All Questions
Tagged with functional-analysis fourier-analysis
1,190 questions
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Prove that Cesaro sums of a Fourier series do not have larger sup-norm
How do I show the sup norm of the Cesaro sum of the Fourier partial sums is less than or equal to the sup norm of $f$ i.e. $\|\sigma(f)\|_\infty\leq\|f\|_\infty$
(Sorry I'm new to StackExchange) I ...
- 19
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0
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32
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Density of a subset of Schwartz space in the p-fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $W^{s,p}(\mathbb{R}^N)$ ($0<s<1$ and $1<p<\infty$) defined as
$$
W^{s,p}(\mathbb{R}^...
8
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1
answer
301
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Defintion of distributions why not define with complex conjugate
For a complex valued locally integrable function $f$ on an open set $U \subset \mathbb{R}^{n}$, I saw many sources defined distribution induced by $f$ as $\phi \mapsto \int f\phi \, dx$.
If $\phi$ is ...
- 1,948
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54
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$\frac{1}{p}+\frac{1}{q}=1, 1\leq p\leq \infty$, a multiplier is bounded on $L^{p}$ if it is bounded on $L^{q}$
On wikipedia, it was mentioned that for $\frac{1}{p}+\frac{1}{q}=1, 1\leq p\leq \infty$, a multiplier is bounded on $L^{p}$ if it is bounded on $L^{q}$. I wonder what proof of this would be.
For any $...
- 1,948
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1
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39
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$\mathcal{F}{\left( \int _{\epsilon}^{R}Q_{t}^{2}f \, dx \right)}=\int _{\epsilon}^{R}\widehat{Q_{t}^{2}f} \, dx$?
Let $\psi \in \mathcal{S}(\mathbb{R}^{n})$ be radial, $\int_{\mathbb{R}^{n}} \psi(x) \, dx =0$, $\int _{0}^{\infty}|\hat{\psi}(t)|^{2} \frac{1}{t} \, dt = 1$. Define $Q_{t}f(x)=\psi_{t} * f(x)=t^{-n}\...
- 1,948
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1
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61
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Density of a subset of Schwartz space in the fractional Sobolev space
It is known that the Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is dense in the fractional Sobolev space $H^s(\mathbb{R}^N)$ ($0<s<1$), as $C_{c}^{\infty}(\mathbb{R}^N) \subset \mathcal{S}(\...
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38
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Fourier transform of 2D Hilbert transform
I would like to rigorously evaluate the Fourier Transform of the 2D analogue of the symbol of the Hilbert transform $s:\mathbb R^2 \setminus \{0\} \to \mathbb C$ with
$$ s(x) = \frac{(x_1+ix_2)^m}{\...
1
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1
answer
42
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Applying of Sobolev Trace Theorem
THEOREM 1 (Trace Theorem). Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exists a bounded linear operator
$$
T: W^{1, p}(U) \rightarrow L^p(\partial U)
$$
such that
(i) $T u=\left.u\...
- 769
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52
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$\phi * Hf = H\phi * f$ (almost) everywhere.
When teaching about Hilbert transform, it seems that my professor used the following:
Let $\phi \in C_{c}^{\infty}$, $f \in L^{p}$ for $1<p<\infty$ (or I guess assume $f\in S(\mathbb{R})$ for ...
- 1,948
2
votes
1
answer
293
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Topology generated by seminorm, why not just take supremum
Let $\{p_{a}\}_{\alpha \in A}$ be a family of seminorms on the vector space on the vector space $\mathcal{X}$, the canonical definition of the topology generated by these seminorms is the topology ...
- 1,948
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52
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$\lim_{ t \to 0} \int Q_{t} * f(x)e^{-2\pi i x \xi} \, dx = \int \lim_{ t \to 0 } Q_{t} * f(x)e^{-2 \pi ix \xi} \, dx $?
Let $Q_t(x)=\frac{1}{\pi} \frac{x}{t^{2}+x^{2}}$, assume $f \in \mathcal{S}(\mathbb{R}^{n})$, my professor claimed that
$$
\lim_{ t \to 0} \int (Q_{t} * f)(x)e^{-2\pi i x \xi} \, dx = \int \lim_{ t \...
- 1,948
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42
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Show that fourier coefficient map is injective
Suppose that we take a function $f \in L^1(S^1)$, where $S^1$ is the unit circle, and we define the $n$-th Fourier coefficient as $$\hat{f}(n) = \langle f, z^n \rangle _{L^2(S^1)}$$
where the $L^2$ ...
