All Questions
Tagged with lp-spaces fourier-transform
90
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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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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
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Weighted inequality on torus
In the Torus (circle). Let $[0,2\pi]\to\mathbb ]0,\infty[\colon \theta\mapsto w(\theta)$ a weight function, i.e. nonnegative and integrable on $[0,2\pi]$. If $\mathbb{Z}\to\mathbb{R}\colon k\mapsto m(...
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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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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
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1
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55
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Integration of discontinuous functions
In order to evaluate its Fourier transform, I want to determine whether $f(x)=\arctan(\frac{1}{x})$ belongs in $L^1(\mathbb{R})$, $L^2(\mathbb{R})$ or both. Therefore, we have to check the continuity ...
- 165
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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 ...
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83
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Limit of a Fourier Trasformation
Knowing that $f \in L^2(\mathbb{R^n})$, then $$\lim_{\epsilon\to0+}\int_{\mathbb{R^n}}e^{-i\langle x,\xi\rangle -\epsilon|x|}f (x)dx=\mathcal{F}_2f(\xi)$$ in $L^2(\mathbb{R^n})$, where $\mathcal{F}_2f$...
- 195
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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
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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
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1
answer
57
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The set $\{f \in \mathcal L^2(\mathbb R) : \hat f \in \mathcal C_c(\mathbb R)\}$ is dense in $\mathcal L^2(\mathbb R)$
Let $\mathcal C_c(\mathbb R)$ denotes the set of all continuous functions with compact support on $\mathbb R$. We know that $\mathcal C_c(\mathbb R)$ is dense in $\mathcal L^2(\mathbb R)$. Now ...
- 127
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55
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Why is the complex exponential a Fourier multiplier?
Example 2.5.12. The function $m(\xi)=\mathrm{e}^{2\pi i\xi \cdot b}$ is an $L^p$ multiplier for all $b \in R^n$ since the corresponding operator $T_m(f)(x)=f(x+b)$ is bounded on $L^p(R^n)$. Clearly $\|...
- 3,650
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Sufficient conditions on $f_n\longrightarrow f$ so that $\hat{f_n}\longrightarrow\hat f$ in $L^1$ sense
Let $f:\mathbb{R}\longrightarrow\mathbb{C}$ be a Schwartz function, and suppose $(f_n)_n\subset\mathcal{C}_c^{\infty}$ is such that $f_n\xrightarrow{n\to\infty}f$ in the $L^1(\mathbb{R})$ norm.
Is it ...
- 105
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83
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On the charateristic function of random variables
For all random variables that admit a probability density function (PDF), their characteristic function provides an alternative way to completely define its probability distribution.
Why is that? The ...
- 141
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Find function $f(x)$ so $x\hat{f}(x)\in L^1(\mathbb{R})$
Suppose we have $f\in L^1(\mathbb{R})$ so $\xi\mapsto \xi\hat{f}(\xi)$ is in $L^1(\mathbb{R})$, where $\hat{f}$ is the Fourier Transform for the function $f$. I'm trying to show that there exists some ...
