# 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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### Approximating a subclass of $L^2(\mathbb{R})$ by Schwartz functions within similar subclass

It is well-known that real valued Schwartz functions on $\mathbb{R}_+$ $\mathcal{S}(\mathbb{R}_+)$ are dense in the set of square integrable functions on $\mathbb{R}_+$ $L^2(\mathbb{R}_+)$. We can ...
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### 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 ...
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### 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 ...
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### 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 ...
1answer
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### 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.
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### 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 $\hat{f}$ ...
1answer
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### A problem regarding a generalization of Schwartz space

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 ...
1answer
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### 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 ...
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### Does $fT_n \to fT$ for a Schwartz Function and Tempered Distribution?

Let $f \in \mathcal{S}(\mathbb{R})$ and $T_n$ be a sequence of tempered distributions converging to $T$, which is also a tempered distribution. Is it true that $fT_n \to fT$? My solution to this seems ...
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### Riesz Interpolation for Bourgain space, and a question about Schwartz space

1)If I have an operator $T$ s.t $||Tv||_{X_{-s,0}} \le c_1 ||v||_{X_{-s,1-b}}$ and $||Tv||_{X_{-s,1-b}} \le c_2 ||v||_{X_{-s,1-b}}$. Where $X_{s,b}$ is the Bourgain space .Then how to apply Riesz ...
1answer
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### A function in the Schwartz space composed with a linear transformation.

Let $A$ a $n\times n$ matrix such that $\det(A)\neq0$, $f\in S(\mathbb{R}^n)$ in the Schwartz space, then $f\circ A$ is in the Schwartz space.
1answer
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### If $f$ is Schwartz, does $f=\mathcal{O}\left(e^{-a|x|^{\epsilon}}\right)$ for some $a,\epsilon>0$?

If $f$ is Schwartz, i.e. $f\in\mathcal{S}\left(\mathbb{R}\right)$, is it true that$f=\mathcal{O}\left(e^{-a|x|^{\epsilon}}\right)$ for some $a,\epsilon>0$ as $|x|\to\infty$? Functions in the ...
1answer
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### Exchanging limit and integral

If $\sup_{x \in \mathbb{R}^n} |f_m(x)|\rightarrow 0$ for $f$ a Schwartz function (i.e. $f$ is smooth and decays rapidly), is it true that $\|f_m\|_{L^1}=\int_{\mathbb{R}^n} |f_m(x)| dx \rightarrow 0$ ...
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### Convergence of fractional laplacian in $\mathcal{S}_s$

Let $s\in(0,1)$, let: $$\mathcal{S}_s=\biggl\{ f\in C^\infty(\mathbb{R}^n): \sup_{x\in\mathbb{R}^n}(1+|x|^{n+2s})|\partial^\alpha f(x)|<\infty,\,\forall \alpha\in\mathbb{N}_0^n\biggr\}.$$ The ...
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### Understanding the identity $f(x) \delta'(x-\xi) = - f'(\xi) \delta(x - \xi) + f(\xi) \delta'(x-\xi)$ in Fourier space

Consider the distributional identity (see Chapter 2.6 of Kanwal's Generalized Functions, for example) $$f(x) \delta'(x-\xi) = - f'(\xi) \delta(x - \xi) + f(\xi) \delta'(x-\xi) \tag{1}$$ I can see ...
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### Proving convolution of two Schwartz functions is again a Schwartz function on $\mathbb{R}^n$

This is something I have encountered in my analysis class. So suppose we take the space of Schwartz functions $S(\mathbb{R}^n)$ which are the rapidly decaying functions, or more formally the functions ...
1answer
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### Integral Inequality for $u\in\mathcal{S}(\mathbb{R}^n)$

Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that: $$\int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s},$$ ...
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### Extension of Fourier transform to $L^2$ by density of Schwartz functions

The Fourier transform is usually extended to the $L^2(\mathbb{R})$ space by invoking an argument that relies on the density of Schwartz functions in $L^2$. Often, this extension is explicitly written ...
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### On the boundary conditions of black body problem

The black body problem tries to find the spectral energy density of electromagnetic field per unit volume in a isolated cavity with electromagnetic free radiation (i.e., with no sources) The free ...
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### Uniform estimate on Schwartz functions away from support of Fourier transform

This question is a follow-up to this post, but hopefully it's a better attempt at formulating the same idea. Roughly speaking, I would like to obtain a uniform estimate on the size of a certain class ...
1answer
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### Uniform estimate on Schwartz functions with compactly supported Fourier transform

Let $\mathcal{C}$ be the class of all even Schwartz functions $f:\mathbb{R}\to\mathbb{R}$ satisfying the following conditions: The Fourier transform $\hat{f}$ is compactly supported; $f$ is non-...
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### Proving that a $C^\infty$ regularizing contraction semigroup leave a subspace invariant.

In Villani's Hypocoercity, I am faced with proving a statement which is indicated by the title. Consider a $C^\infty$ function $V: \mathbb{R}^n \to \mathbb{R}$, convergeing to $+\infty$ fast enough ...
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