- 41
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Fourier transform for function restricted to a line
Given $f \in \mathcal{S}(\mathbb{R}^{n})$, we can define the fourier transform of $f$, $\hat{f}$ by $\hat{f}(\xi) := \int _{\mathbb{R}^{n}}f(x)e^{-2\pi ix\cdot \xi} \, dx$.
Now consider the function $...
- 1,948
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43
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Proof mistake of: $M_0A(G) = B(G)$ for a locally compact group
Let $G$ be a locally compact group with Haar measure $\mu$, and $B(G),A(G),C_r^*(G),L(G)$ be its Fourier-Stieltjes algebra, Fourier algebra, group $C^* $-algerba and von Neumann algebra respectively. ...
- 539
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52
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Convergence of $p$-series for p=2 in the case of multi-indices.
We all know that the series $\sum_{1}^{\infty} 1/k^{2}$ converges. But is it true that $\sum_{|k|=1}^{\infty} 1/|k|^{2}$ converges when $k\in \mathbb{N}_{0}\times \cdots \times \mathbb{N}_{0}$ (n-...
- 71
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99
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Operator strong $(p,p)$ implies adjoint strong $(p',p')$
Let $T$ be a linear and continuous operator $T: L^{2}(\mathbb{R}^{d}) \to L^{2}(\mathbb{R}^{d})$. I wonder to what extent we can conclude if $T$ is strong $(p,p)$, we will obtain that $T'$, the ...
- 1,948
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47
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Fourier Transformation of Heaviside Function
Let be $H(x)$ Heaviside function. I want to show that $H$ defines a tempered distribution and compute is Fourier transformation.
The exercise has two steps:
I have alredy proof that $\xi\left(i\hat{H}...
1
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1
answer
168
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$B=\{\frac{1}{\sqrt{2\pi}},\frac{\sin(x)}{\sqrt{\pi}},\frac{\cos(x)}{\sqrt{\pi}},\frac{\sin(2x)}{\sqrt{\pi}},\frac{\cos(2x)}{\sqrt{\pi}},\dots\}$
I know that $B = \left\{ \frac{1}{\sqrt{2\pi}}, \frac{\sin(x)}{\sqrt{\pi}}, \frac{\cos(x)}{\sqrt{\pi}}, \frac{\sin(2x)}{\sqrt{\pi}}, \frac{\cos(2x)}{\sqrt{\pi}}, \dots \right\}$ is a countable ...
user1390718
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40
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When does a function $g\colon\mathbb R^d\to\mathbb R$ $g(-i\nabla)$ define an operator on $L^2(\mathbb R^d)$?
When does a function $g\colon\mathbb R^d\to\mathbb R$ $g(-i\nabla)$ define an operator on $L^2(\mathbb R^d)$?
The operator $g(-i\nabla)$ is defined on the Fourier side. For any $\psi\in L^2(\mathbb R^...
- 329
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58
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Why are Besov norms weighted by powers of 2?
For $p, r \in [1, \infty]$, $s \in \mathbb{R}$ and $u$ a tempered distribution we can define the Besov norm $\|\cdot\|_{B_{p,r}^s}$ as follows:
$$\|u\|_{B_{p,r}^s} := \Big\|\big(2^{js}\|\Delta_ju\|_{L^...
- 6,641
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90
views
What does the $\dot{H}^s$ norm measure?
For integer integers $k \geq 0$ and $p \geq 1$ the Sobolev space $W^{k,p}(\mathbb{R}^d)$ is defined as the functions belonging to $L^p(\mathbb{R}^d)$ and whose weak derivatives up to order $k$ are ...
- 6,641
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41
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An inequality including maximal function
Let $ f \in L^1\left(\mathbb{R}^n\right) $. Prove that for any $ \lambda > 0 $,
$$
\left|\left\{x \in \mathbb{R}^n : M f(x) > \lambda \right\}\right| \leq \frac{C}{\lambda} \int_{(|f| > \...
- 769
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43
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Definition of the space $\mathcal{S}'_h(\mathbb{R}^d)$
I am reading the book Fourier Analysis and Nonlinear Partial Differential Equations by Bahouri, Chemin, and Danchin, in which they give the following definition:
We denote by $\mathcal{S}_h'(\mathbb{...
- 6,641
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1
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167
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Do function in $u\in C^{1}_c(\mathbb{R}^n)\cap \dot{H}^1$ satisfy the following decay estimate?
Let $u\in C^{1}_c(\mathbb{R}^n)\cap \dot{H}^1$ be a smooth compactly supported function. Then is the following decay estimate correct?
\begin{align}
|u|(r,\theta) &\leq \int_r^\infty |\partial_{\...
