \hat{f}$ in $\mathcal{S}(\mathbb{R}^n)$. This is the Exercise 2.2.2 in Loukas ...
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33 views

Relationship between Fourier transforms when $\varphi(x):=f(x,0)$

Suppose we are given a Schwartz function $f\in\mathcal{S}(\mathbb{R}^2)$ and we define $\varphi\in\mathcal{S}(\mathbb{R})$ by $\varphi(x):=f(x,0)$. I am wondering if there is a nice relationship ...
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54 views

even Schwartz function produces radial Schwartz function

suppose a Schwartz function $f\in\mathcal{S}(\mathbb{R})$ is given, and it is even, i.e. $f(-x)=f(x)$ for all $x\in\mathbb{R}$. Then the function $\mathbb{R}^n \ni x\mapsto f(\vert x\vert)$ is a ...
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29 views

Pointwise Functional Derivative?

Let $\phi$ be an element of some function space, say $\phi \in \mathcal{S}(\mathbb{R}^{d})$. Define: $$ f(\phi) = e^{\alpha \phi(x)}$$ for some fixed $x \in \mathbb{R}^{d}$ and $\alpha$ scalar. I'd ...
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Derivative as an Integral Kernel

Let $E, F$ be Fréchet spaces and $f: E \to F$. We say that $f$ is differentiable at $x$ in the direction of $h \in E$ if the following limit exists. \begin{eqnarray} Df[x](h) := \lim_{t \to 0}\frac{f(...
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Proving $e^{1/x}$ is not a distribution.

Let $\mathcal{S}(\mathbb{R})$ denote the space of Schwartz functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. By definition, any $u\in\mathcal{S}'$ satisfies that $...
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Relation between two definitions of differentiability

I'm working on my research project and I have two notions of differentiability. I'll state both in what follows. Definition 1: [Locally Convex Spaces] Let $E, F$ be locally convex spaces over some ...
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40 views

Fourier transform gives rise to the delta function

Let $\varphi$ be a real smooth function with sufficient decay and vanishing mean. Then there exist a functions $\psi$ such that: $$\psi(x)'=\varphi(x)$$ then $$\hat{\psi}(\xi) = \frac{\hat{\varphi}(\...
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Exponential Dominating any continuous Function?

Let $f \in C(\mathbb{R}^n,\mathbb{R}^k)$. For most familiar such functions we can find some $\lambda>0$ such that $\lim_{\|x\|_{\mathbb{R}^n}to \infty} e^{-\lambda x} \|f(x)\|_{\mathbb{R}^k} = 0$. ...
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25 views

Radon transform maps a Schwartz function to a differentiable function

This isn't supposed to be hard, but I just can't seem to be able to write this down. My Radon transform is defined as follows: Let $f\in C_c^{\infty}(\mathbb{R}^2)$, then $$Rf(s,\omega)=\int_{\mathbb{...
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Let $S$ denote Schwartz space , $T:L^p \to L^p$ be a linear operator . If the restriction of $T$ on $S$ is bounded , can we show that $T$ is bounded?

Let $S$ denote Schwartz space $1 \le p \lt \infty$, can we construst a linear (sublinear) operator $T:L^p \to L^p$ which is unbounded. However , the restriction $T|_S$ is bounded on $S$ ? ...
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Fourier transform of the regular tempered distributions induced by $x^\alpha$, $\sin(\langle a, x \rangle_{\mathbb R^n})$, and $\mathbb 1_{[-1,1]}$ [duplicate]

Compute the Fourier transform os the following tempered distributions on $\mathbb R^n$, which, for $\phi \in \mathcal S(\mathbb R^n)$ are given by $T_1(\phi) := \int_{\mathbb R^n} x^{\alpha} \...
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68 views

Fourier Transform of compactly supported Distribution is actually a Function

If $u$ is a compactly-supported distribution on $\mathbb{R}^n$, how can we prove that its Fourier transform $\mathcal{F}u$ is the tempered distribution given by the function $\xi\mapsto u(e^{-ix\xi})$?...
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42 views

Test Function Space and Schwartz Function Space

I understand that convergence in the set of test functions is that there needs to exist a null sequence ${\varphi}_{m}$ for a sequence of functions in $C^{\infty}_{0}(\mathbb{R}^{n})$ and the same is ...
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29 views

An operator invariant under translations is a convolution with a tempered distribution.

It is possible to extend a continuous operator defined in Schwartz space $S$ to $L^p,\ 1\leq p<\infty$? Example: Let $T$ a bounded operator defined on $L^p,\ 1\leq p<\infty$ invariant under ...
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42 views

Density of $\mathscr{S}$ in $H^s(\mathbb{R}^n)$

As usual, we define $H^s(\mathbb{R}^n)$ as the set of distributions $f\in\mathscr{S}'$ such that $\hat{f}\in L^2_{\text{loc}}$, and $$ \|f\|_{H^s}:=\left(\int \left(1+\left|\xi\right|^2\right)^s\left|\...
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Regard Schwartz space as a subspace of $L^2$ , then can it be written as a countable union of nowhere dense subset in $L^2$?

Let $\{T_n\}$ be a sequence of functional $T_n: L^2 \to C$ ; $S$ denote the schwartz space . If we can show that $$\lim_n T_n(g)$$ Converges for every $g \in S$ and $S$ is of second category (which ...
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Interpreting $f \in W_{k}^{p}(\mathbb R^n)$ as tempered distributions

In our functional analysis II lecture we defined to classical Sobolev space like this: Let $k \in \mathbb N_0$ and $p \in [1, \infty)$ . $$ W_p^k(\mathbb{R}^n) := \{ f: L^p(\mathbb R^n): D^{\...
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Product of a convolution by a monomial in Schwartz space.

