# 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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### Interaction of pullback and the Fourier transform

I'd like to understand how the spectra of functions on a given domain are affected by (different kinds of) maps of that domain. Specifically, consider the Schwartz space $S(\mathbb R^n)$ of test ...
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### Is it really true that $\mathcal{S}(\mathbb{R}^n)$ is identified with smooth functions on $S^n$ vanishing at a fixed point?

Let $\mathcal{S}(\mathbb{R}^n)$ be the Schwartz space on $\mathbb{R}^n$ and $C^\infty(S^n)$ be the space of smooth functions on $n$-sphere. Now fix a point $x \in S^n$ and define C^\...
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### Formal Demonstration of an Integral Inequality Involving Nested Sets

I am delving into an analysis involving a specific integral inequality that arises within the context of Fourier analysis, particularly focusing on expressions involving a series of integrations over ...
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### $\sqrt{ f(x) + \alpha e^{-\lVert x \rVert^2}}$ is a Schwartz function for any nonnegative Schwartz $f$ and $\alpha>0$?

This question has been motivated by original and comments therein. Let $f$ be a nonnegative Schwartz function on $\mathbb{R}^n$ and let $\alpha \in (0,\infty)$ be fixed. Then, I wonder if \begin{...
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Let $f$ be a nonnegative Schwartz function on $n$-dimensional Euclidean space. Then I wonder if it is possible to find a sequence of complex-valued Schwartz functions $f_n$ such that $\lvert f_n \... • 7,715 1 vote 0 answers 28 views ### Convolution of Schwartz and$C^\infty$function with bounded increasing [duplicate] I have the following problem: Let$f\in\mathcal{S}(\mathbb{R}^n)$and$g\in C^\infty(\mathbb{R}^n)$such that exists$\alpha>0$$$\left|g(x)\right|\leq \frac{1}{1+|x|^\alpha} \quad \forall x\in\... • 355 1 vote 1 answer 43 views ### Question regarding the indexes \alpha and \beta in the definition of Schwartz space I was asked to prove that the Fourier transform of any Schwartz function belongs to the Schwartz class, i.e:$$||\xi^{\alpha}\frac{d^{|\beta|}\hat{f}}{d\xi^{\beta}}||<\infty$$For \alpha and \... • 13 0 votes 0 answers 74 views ### Is the Schwartz space "sequentially dense" in the space of tempered dsitributions with respect to some topology? Let us consider the Euclidean space \mathbb{R}^n of any n \in \mathbb{N}. Then, I somehow vaguely recall that the Schwartz space \mathcal{S}(\mathbb{R}^n) is dense in the space of tempered ... • 7,715 1 vote 0 answers 35 views ### For f \in H^s, then \exists g \in C_c such that f=g a.e. Consider the space H^s(\mathbb R^d) (f \in L^2 not in Schwartz class), s \in \mathbb R. Apply Riemann-Lebesgue Lemma to \hat{f} to show that for some s>s_0 then there is a continuous ... • 1,525 1 vote 0 answers 136 views ### Paley-Wiener theorem conversion proof Let v:\mathbb{C}\rightarrow \mathbb{C} be entire function with an estimate |v(z)| \leq C_N(1+|z|)^{-N}e^{R|Imz|} for all z \in \mathbb{C} then there is u \in S(\mathbb{R}) such that supp(u) \... • 169 0 votes 0 answers 54 views ### How to estimate schwartz space functions? I'm currently learning about Schwartz spaces and I'm having difficulties when I'm trying to estimate the derivatives of the functions. For example when you have a function e^{-ln(1+x^2)^N} and you ... • 169 0 votes 1 answer 54 views ### Confusion with identities in different function spaces Let \mathcal{S}(\mathbb{R}^3,\mathbb{R}^3) be the space of vector fields on \mathbb{R}^3 with Schwartz function components. Next, let \mathcal{S}_{div}(\mathbb{R}^3,\mathbb{R}^3) be the space of ... • 7,715 1 vote 2 answers 79 views ### Proving that \frac{x}{x^2+\varepsilon^2}\to \mathsf{p.v.}\left(\frac1x\right) in \mathscr{S}' Let \mathscr{S}' be the space of tempered distributions. I've to prove that:$$\frac{x}{x^2+\varepsilon^2}\to \mathsf{p.v.}\left(\frac1x\right) \ \ \text{ in$\mathscr{S}'$}$$My book says that it'... • 3,654 0 votes 0 answers 23 views ### Proving that \rho_\varepsilon\ast U\to U as \varepsilon\to 0^+ I'm trying to understand the proof of the following fact about tempered distributions. Let U\in \mathscr{S}' be a tempered distribution, and let \{\rho_\varepsilon\} be the family of standard ... • 3,654 0 votes 0 answers 67 views ### Schwartz kernel theorem and musical isomorphisms I am reading about the Schwartz kernel/nuclear theorem which states for any bilinear form B on \mathcal{S}(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^n) (where \mathcal{S}(\mathbb{R}^n) is ... • 6,031 8 votes 3 answers 565 views ### How can tempered distributions be identified with functions? In Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, he defines tempered distribution (\mathscr S') as continuous linear functionals from the Schwartz class. ... 0 votes 0 answers 55 views ### Tempered distribution locally integrable function Let f \in L^1_{\text{loc}}\left(\mathbb{R}\right) be a nonnegative function. Show that if \Lambda_f is a tempered distribution, then there exist C > 0, N \in \mathbb{N}_0 satisfying$$\forall ... • 371 2 votes 0 answers 39 views ### Given a Schwartz function$f(x,y)$on$\mathbb{R}^{2n}$, is$f(x,x)$necessarily a Schwartz functoin on$\mathbb{R}^n$? I am suddenly a bit confused by the question described in the title. Let$f(x,y)$be a Schwartz function on$\mathbb{R}^{2n}$with$x,y \in \mathbb{R}^n$. Then, is it necessarily true that$x \to f(x,...
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Let $\mathscr{S}(\mathbb{R})$ denote Schwartz space, $T \in \mathscr{S}'(\mathbb{R})$, and $f \in \mathscr{S}(\mathbb{R})$. How can we make sense of the product $fT$? In the second volume of Reed &...