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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Boundedness of singular integral operator on normalized bump functions implies boundedness on Schwartz functions controlled by suitable seminorms
In the book "Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory integrals" by E. Stein, the author considers on page 294 a singular integral operator $T:\mathcal{S}(\...
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Spherical Mean of Schwartz Function Decreases Rapidly - Stein, Shakarchi, Fourier Analysis.
I'm studying Stein and Shakarchi's Fourier Analysis book, and I'm stumped on page. 189, where it talks about the spherical mean of a function.
The book defines the spherical mean of a complex-valued ...
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Schwartz Function Convergence
I am a little bit confused about how convergence in the Schwartz space is defined. I am aware that the topology on the Schwartz space $S(\mathbf{R}^N)$ is metrizable by
$$
d(f, g) := \sup_n \frac{1}{n}...
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If $\psi(t)=\displaystyle\int\limits_{-\infty}^{t}\varphi(x)dx$ then $\psi\in\mathbb{S}(\mathbb{R}) $
Exercices :
Let $\varphi\in\mathbb{S}(\mathbb{R}) $ such that : $\displaystyle\int\limits_\mathbb{R}\varphi (x)dx=0 $
Let defined :
$$\psi(t)=\displaystyle\int\limits_{-\infty}^{t}\varphi(x)dx$$
Then ...
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Proving or disproving that if $0<\epsilon<1$ then $\frac{g(x)}{g(x/\epsilon)}$ is in Schwartz space when $\frac{1}{g(x)}$ is a Schwartz function.
I have a question about functions in Schwartz space.
$f(x)=\frac{1}{e^{x^2}}$ is in $\mathcal{S}(\mathbb{R})$ (Schwartz space)
If $0<\epsilon<1$, then $\frac{f(x)}{f(x/\epsilon)}=e^{x^2(\frac{\...
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Shifted Schwartz Functions are Schwartz?
I am asking about the same question as this post: Translation of a Schwartz function is a Schwartz function?
Namely: If $f\in S(\mathbf{R}^n)$ and for all $y\in\mathbf{R}^n$ then $\tau_{y}f\in S(\...
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Bound of norm in Schwartz class of functions
I am currently working my way through Stein's Functional Analysis and at one point in the text he claims without proof that for all $\phi \in\mathcal{S}$, where $\mathcal{S}$ is the Schwartz class of ...
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How can I show a Schwartz function is identical 0? [duplicate]
I have the following problem:
If a Schwartz funcetion $f\in\mathcal{S}(\mathbb{R})$ satisfies $f(2\pi n)=0$ and $\hat{f}(n)=0,\forall n\in\mathbb{Z}$, where $\hat{f}(\xi)=\int e^{-ix\xi}f(x)dx$ is the ...
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Solving Transport Equation with Velocity Switch using Tempered Distributions
Imagine we have the following two transport problems. The first is well known and can be solved using the Fourier transform, but I do not know how to solve the second one.
Problem 1: With Constant ...
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Clarification of the definition of tempered growth.
This is a rather simple question, but I don't have many resources I've been able to consult in this regard (meaning that a Google search hasn't yielded any meaningful results). I just want to clarify, ...
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Equivalence of topologies on the Schwartz space
I am dealing with the Schwartz space: set of all inifinite differentiable functions on $\mathbb{R}$ such that for all $a, b\geq 0$, $x^{a}f^{(b)}(x)$ is a bounded function. The topology is generated ...
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Show that exp is not a tempered distribution
I'm trying to show that $x \mapsto e^x$ is not a tempered distribution. For that I want to find a schwartz function $\varphi \in S(\mathbb{R}) $ such that the support of $\varphi$ is contained in $[-1,...
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Does this integral inequality hold?
Take the Schwartz space $\mathcal{S}(\mathbb{R}^{n})$ with family of norms:
$$\|f\|_{\alpha,\beta} = \sup_{x\in \mathbb{R}^{n}}|(\partial^{\alpha}f)(x)x^{\beta}|.$$
Now consider the following mapping:
...
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Question regarding rapidly decreasing in complex analysis
I have encountered some problem when trying to prove this theorm,
where the definition for rapidly decreasing is that We say that φ is rapidly decreasing if the function φ(x)p(x) is bounded for all ...
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An identity regarding Dirac delta function
From the Wolfram Mathworld website, I have it that:
$$
\int f(x)\delta^{(n)}(x) dx \equiv - \int \frac{\partial f}{\partial x} \delta^{(n-1)}(x)dx
$$
Now, taking $f(x)=x^2, n=1$ on the LHS, and ...
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Multi-variable Schwartz functions can be controlled by one variable Schwartz functions
I met the following concrete problem in my study, which I have no idea to write down a precise proof. So let $f(x_1,...,x_n)$ be a Schwartz function defined on $\mathbb{R}^n$. I want to show that ...
