All Questions
389 questions
14
votes
0
answers
3k
views
Tanh version of a Fourier Transform?
I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...
- 3,979
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
7
votes
2
answers
1k
views
Carleson's Theorem (on the Adeles and other exotic groups)
I have redone this question:
On $\mathbb R^n$ the Carleson Operator if defined by
$$Cf(x) = \sup_{R>0} \left \vert \int_{B_R(0)} e^{2\pi i x\cdot \xi} \widehat{f}(\xi) d \xi \right \vert. $$ (...
59
votes
7
answers
29k
views
Learning roadmap for harmonic analysis
In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas ...
0
votes
1
answer
901
views
Schwartz space inequality
Let $g$ be a function in the Schwartz space $\mathscr S (\mathbb R)$. Show that for any $l \ge 0$, we have $\sup_x |x|^l |g(x-y)|\le A_l (1+|y|)^l$ by considering separately the cases $|x|\le 2|y|$ ...
1
vote
1
answer
977
views
Fourier transform of distributions with non-standard test functions
This might be a quite simple question for function analysis standards, but it has some obstacles. I'll try to improve the readability a bit by not using the full tex code. A short motivation:
Given a ...
- 278
17
votes
2
answers
5k
views
Positive-Definite Functions and Fourier Transforms
Bochner's theorem states that a positive definite function is the Fourier transform of a finite Borel measure. As well, an easy converse of this is that a Fourier transform must be positive definite.
...
- 4,952
2
votes
1
answer
508
views
Fractional integration lemma
Hello everyone.
I am trying to establish a fractional integration lemma of this form.
For $\alpha\geq 0$, and
$1\leq p,q<\infty$ and $0\leq \frac{1}{q}-\frac{1}{p}=\frac{\alpha}{d}$
or $1\leq p,...
- 23
24
votes
3
answers
3k
views
Can Hölder's Inequality be strengthened for smooth functions?
Is there an $\epsilon>0$ so that for every nonnegative integrable function $f$ on the reals,
$$\frac{\| f \ast f \|_\infty \| f \ast f \|_1}{\|f \ast f \|_2^2} > 1+\epsilon?$$
Of course, we ...
- 9,756
2
votes
2
answers
1k
views
Characterization of closed subspaces of $ L^2(R)$
Natural way to find an example of banach spaces is to look at closed subspaces of Banach spaces. Initially, It was really hard to find examples of closed subspaces of $L^2(R)$. Then, my professor gave ...
- 97
48
votes
6
answers
7k
views
Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway...
Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "...
8
votes
1
answer
2k
views
Recent progress on Bochner-Riesz conjecture
Consider the family of operators $T_\delta$, $\delta \geq 0$, defined on $\mathbb{R}^n$ by
$
\widehat{T_\delta f}(\xi) = (1-|\xi|^2)_+^\delta \widehat{f}(\xi).
$
($(1-|\xi|^2)_+^\delta$ are known as ...
- 505
7
votes
1
answer
823
views
On a decomposition of L^1(G)
[EDITED by Y. Choi - I have attempted to paraphrase the original question into something a bit terser and more precise; if this is not what the original poster intended, they should make corrections ...
- 643
5
votes
2
answers
958
views
L1 distance from a trigonometric susbspace
How to check, whether the $L^{1}$ distance between a finite exponential sum $S_{F}(x)=\sum\limits_{n\in F} \exp(inx)$ and the $L^{1}$-closure of subspace $\mathrm{span}\left(\exp(inx): n\in \mathbb{Z}\...
- 696
6
votes
1
answer
727
views
Is this method of "fractional sums" using a Fourier series viable?
Hi.
I have this idea about developing what I call a "continuum sum", that is, a method to "add up a non-integer number of terms", i.e. to see if there is a "natural" way to assign a meaning to the ...
- 1,687
2
votes
2
answers
528
views
Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
Let $G$ be a discrete group.
Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution?
I only know Johnson's theorem:
A group is amenable if and only if the ...
44
votes
10
answers
47k
views
Is square of Delta function defined somewhere?
I am wondering whether anyone knows if the square of Dirac Delta function is defined somewhere.
In the beginning, this question might look strange. But by restricting the space of the test functions, ...
4
votes
1
answer
2k
views
Characterizations of a linear subspace associated with Fourier series
Let $c_0$ be the Banach space of doubly infinite sequences $$\lbrace
a_n: -\infty\lt n\lt \infty, \lim_{|n|\to \infty} a_n=0 \rbrace.$$ Let $T$ be the space of $2\pi$ periodic functions integrable ...
