All Questions
Tagged with lp-spaces fourier-analysis
213
questions
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14
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Equivalence of Fourier Transform on $\ell_2(\mathbb{Z}_+)$ and $L_2(\mathbb(R)_+)$ via equivalence of $H_p( \mathbb{D})$ and $H_p(\mathbb{C}_+)$?
Throughout I'll use the fact that the Hardy space $H_2$ is the set of $L_2$ functions on the boundary with vanishing Fourier coefficients.
We know that the Fourier Transform is an isometric ...
- 286
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38
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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 ...
- 3,650
1
vote
0
answers
21
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Mikhlin, Marcinkiewicz theorem on weighted space $L^p$ spaces.
This theorem is in D. Guidetti, “Vector valued Fourier multipliers and applications,” Bruno Pini Mathematical Analysis Seminar, Seminars 2010 (2010). It is a variant (I think, a easier variant) of ...
- 3,650
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17
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Prove that the derivative of the mollification approaches the strong $L^p$ derivative
Let $g \in C_c(\mathbb{R})$ be such that $\int_\mathbb{R} g dx = 1$. Suppose that $f \in L^p(\mathbb{R}$ has a strong $L^p$ derivative, i.e. there is some $h$ such that $\displaystyle\lim_{y \to 0} \...
- 3,800
2
votes
1
answer
93
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Prove that Fourier transform is $L^p$
I am working on a problem, and I need to show that, if $f : \mathbb{R}^n \to \mathbb{C}$ is given by
$$f(x) = \frac{|x|^a}{(1+|x|^2)^{(a+b)/2}}$$
and $2a>-n$, and $b > n$, then the Fourier ...
- 591
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65
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If $f\in L^2(\mathbb{R^3})$, then $\frac{\hat{f}(\xi)\cdot \xi}{2\pi i |\xi|^2} \in L^1_{loc}(\mathbb{R^3})$.
I have that $f$ is an $L^2(\mathbb{R^3})$ function, so I know that $\hat{f}$ its Fourier transform is in $L^2(\mathbb{R^3})$ too. I want to prove that:
\begin{equation}
\frac{\hat{f}(\xi)\cdot \xi}{2\...
- 61
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votes
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answers
8
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When an operator with variable coefficient is bounded?
When the operator $Au(x):=\mathcal{F}^{-1}(m(x,\xi)\widehat{u}(\xi))(x)$ is bounded on $L^p(\mathbb{R}^n)$?
I know that, by Mikhlin theorem, if $m(x,\xi)=m(\xi)$ (not dependence on $x$) is a fourier ...
- 3,650
3
votes
1
answer
98
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Summability of the Fourier Transform.
I was reading the notes "Introduction to Complex Analysis" by Michael E. Taylor, and I am stuck on the following exercise about the Fourier Transform and the space $\mathcal{A}(\mathbb{R})$ ...
4
votes
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answers
278
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$L^1- L^\infty$ estimate for the semi group of wave equations
I am looking for a proof of the following lemma
for the case where:
$y= (y_1,\cdots, y_n)\mapsto P(y) = \|y\|_2= \sqrt{y_1^2+ \cdots + y_n^2}.$
In this case the rank of the mentioned matrix is $n-1$...
- 639
1
vote
1
answer
62
views
A lemma on the $L_p$ estimates
I found the following lemma in some old paper
with proof referred in unavailable reference
I am trying to check why point (iii) is true. I guess the constants $C_0, C_1$ were
exchanged.
I tried to ...
- 639
1
vote
1
answer
49
views
Showing that Sobolev Space $H^m$ is in $L^\infty$
I'm very new to Fourier analysis/Sobolev spaces and am stuck on this exercise. I found proofs of more general embedding theorems for Sobolev spaces and some similar questions on here, but they are too ...
8
votes
1
answer
115
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weakened $L^p$ interpolation using the Fourier transform?
It is easy to see that $$f\in L^\infty(\mathbb R^d), \ f\in L^1(\mathbb R^d) \implies f\in L^p(\mathbb R^d) \ \text{for all}\ p\in[1,\infty] \label{1}\tag{A}$$
Indeed, $ \int|f|^p \le \|f\|_{L^\...
- 35.1k
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votes
1
answer
108
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Is Plancherel's theorem on a Weighted $L^2$-space valid?
In Rudin, Real and complex, (Plancherel's Theorem) $$\left\|f\right\|_{L^2(dx)}=\left\|\widehat{f}\right\|_{L^2(dx)},\quad f\in L^2(\mathbb{R})$$ with $dx$ Lebesgue measure.
In Function, spaces, and ...
- 3,650
1
vote
1
answer
59
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If Fourier transform of a simple function $f$ satisfies inequality $\| \hat{f} \|_q \leq C\|f\|_p$ then $1/p + 1/q = 1$
I have some difficulties of getting started with this question. Namely,
If the Fourier transform $\hat f$ of a simple function $f$ satisfies
$$\|\hat f \|_q \leq C\|f\|_p$$
then $1/q + 1/p = 1$.
Could ...
- 25
1
vote
0
answers
44
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Solution verification of continuity of a function in $L^p(\mathbb R)$
We are asked to prove the following:
Let $f\in L^p(\mathbb R)$, with $1\leq p<\infty$. Show that
$$g(x)=\int_{x}^{x+1}f(t)\,dt$$
is continuous. We did the following:
From Hölder's inequality, for $...