- 9,326
3
votes
1
answer
243
views
Proof of Dirac delta representation as sum of exponentials
Physicists sometimes use the formula $\delta(x)=\frac1L\sum_{n\in\mathbb Z} e^{2\pi i nx/L}$, where $\delta(x)$ is the Dirac delta "function." I think the rigorous way to interpret this ...
- 6,898
5
votes
2
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551
views
Can the Fourier transform of a test function vanish on an interval?
Suppose $f$ is a test function and thus compactly supported on some set $S$.
Its Fourier transform is
\begin{equation}
\widehat{f}(p) = \int_{-\infty}^{\infty} dx \, e^{ipx} f(x) \, .
\end{equation}
...
- 246
2
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1
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78
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Confusion regarding Amenability and Weak Amenability
I was reading Todorov's notes about Herz-Schur Multipliers: Herz-Schur Multipliers and there is something I am finding weird related to (weak) amenability and the Fourier algebra.
On page 8, after ...
2
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1
answer
116
views
Identity involving $\sinh(l\cdot\nabla_v)$
Observe that we have the identity
$$\nabla_v(v^\alpha g)=v^\alpha\nabla g+\nabla_v(v^\alpha)g.$$
Do we have something similar for the operator $\sinh(l\cdot\nabla_v)$? That is, does $$\sinh(l\cdot\...
- 329
2
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0
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104
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Is there a coherent notion of "weak orthogonality" of functions/distributions?
Recently I gave an answer to Laplace transforms of non-exponential, non-sinusoidal functions that claims the following:
Our [change-of-basis] interpretation doesn't quite survive [the generalization ...
- 1,385
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0
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55
views
Is there a bounded discontinuous function in the Sobolev space $H^{1/2}(\mathbb{S}^1)$?
I want to know, whether there is an example of an bounded, discontinuous function in $H^{1/2}(\mathbb{S}^1)$.
For the general question for $W^{s,p}(\mathbb{R}^n)$, if $sp>n$ one can use the ...
- 36
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0
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30
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Proving a set defined in terms of Fourier transform is dense in $L^2$.
I recently came across this claim in one of the papers on Sobolev estimates for Radon transform (see here).
The set $\left\lbrace \| \cdot \|^t \left( 1 + \| \cdot \|^2 \right)^{\frac{s - t}{2}} \hat{...
- 4,065
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34
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Strictly positive definite kernel if all eigenvalues positive in Hilbert-Schmidt expansion?
Suppose I have a positive-definite kernel $K:X^2\to\mathbb{R}$. Mercer's theorem says there is a complete orthonormal system $\{\phi_i\}_{i=1}^\infty$ for $L^2(X)$ such that each $\phi_i$ is an ...
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Kernel feature and derivative of kernel feature linearly independent?
Suppose we have a strictly positive definite symmetric kernel $k$ on an open set $\Omega\subset\mathbb R$. By "strictly" I mean that all kernel matrices $(k(x_i,x_j))_{i,j}$ with distinct $...
- 9,889
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1
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44
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Show that $\exp(D_s)$ converges strongly on $L^2$ to $T_1$ as $s \to 0$.
Consider $L^2 = L^2(\mathbb{R})$. Given any real number $s$, define $T : L^2 \to L^2$ by setting
$$(T_s x)(t) = x(t+s) \quad \text{for every} \ x \in L^2 \ \text{and} \ t \in \mathbb{R}.$$
For $s \neq ...
- 1,682
4
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1
answer
86
views
Is it true that $\widehat{(\delta_{x_{0}}\otimes T)} = \hat{\delta}_{x_{0}}\otimes \hat{T}$?
For a fixed $x_{0} \in \mathbb{R}$ consider the Dirac delta distribution $\delta_{x_{0}}$. Its Fourier transform is given by $\hat{\delta}_{x_{0}}(p) = e^{-px_{0}}$, in the sense that $\hat{\delta}_{...
- 999
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0
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107
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Why is a lot of Fourier analysis done on an annulus?
I am studying harmonic analysis from these lecture notes and a lot of results and definitions always assume that the Fourier transform of a function has support in an annulus or a ball. The same ...
- 6,641
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Fourier transform of real exponential
I'm currently reading Zworski's semiclassical analysis and have some question regarding the following example:
In the second equality, I verified the computations by expanding the inner products, and ...
