3
votes
1answer
26 views

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 ...
0
votes
1answer
24 views

$f, \hat {f} \in L^{p} \cap L^{\infty} \implies f\in A(\mathbb R)$?

$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 ...
2
votes
1answer
44 views

Why are function spaces typically defined on open sets?

I never really bothered to ask this question and now it seems silly...but why do we always seem to define function spaces $X(U)$ (e.g. $L^2(U), BV(U)$ for open sets $U\subset\Bbb{R}^n$? What breaks if ...
2
votes
0answers
36 views

Essential hypothesis of Fourier Inversion formula

Let $f\in L^{1}(\mathbb R)$ and we define its Fourier transform as follows: $\hat{f}(\xi)=\int_{\mathbb R} f(x)e^{-2\pi i \xi\cdot x} dx, (\xi \in \mathbb R);$ and we define $f^{\vee}(x):=\hat{f}(-x)= ...
3
votes
0answers
35 views

How to show the completeness of the space of Fourier transforms $\mathcal{F}L^{1}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
1
vote
2answers
38 views

$f_{n}\to f$ in $L^{1}\implies \hat{f_{n}}\to \hat{f}$ in $L^{1}$?

Let $f\in L^{1}(\mathbb R)$ and it Fourier transform, $\hat{f} (y) : = \int _ {\mathbb R} f(x) e^{-2\pi i x\cdot y} dx ; y \in \mathbb R .$ Suppose there exists $f_{n}, \in L^{1}(\mathbb R)$ such ...
1
vote
0answers
33 views

Differentiabilty of certain convolution

Let $\Gamma$ be a smooth closed curve. Suppose $f\in L^2(\Gamma,ds)$ and $g$ is defined everywhere on $\mathbb{C}$ with compact support. Moreover $\frac{dg}{dz}$ exists everywhere but point $z=0$. ...
2
votes
1answer
68 views

$g, f, \hat {f} \in L^{1}(\mathbb R)\cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R) \implies \widehat{(fg)}= \hat{f} \ast \hat{g} ? $

Let $f, g\in L^{1}(\mathbb R)$ and it Fourier transform of $f$, $\hat{f} (y) = \int _ {\mathbb R} f(x) e^{-2\pi i x \cdot y} dx, \ (y\in \mathbb R)$ and the convolution of $f $ and $g$; $f\ast g ...
2
votes
0answers
20 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 ...
2
votes
0answers
44 views

“Dual Lorentz characterization of L^q”

I am trying to do Exercise A.4 of Terence Tao's book on nonlinear dispersive equations (available here: http://www.math.ucla.edu/~tao/preprints/chapter.pdf, the exercise is on p.342 in Appendix A). ...
2
votes
1answer
40 views

How to apply the method of stationary phase here?

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, ...
6
votes
1answer
47 views

How to determine measure from the integral equation?

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 $F:\mathbb R \to \mathbb C$ ...
1
vote
0answers
33 views

When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral ...
0
votes
0answers
17 views

How to factorize exponential as a convolution of finite number of functions( series)?

We choose a points $x_{1},..., x_{n}$ in $\mathbb Z$ (not 0), such that $$x_{k+1}\not = x_{1}+x_{2}+...+x_{k} \ \ (k=1,2,...,n-1).$$ Let $\delta_{x}$ denote the measure of total mass $1$, ...
1
vote
1answer
32 views

Does there exists $f\in A(\mathbb T)$ such that $||f||=r$ and $||\mathrm{e}^{if}||= \mathrm{e}^{r}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t)\, \mathrm{e}^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb ...
0
votes
1answer
23 views

How to inter change of norm and limit in the Banach algebra?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
0
votes
1answer
45 views

The dual of L^1(G) for a locally compact group G

I might be missing something, but most literature on topological groups and harmonic analysis that I've encountered mention that $L^\infty(G)$ can be naturally identified with the dual of $L^1(G)$ by ...
1
vote
0answers
29 views

Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
2
votes
2answers
47 views

Boundedness of functional

In the setting of $2\pi$-periodic $C^1$ functions (whose Fourier series converge to themselves), and given a linear functional $D:C^1_{\text{per}}\to\mathbb R$ satisfying ...
2
votes
0answers
47 views

Generalizing the ptwise or $L^1$ ergodic theorem

I was able to write a proof based on the hardy-littlewood maximal function of the following statement: Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, and $d=n \geq 1$ be fixed. Let ...
3
votes
0answers
31 views

