# 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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### On the definition of Schwartz functions

$$\mathcal{S}(\mathbb{R}^{n})=\Big\{f\in C^{\infty}(\mathbb{R}^{n})\,\Big|\; \sup_{\mathbb{R}^{n}}|x^{\alpha}D^{\beta}f(x)|<\infty\, \forall \alpha,\beta\, \text{multi indexes}\Big\}.$$ I can't ...
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### Properties of Schwartz functions

Let $f$ in Schwartz space, and $K, \gamma >0$. Then I guess the following: \begin{align} \int _{ -k }^{ -\gamma }{ \left| f(x) \right| dx } +\int _{ \gamma }^{ k }{ \left| f(x) \right| dx } &=...
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### For which functions/distributions X, $S(\mathbb R) \ast X \subset \ell ^{1}({\mathbb Z})$?

Let $f\in \mathcal{S}(\mathbb R^d)$ (Schwartz space) and define $f_k(x) =f(x-k) \ (k\in \mathbb Z^d)$ (translation of $f$). Question: For which kernel $h:\mathbb R^d \to \mathbb C,$ one can ...
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### Structure of tempered distribution

I was reading a note on distributions. The author left the following Schwartz representation theorem as an exercise: I'm trying to prove the theorem. According to the hints, I've done all the ...
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### Translation invariance of distributions

Let $\mathcal{S}(\mathbb{R}^4)$ denote the Schwartz space on $\mathbb{R}^4$, $\mathcal{S}^{\otimes n}(\mathbb{R}^4)$ the tensor product of $n$ of these spaces and $\mathcal{S}(\mathbb{R}^{4n})$ the ...
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### If $\hat{f}\in L^2(\mathbb{R})$ then $\hat{f}$ is rapidly decreasing.

This is problem 2.3.6 from the book of Mckean, "Fourier Series and Integrals". Problem: Check that if $\hat{f}\in L^2(\mathbb{R}^1)$ then $\hat{f}$ is rapidly decreasing in the sense of \$\gamma^n\...