- 529
1
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1
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93
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Estimate of Fourier coefficients of $x^{-1/4}$
I'm studying whether the function $f(x)=x^{-1/4}$ on $[0,1]$ has $p$-summable Fourier coefficients for some $1<p<2$, i.e., $(\widehat{f}(n))_{n\in \mathbb{Z}}\in \ell^p(\mathbb{Z}).$ Apparently, ...
- 83
2
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0
answers
58
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Rate of Uniform Convergence of Fourier Series to a Smooth Function?
I'm wondering if there are any known results on the rate of uniform convergence of a Fourier partial sum to a smooth function ?.
More specifically, I am wondering ...
user1115096
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0
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49
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Weighted $L^2$ space on Torus.
I'm studying weighted $L^2$ spaces in the circle $[0,2\pi]$
Definition 1
A weight is a function $w\colon [0,2\pi]\to \mathbb{R}^+$ (non negative)
Definition 2 The weighted $L_w^2([0,2\pi])$ is defined ...
- 3,806
0
votes
0
answers
43
views
Exercise about identity regarding a function in Schwartz space which Fourier coefficients are Fourier transforms of another function
The following is a simple yet challenging exercise about Fourier transforms. It is part of an older exam for my undergrad course in Elements of Functional Analysis. The professor usually comes up with ...
- 99
1
vote
1
answer
48
views
Finding a Closed Form Expression for a Distribution Defined by an Integral Involving Sine and Bessel Functions
I am seeking a closed form expression for the following distribution:
$$ D(t,x) = \int_0^\infty d\omega\, \omega^2 \sin(\omega t) J_0(\omega x), $$
where $J_0(x)$ is the Bessel function of the first ...
2
votes
1
answer
98
views
Finding $f$ orthogonal to set.
Let $A \subsetneq \mathbb{Z}$. Consider the following set of functions $$M = \{e^{2 \pi in x}\}_{n \in A} \cup \{xe^{2 \pi i n x}\}_{n \in A^c}$$ belonging to $L^2[0, 1]$. Does there exists a function ...
- 169
1
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0
answers
27
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Classifying bounded linear functions satisfying convolution identity
Find all bounded linear functionals $T$ on $Y= \{f \in W^{1,1}[0,1]: f(0)=0\}$ such that there exists $K>0$ such that $$\int_0^1|Tf(\cdot-t)|\ dt\le K \|f\|_1\qquad \forall\ f\in Y\tag{1}$$
My ...
- 373
0
votes
0
answers
71
views
Chain rule with functional derivatives?
I'd like to make the functional derivative of the functional $S[\phi(x)]$ with respect to the Fourier transform $\widetilde{\phi}(p)$ such that
$$\phi(x)=\int\frac{d^{d}p}{(2\pi)^{d/2}}e^{ip\cdot x}\...
- 187
0
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0
answers
68
views
Is it possible to get a lower bound of mollified function with Sobolev norm?
Let $f\in C_c^\infty(\mathbb{R}^n)$ and $\rho_\epsilon$ be a mollifier with support in $B(0,\epsilon)$. Is it possible to get a lower bound on $||\rho_\epsilon*f||_{H^s}$ in terms of $||f||_{H^s}$ if ...
12
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1
answer
311
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The Fourier transform of $e^{-i/x}$
$\def\R{\mathbb R}$
Question.
Does anyone know what is the Fourier transform of
$$ f(x)=e^{-i/x} $$
on the real line? I would like to compute it explicitly, or to establish some properties to have a ...
- 5,124
1
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1
answer
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Let $A:=\{f\in C^1(\mathbb{R}): f, f'\in L^1(\mathbb{R}\}$. Then are the Schwartz functions dense in $A$ w.r.t. $\|f\|=\|f\|_1+\|f'\|_1$?
Let $A:=\{f\in C^1(\mathbb{R}): f, f'\in L^1(\mathbb{R}\}$. Then is it true that Schwartz functions are dense in $A$ with the norm $\|f\|=\|f\|_1+\|f'\|_1$.?
My guess is the above statement shoud be ...
- 113
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0
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23
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Understanding the transpose of a (generalized) Calderon-Zygmund operator
I have been reading Fourier Analysis by Javier Duoandikoetxea, and I have a question about generalized Calderon-Zygmund operators. For reference, I state the definition as given in the book.
...
- 4,065
0
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0
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96
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Approximating self-maps of $[0,1]$
Let $\mathrm{Inc}([0,1])$ denote the space of continuous, increasing functions $f:[0,1]\rightarrow [0,1]$ such that $f(0)=0$ and $f(1)=1$. I want to find a countable family of functions $f_n\in \...
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