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Convolution in Schwarz Space
Suppose $f,g \in \mathcal{S}(\mathbb{R})$, where $\mathcal{S}(\mathbb{R})$ denotes the Schwarz space on $\mathbb{R}$. Then we need to show that
$$ x (f *g) = (xf)*g \;\;\;\; \mbox{ a.e. }$$ So, ...
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On a property of Schwartz functions in $\mathbb{R}^2$ [closed]
Let $f\in \mathcal{S}(\mathbb{R}^2)$, space of Schwartz functions on $\mathbb{R}^2$ and define
$$g(x)=\int_\mathbb{R} f(x,y) h(y) dy,$$
where $h$ is a compactly supported continuous function on $\...
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Fourier transform of $2\pi$-periodic function
I want to describe the image of $2\pi$-periodic tempered distributions under the Fourier transform
$$F: \mathcal S'(\mathbb R) \to \mathcal S'(\mathbb R), f \mapsto \left(\varphi \mapsto \langle F[f](\...
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Question that regards multiindex; if I have $x \in \mathbb{R}^n$ how may I express $|x^\alpha| = |x|$ for some $\alpha$?
I mean, the notation $x^{\alpha} = x_1^{\alpha_1}x_2^{\alpha_2}...x_n^{\alpha_n}$ for $\alpha = (\alpha_1,..., \alpha_n)$ and $|x|$ is the sum of all (?) So I don't see how that would be possible (if ...
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Periodic sum smooth iff it comes from Schwartz function?
Let's start with a continuous function $f : \mathbb R \to \mathbb R$ with enough decay to make
$$F(x) := \sum_{k \in \mathbb Z} f(x+k)$$
converge absolutely and uniformly (side question: does in this ...
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Fourier coefficients of smooth functions behave like Schwartz functions? [duplicate]
Let $f : \mathbb R \to \mathbb R$ be a $1$-periodic $L^1([0,1])$ function. Hence, the Fourier coefficients
$$a_n = \int_0^1 f(x)\exp(-2\pi i nx) dx$$
are well definied. Do we now have the equivalence
$...
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How to apply the Differential Operator $D^\beta$ to a map of Schwartz functions $F:\mathcal S^k \to \mathcal S$
I'm currently trying to prove, that we can use any function $F \in C^\infty (\mathbb{C}^k , \mathbb C)$ with $F=0$ to map Schwartz functions $f_j \in \mathcal S(\mathbb R ^d)$ to themselves. So
$$F \...
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For which $s \in \mathbb{R}$ is the surface measure an element of $H^{s}(\mathbb{R}^{m})$
I'm working on Q17 of sheet 2 of the course Distribution Theory and applications of part iii of the mathematical Tripos. I've left the link for the question out due to copyright reasons but you may ...
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Product of a Schwartz function and a function with polynomial bonuded derivatives is schwartz?
Is true the following?: If $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in\mathcal{C}^{\infty}(\mathbb{R})$ with $|g^{k}(|x|)|\leq p_k(x)$ all $k$ where $p_k(x)$ is a polynomial then $f(x)g(|x|)$ is in $\...
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Prove smooth functions vanishing at infinity belong to Schwartz space
I believe the definition of $C^{\infty}_0(\mathbb R^n)$ is
$$\{f(x):\forall\alpha\in\mathbb N^n,D^{\alpha}f(x)\in C_0 \cap C^{\infty}\}$$
note that $\alpha$ is a multi-index with nonnegative integer ...
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Confirm if this is correct: Proof of convergence with respect to the Schwartz metric implies convergence in Schwartz space
This is the definition of convergence in Schwartz space:
We say that $f_n \rightarrow f$ in $S$ if $\lim_{n \to \infty} ||f-f_n||_{ \alpha , \beta} \rightarrow 0$ for all $ \alpha , \beta \in \mathbb{...
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When can the Fourier transform change order in the inner product of $L^2? $
In $L^2(\mathbb{R}^n)$. Let $(f,g):=\int fg$.
If $f\in\mathcal{S}(\mathbb{R}^n)$ and $g\in L^2(\mathbb{R}^n)$.
When $(\mathcal{F}^{-1}(f),g)=(f,\mathcal{F}(g))$? This always holds in this case?
Thanks....
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Does this integral operator preserve the Schwartz space?
Let $\varphi_\mu$ denote the characteristic function of an infinitely divisible probability measure $\mu$ on $\mathcal B(\mathbb R^d)$ and $\mu^{\ast t}$ denote the convolution power for $t\ge0$. We ...
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It is true that $L^p(\mathbb{R}^n)\subset \mathscr{S}'(\mathbb{R}^n)$, $1\leq p\leq\infty$?
Remember that $\mathscr{S}'$ is the space of tempered distributions.
In a certain text they suggest that this statement is true, and that he uses Holder's inequality to prove it. The question is, how ...