How proves the next formula? Let $f,g\in S(R^n)$ Schwartz space. Then for all multi index $\alpha=(\alpha_1,\ldots, \alpha_n),\quad x^{\alpha}(f*g)(x)=\sum_{\gamma+\overline{\gamma}=\alpha}\frac{\...
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31 views

Translation of a Schwartz function is a Schwartz function?

If $f\in S(\mathbb{R}^n)$ and for all $y\in\mathbb{R}^n$ then $\tau_{y}f\in S(\mathbb{R^n})$? $\sup_{x} |x^{\alpha} \partial_{x}^{\beta} \tau_{y}f(x)|<\infty$?
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33 views

Finding a Schwartz function which makes the integral nonzero.

Let $f(\not\equiv 0)$ be an arbitrary continuous function from $\mathbb{R}^n \to \mathbb{R}$. Can we find/construct a Schwarts function $g \in \mathcal{S}(\mathbb{R}^n)$ such that $$\int_{\mathbb{R}^...
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Schwartz function with bounded Schwartz seminorms

Suppose $f:\mathbb R\to \mathbb C$ is a Schwartz function such that there is $C>0$ such that all of its Schwartz seminorms are bounded by $C$: $$ \sup_{m,n}\sup_x |x|^n |f^{(m)}(x)|\leq C. $$ Must $...
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48 views

Fourier transform of the regular distribution $ \lvert x \rvert^2 $, $ x \in \mathbb{R}^n $

Question 1: It is not complicated to show that the regular distribution $ \lvert x \rvert^2 = x_1^2 + ... + x_n^2 $, $ x \in \mathbb{R}^n $, is tempered. What is the Fourier transform of this ...
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Fourier transform of delta distribution satisfies $ \hat\delta_{x-x_0} = \hat\delta_{x_0} $

Let $ \delta_{x_0} $ be the distribution defined by $ \delta_{x_0} = \varphi (x_0) $, $ x_0 \in \mathbb{R^n} $. One can show that it has compact support, so in particular $ \delta \in \mathcal{S}'$, ...
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179 views

Weak convergence in the space of tempered distributions and weighted Sobolev spaces

It is well known that, at least as sets, $$\mathcal{S}'(\mathbb{R}^n)=\bigcup_{m\in\mathbb{N}}(1+|x|^2)^mH^{-m}(\mathbb{R}^n)$$ where $\mathcal{S}'(\mathbb{R}^n)$ is the space of tempered distribution ...
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1answer
127 views

Exercise $2.3.2$ of Grafakos, Classical Fourier Analysis.

I am trying to prove the exercise $2.3.2$ of "Grafakos-classical Fourier Analysis". Exercise $2.3.2.$ Let $\varphi,\ f\in \mathcal{S}(\mathbb{R}^n)$, and for $\epsilon>0$ let $\varphi_{\epsilon}(x)...
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41 views

Schwarz function if $|f(x)|\leq\frac{C}{(1+|x|^2)^N}$ for all $x$?

Knowing that $f:\mathbb{R}^n\longrightarrow\mathbb{C}$ is a measurable function such that for all $N\in\mathbb{N}$, there exists a constant $C=C_N$ such that $|f(x)|\leq\frac{C}{(1+|x|^2)^N}$ for all $...
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28 views

Prove that $x^2\cdot e^{-\frac{x^2}{2}}$ is a Schwartz function over $\mathbb{R}$

I need to show that the function $x^2\cdot e^{-\frac{x^2}{2}}$ belongs to the Schwartz space $\mathcal{S}(\mathbb{R})$. The fact that it is smooth is clear, but I am stuck when trying to prove that ...
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29 views

Proof of an inequality involving $Q\in H^{1}(\mathbb{R}^d).$

I am trying to understand a special case of Diamagnetic inequality mentioned in Appendix [B] of the book Nonlinear Dispersive Equations by Tao. Here is the statement, Let $Q\in H_x^{1}(\mathbb{R}^d)...
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1answer
66 views

Schwartz function exponential decay

Let $f$ be a Schwartz function in $\mathbb{R}^n$. I would like to know if there is some constant $\alpha > 0$ such that the function defined by $g(x) = f(x)e^{\frac{\lvert x \rvert^2}{\alpha}}$ is ...
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5 views

Dense subspace of $\mathcal{S}(G)$ for Nilpotent $G$

Let $G$ be a connected and simply connected nilpotent Lie group and let $\mathcal{S}(G)$ be it space of Schwartz functions, i.e. the image under $\exp$ of the Schwartz functions on its Lie algebra. ...
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46 views

Justify the idea of the Fourier transform of a tempered distribution

Let's say that I want to calculate the Fourier transform of $\frac{1}{|x|^2}$ in $\mathbb{R}^3$. This should not be defined as a function, because $\frac{1}{|x|^2} \notin L^1$, but it should define a ...
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30 views

Reference request on the Schwartz class on integers

It's a well known fact that the Schwartz class over integers defined by $$S(\mathbb{Z}^n)=\{\{a_m\}_{m\in\mathbb{Z^n}}\mid \sup_{m\in\mathbb{Z}^n}|m^\alpha a_m|<\infty, \forall \alpha\in\mathbb{N}^...

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