Matrix-valued functions with lacunary Fourier series

This question is motivated by investigation of the operator space structure of Hankel matrices (which is surely well-known to the experts). Consider a lacunary Hankel matrix, i.e. a matrix ...
2
votes
0answers
92 views

A problem concerning measures on locally compact spaces

I am stuck on a question for quite sometime now, although in the text (http://www.latp.univ-mrs.fr/~chaabi/ARTICLES%20IMPORTANTS/Autres%20articles%20interessant/Jewett.pdf , Pg. 10, 2.3E ) it is said ...
3
votes
1answer
129 views

Corollary from Khintchine's inequality

Let $z_1,\dots,z_n\in\mathbb{C}$ and $\epsilon_j\in\{-1,1\}$ for $j=1,\dots,n$ independent random variables with $P(\epsilon_j=\pm 1)=1/2$. Khintchine's inequality states that ...
2
votes
1answer
62 views

Application of Green's theorem to probability

I encountered this problem while reading a statistic text. Since I am not quite familar with the background knowledge. Wonder can someone help me to explain the details of the following proof? ...
5
votes
1answer
59 views

Function invariant under Hilbert transform

Let $f\in C_0^\infty(\mathbb{R}^n)$ and $$Hf(x)=p.v.\int_{\mathbb{R}}\frac{f(x-y)}{y} dx$$ the Hilbert transform of $f$. Is it possible that $Hf=f$ (a.e. and possibly after extending $H$ to $L^p$ ...
4
votes
1answer
274 views

Example of an unbounded operator

Can somebody give me an easy example of a linear operator which maps $L^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$ and $L^\infty(\mathbb{R}^n)$ to $L^\infty(\mathbb{R}^n)$ (but not boundedly) but does ...
3
votes
1answer
74 views

Weak $L^{p}$ spaces are quasi-normed?

Let $(X,\mathcal{B}, \mu)$ be a measure space. Then for $0< p < \infty$ by definition $L^{p,\infty}(X,\mathcal{B}, \mu)$ is the class of all measureable functions $f$ such that ...
3
votes
1answer
57 views

Are the continuous functions on $G$ dense in $L^{1}(G)$?

If $G$ is a locally compact group, is the set $C_{c}(G)$ of all continuous functions on $G$ with compact support dense in $L^{1}(G)$?
2
votes
1answer
71 views

Why does this inequality for all characteristic functions imply it for simple functions?

This question is probably obvious, but I'm not seeing how to obtain it. A simple function is said to be finitely simple if its support is of finite measure. Let $(X_1,\mu_1)$, $(X_2,\mu_2)$, and ...
2
votes
1answer
49 views

Does $|T(f) - T(g)| \leq |T(f-g)|$ hold for a sublinear operator $T$?

Let $X,Y$ be function spaces with functions taking values in $\mathbf{C}$. An operator $T:X\to Y$ is called sublinear if for all $f,g \in X$ and all $\lambda \in \mathbf{C}$, we have $$ |T(\lambda f)| ...
1
vote
0answers
59 views

Fourier algebra and multipliers of a finite discrete group

Let $G$ be a finite discrete group. We denote by $A(G)$ the Fourier algebra of $G$ and $M_{cb}A(G)$ the space of completely bounded multipliers of $A(G)$. Is is true that $A(G)=M_{cb}A(G)$ ...
1
vote
0answers
38 views

How to show that $ν(z)$ is Carleson measure

If $ν(z)=|1+z|^ β dμ(z)$, $β\in \!\ R^-$. How to show that $ν(z)$ is Carleson? I know that $ν$ is Carleson measure if $ν(Q_I)\leqq \!\ c.I$ But how to apply this?
2
votes
1answer
178 views

Are nilpotent Lie groups unimodular?