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Leibniz Rule in $\mathscr{S}'(\mathbb{R}^n)$
I am interested in proving Leibniz's rule in $\mathscr{S}'(\mathbb{R}^n)$:
$$\partial_{j}^m(fu)=\sum_{k=0}^m\displaystyle{m\choose k}\left(\partial_{j}^k f\right)\left(\partial_{j}^{m-k} u\right),\...
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prove that $C_{c}^\infty\subset \mathcal{S}$.
In this exercice $C_{c}^\infty$ denotes the set of all functions defined in $\mathbb{R}^n$ that are infinitely differentiable and compactly supported and let us remember how the space of Schwartz. A ...
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If $f\in\mathcal{S}$, then the functions $\frac{\partial f}{\partial x_j}$ and $x_jf$ are in $\mathcal{S}$
Let us remember how the space of Schwartz. A multi-index $\alpha=(\alpha_1,...,\alpha_d)$ is an element of $\mathbb{N}_{0}^d$, where $\mathbb{N}_0=\mathbb{N}\cup \{0\}$, and the order of the multi-...
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If $u$ is a Schwartz function then so is $\exp(-\partial_{x}^2)u$?
If $u\in \mathcal{S}:=\mathcal{S}(\mathbb{R})$ (Schwartz space) then $\exp(-\partial_{x}^2)u\in \mathcal{S}$?
My attempt is the following
$\exp(-\partial_{x}^2)u=\sum_{k=0}^{\infty}\frac{(-\partial_{...
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A Schwartz function is measurable
prove in details that if $f \in \mathcal{S}$ (a Schwartz function) where$$
\mathcal{S}\left(\mathbb{R}^{n}\right):=\left\{f \in C^{\infty}\left(\mathbb{R}^{n}\right): \forall \alpha \in \mathbb{N}_{0}...
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Approximate family of tempered distributions by Schwartz functions
I write $S(\mathbb R^d)$ the space of Schwartz functions and $S'(\mathbb R^d)$ the space of tempered distributions.
Assume that I have a family $(u(t))_{t=0}^T \subset S'(\mathbb R^d)$ of tempered ...
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Estimate $N_p$ norm of $\psi : x \longmapsto \int_{-\infty}^x \phi(t)\ \mathrm dt$
Context and Assumptions : we consider the case of a very standard exercice in Schwartz functions theory: Let's suppose that $\phi$ is a $S(\mathbb{R})$ function, and let's suppose additionally that $\...
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Is the Hilbert space of quantum physics the space of tempered distributions?
I am currently studying quantum mechanics and I was introduced to Dirac notation (bra-ket notation). I have read multiple times that quantum states are represented by vectors called kets ($|\psi\...
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If $\langle u(t), \phi \rangle \in L^1\ \forall \phi \in S(\Bbb R^d)$, does $\langle u(t), \phi(t) \rangle \in L^1\ \forall \phi \in S(\Bbb R^{1+d})$?
Let $S(\mathbb R^d)$ be the space of Schwartz functions and $S'(\mathbb R^d)$ be the space of tempered distributions (the continuous linear functionals from $S(\mathbb R^d)$ to $\mathbb R$) endowed ...
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Symbol of the conjugate of a partial differential operator
Let $P=\sum_{\alpha}p_{\alpha}(x)D_x^{\alpha}$ be a partial differential opperator on $\mathbb{R}^m$ where ${\alpha}\equiv(\alpha_1,...,\alpha_m)$ is a multi-index and $D_x^{\alpha}=(-i)^{|\alpha|}\...
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Grafakos, Modern Fourier Analysis, Third Edition, Exercise 2.1.4. Bounded Tempered Distributions
In Grafakos book, Modern Fourier Analysis, the exercise 2.1.4 is as follows:
Let $P$ be the Poisson Kernel. Show that for any bounded tempered distribution $f$ we have $P_t \ast f \to f$ in $\mathcal{...
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Fourier transform in Schwartz space
Reading Rudin, I've got several questions related to the Fourier transform in Schwartz space (some not clear consequences or omitted proofs).
$ $ We define $\mathcal{S}\left(\mathbf{R}^{2}\right)$ the ...
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Convolution between a Bounded Tempered Distribution and a $L^1$ function is a distribution.
In this question we can see the definition of a convolution between a bounded tempered distribution and a $L^1$ function. In the comments, there are two possible ways to show that this is in fact a ...
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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 ...
user917818
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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 ...
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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} $ ...
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A problem regarding a generalization of Schwartz space [closed]
Consider a space of functions $f:\mathbb{R}^n\to \mathbb{C}$ satisfying
$$\sup_{x\in \mathbb{R}^n}|x^\alpha f(x)|<\infty, $$
for any multi index $\alpha$.
This space contains Schwartz space. Does ...
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Contraction principle and ODE
I would like to find a bounded $C^{\infty}$ solution of the following equation:
$$- \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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