# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...