The continuous homomorphism $\Delta:G \rightarrow \mathbb{R}^{\times}_+$ is defined by \begin{equation*} \int_G f(xy)dx = \Delta(y)\int_Gf(x)dx \end{equation*} where $dx$ is a left Haar measure on ...
1
vote
1answer
151 views

Orthonormal basis in Hilbert spaces

I have a general question but I'm going got ask it in a very restrictive setup. It is known that an equivalent condition for a system $\left\{e^{i\lambda t}\right\} _{\lambda\in\Lambda}$ being an ONB ...
4
votes
1answer
131 views

Canonical field of Hilbert spaces in Dixmier; Plancherel Theorem

I'm working through the proof of Plancherel's Theorem in $C^{*}$-algebras by Dixmier, section 18.8. For the most part, I'm happy with it, although I have one problem. From Dixmier, I have the proof ...
1
vote
0answers
197 views

Find a sequence of function in the Schwartz space $S(\mathbb R)$ which does not converge in $S(\mathbb R)$

Show there exists a sequence $\{f_n\}$ in the Schwartz space $S(\mathbb R)$ with limit $f$ for which $$ \lim \|f_n\|_{u,v} \text{ induced that } f \not\in S(\mathbb R) \text{ for some } u,v. $$ But ...
0
votes
0answers
46 views

Amenability of finite dimensional norm algebras

Let $(\cal A,\|\cdot\|)$ be a finite dimensional norm algebra (Banach Algebra). Can we say any thing about the amenability of $\cal A$. What if we impose some extra conditions on $\cal A$, say ...
3
votes
1answer
110 views

Necessary and sufficient conditions for $G$ to have $A(G)$ isomorphic to an operator algebra?

Let $G$ be a locally compact group. We denote by $A(G)$ the Fourier algebra of $G$. An operator algebra is a closed subalgebra of $B(H)$ where $H$ is a Hilbert space. What are the necessary and ...
3
votes
1answer
298 views

Nirenberg-Gagliardo- Sobolev inequalities

I need a small help in understanding the following that how "Nirenberg -Gagliardo-Sobolev inequalities" were used. This is a part of the paper. Denote $$ H^1=W^{1, 2}(\Omega)\\ V_1=\{ f\in H^2 ...
3
votes
1answer
78 views

Reference for: $G$ discrete iff the measure algebra $M(G)$ is weakly amenable.

I search the reference for the proof of the following theorem: Let $G$ be a locally compact group. Then the group $G$ is discrete if and only if the measure algebra $M(G)$ is weakly amenable. The ...
1
vote
0answers
52 views

if the system localized

Let g be integrable, even function. Let $\Lambda$ be uniformly discrete sequaence (i.e. $\inf|\lambda_i-\lambda_j|>0, \lambda_i, \lambda_j \in \Lambda$). We say that system $\Psi=\{\psi_n\}$ is ...
3
votes
0answers
53 views

$\Lambda_p$-set for compact abelian group

We denote by $|A|$ the cardinal of a set $A$. Let $S$ be a subset of $\mathbb{Z}$. Denote $S_N=S\cap [0,N]$ where $N$ is an integer. Suppose $2<p<\infty$. There is well-known that if $S$ is a ...
2
votes
1answer
121 views

Truncation in Lorentz spaces

I am reading a paper, whose author state the following: if $f \in L^{(q,\infty)}(\mathbb{R}^N)$, then $f_\delta \in L^p(\mathbb{R}^N)$ for every $p \in [1,q)$, where $\delta > 0$ and $$ f_\delta = ...
7
votes
1answer
440 views

A net version of dominated convergence?

Let $G$ be a locally compact Hausdorff Abelian topological group. Let $\mu$ be a Haar measure on $G$, i.e. a regular translation invariant measure. Let $f$ be fixed in $\mathcal{L}^1(G, \mu)$. ...
7
votes
3answers
327 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in ...
4
votes
1answer
137 views

Product of Sidon sets

Let $G$ be a compact abelian group with dual $\Gamma$. Let $\Lambda \subset \Gamma$ a Sidon set (see the book of Rudin: Fourier Analysis on Groups for the definition). Consider the set ...
3
votes
1answer
255 views

Left regular representation of $L^1(G)$ for a locally compact group $G$

Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
10
votes
1answer
316 views

Stone-Weierstrass implies Fourier expansion

To prove the existence of Fourier expansion, I have to solve the following exercise, which supposedly follows from the Stone-Weierstrass theorem: Let $G$ be a compact abelian topological group ...
2
votes
1answer
613 views

Hilbert transform of white noise

What is the Hilbert transform of a white noise $\xi(t)$? By the Hilbert transform I mean: http://mathworld.wolfram.com/HilbertTransform.html Thank you.
4
votes
1answer
513 views

Reference request: Fourier and Fourier-Stieltjes algebras

I would like to learn the basic theory of Fourier algebras and Fourier-Stieltjes algebras. In particular, I want to know how these two objects are defined in the case of not necessarily abelian ...