# Tagged Questions

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### Is $C(\mathbb R) \subset H_{s}^{loc}$ (localized Sobolev space)?

We put, Sobolev space $$H_{s}=H_{s}(\mathbb R)=\{f\in L^{2}(\mathbb R):[\int_{\mathbb R} |\hat{f}(\xi)|^{2}(1+|\xi|^{2})^{s}d\xi]^{1/2}<\infty \}.$$ If $U$ is an open set in $\mathbb R,$ the ...
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### $2\pi$ in the Definition of Fourier Transform

Most textbooks I read define Fourier transform of a function $f \in L^2(\mathbb R)$ as $$\hat f (\xi) := \int_\mathbb R f(x) e^{-2\pi i x \xi} dx.$$ However, in class my teacher defines it without ...
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### Does the Fourier coefficients of a function $f\in H^1(0,L)$ (the first order Sobolev space) are absolutely summable?

My precise question: Let $f\in H^1(0,L)$ and let $\{f_n\}$ be its Fourier sine series coefficients on $(0,L)$, is it true that $\{f_n\}\in l^1$, i.e. $$\sum_{n}|f_n|< \infty .$$ Thanks
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### An exercise about Fourier transform and $H^s$ in Treves

In Treves, the Fourier transform is defined by $\hat{f}(\xi)=\int {e^{-i\langle\xi,\space x\rangle} f(x) dx}$. The following is the problem, where I have figured out almost all of the questions except ...
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### characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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### What does local space of a given Banach space says intuitively?

We put, $\mathcal{D}(\mathbb R)=$ The space of $C^{\infty}-$ functions on $\mathbb R$ with compact support Example: For instance bump function is in $\mathcal{D}(\mathbb R)$ Let $E$ is a Banach ...
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### Fourier transform and inverse transorm

I have to prove that for $f\in L^1(\mathbb{R})$ $$\check{\hat{f}}=\hat{\check{f}},$$ where $$\hat{f}(\xi):=\int\limits_{\mathbb{R}}e^{-i\xi x}f(x)\mathrm{d}x$$ and ...
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### Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
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### Eigenvectors of Laplace operator on a particular functiona space

Exercise 8.15 in [1] is: "Show that $\Delta u=\lambda u$ has no solutions of polynomial growth if $\lambda > 0$, but does have such solutions if $\lambda < 0$." How should I make sense of this? ...
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### Fourier transform and sufficient condition…

Does anyone could give me a sufficient condition on $f$ so that the Fourier transform of $f$ (denoted as $\hat{f}$) is in $L^{1}(\mathbb R)$. The Fourier transform here is the linear operator ...
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### Complex exponential is not Fourier multiplier on $L^p$

I am having difficulty to show that the function $m(\xi):= e^{i|\xi|^2}$ is not a Fourier multiplier on $L^p$ when $p\neq 2$. Note that $m:\mathbb{R}^n\to \mathbb{C}$ is called an $L^p$ Fourier ...
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### Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
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### $f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$1\leq p \leq \infty$. We put, $$X_{p}= \{f\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) :\hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R)\};$$ and we consider the algebra of Fourier ...
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Define the Fourier transform on $L^2(\mathbb{R})$ in the usual way by a limiting process from \begin{eqnarray*} \hat{f}(y)=\int_{-\infty}^\infty e^{-2\pi i xy}f(x)dy. \end{eqnarray*} Fix $m\in ... 1answer 57 views ### Range of Fourier transform: criterion to belong to the range of the Fourier transform on L1 I would like to show that a given function belongs to the range of the Fourier transform on L^1(R). More specifically: given a partition of unity (g_0, g_1) on R, where g_0 has compact support ... 0answers 22 views ### Can we expect,$\left\|f|f|- g|g|\right\|\leq C \left \|f-g\right \|$in the Banach algebra$A(\mathbb R)$? Let$f\in L^{1}(\mathbb R)$and it Fourier transform,$\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R ;$and consider Fourier algebra $$A(\mathbb R):= \{f\in ... 3answers 605 views ### Does f(x)\in L^1 imply that \lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x \omega } \, dx=0? Suppose that f(x) is L^1 and R- integrable function, problem is to resolve if it is possible existence of such a f(x) that:$$\lim_{\omega \to \infty } \, \int_{-\infty }^{+\infty } f(x) e^{-i x ... 0answers 55 views ###$f, f|f| \in L^{1} (\mathbb R) \cap C_{0} (\mathbb R) \implies |f| \in H_{1} (\mathbb R)$? Suppose$f \ \text{and} \ f|f|\in L^{1}(\mathbb R).$Then, clearly,$|f|\in L^{2}(\mathbb R)$and therefore by Plancheral theorem, we get,$\widehat{|f|} \in L^{2}(\mathbb R).$Also, assume,$f, ...
Consider the following oscillating integral  I(\xi;t) = \int\limits_{\mathbb R}\int\limits_{\mathbb R^n} e^{it(\xi y - \theta f(y))}a(y) \, dy \, d\theta, \quad \xi \in \mathbb R^n \setminus 0, ...
Let $\{c_{n}\}_{n\in \mathbb Z}\subset \mathbb C$ and $\sum_{n\in \mathbb Z} |c_{n}| < \infty$ (that is, the series $\sum c_{n}$ is absolutely converges); we define $F:\mathbb R \to \mathbb C$ ...