- 744
7
votes
3
answers
1k
views
A Question concerning the Fourier Transform of $\mathbb{R}$
Consider the classical Schwartz space $\mathcal{S}(\mathbb{R})$ together with the Fourier transform $\mathcal{F} : \mathcal{S}(\mathbb{R}) \rightarrow \mathcal{S}( \mathbb{R})$.
Consider the subspace ...
- 11.2k
2
votes
1
answer
1k
views
Range of the Radon Transform
Let us consider the Radon transform in two dimensions:
$$\tag{1}Rf(r,\theta):=\int\limits_{-\infty}^{\infty} f(r\cos\theta-t\sin\theta,r\sin\theta+t\cos\theta) dt,$$
where $r\in\mathbb{R}$ and $0\...
- 931
29
votes
1
answer
4k
views
Furstenberg's Conjecture on 2-3-invariant continuous probability measures on the circle
Hillel Furstenberg conjectured that the only $2$-$3$-invariant probability measure on the circle without atoms is the Lebesgue measure. More precisely:
Question: (Furstenberg) Let $\mu$ be a ...
- 25.5k
0
votes
2
answers
2k
views
fundamental solution of radial wave equation
i am trying to find resources on the derivation of the fundamental solution to the radial wave equation. any suggestions of or links to books, papers, and/or notes would be much appreciated. i have ...
- 1
3
votes
1
answer
2k
views
fourier transform of radon measure
hi,
assume that I have a function $q$ which is a Fourier Multiplier of order zero, i.e.
$$
\left|\left( \frac{d}{dx}\right)^nq(x)\right|\lesssim \left(\frac{1}{1+|x|}\right)^n\quad \mbox{for all ...
- 979
3
votes
1
answer
235
views
Odd element of L^1 group algebra of the integers
Giving some motivation is hard here, so I'll just ask the question. I want an element $a=(a_n)\in\ell^1(\mathbb Z)$ such that:
$\|a\|>1$
a is power bounded (turn $\ell^1(\mathbb Z)$ into a Banach ...
- 18.7k
2
votes
1
answer
1k
views
Elliptic regularity on bounded domains
I'm concerned with a generic uniformly elliptic operator $L$ on $\mathbb{R}^n$. If $L$ is uniformly elliptic and I am studying the equation $Lu=f$ then the way I can deduce regularity on $\mathbb{R}^n$...
- 2,641
2
votes
3
answers
891
views
Fourier Transforms restricted to mass shell
Hello,
I am stuck with the following (hopefully not too trivial) problem.
I want to know, if the map
$${\cal D}(\mathbb{R}^2)\to L^2(H_m,d\Omega_m)\qquad f \mapsto \hat{f}|_{H_m}$$
has dense range.
...
- 23
1
vote
1
answer
706
views
Plancherel-Polya Type Inequality for non-compactly Fourier-supported Functions??
Hi!
The Plancerel-Polya inequality can be stated as follows:
Let $0 < p\le \infty$ and $ \nu \in \mathbb{Z}$. Suppose that $g$ is a (smooth) function satisfying $\mbox{supp }\hat g \subset \...
- 979
18
votes
1
answer
3k
views
Let a function f have all moments zero. What conditions force f to be identically zero?
Throughout, let $f$ be a Lebesgue measurable function (or continuous if you wish, but this is probably no easier). (Questions with distributions etc. are possible also but I want to keep things simple ...
- 1,990
11
votes
0
answers
1k
views
Is the Fourier-Transform a bounded operator on Lorentz spaces L(2,q)?
It is well known that the Fourier transform $\mathcal{F}$ maps $L^1(\mathbb{R}^n)$ continuously into $L^\infty(\mathbb{R}^n)$ and $L^2(\mathbb{R}^n)$ continuously into $L^2(\mathbb{R}^n)$.
Then, by ...
- 111
25
votes
1
answer
8k
views
Convergence of Fourier Series of $L^1$ Functions
I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
- 871
7
votes
3
answers
1k
views
Can Stein's maximal principle be strengthened?
Let $T$ be an operator on $S(G)$ where $G$ is the line $R$ or the circle $T$, and $S(G)$ denotes the Schwartz space of functions on $G$.
We can ask if the operator T is bounded (as an operator from $...
- 13k
1
vote
0
answers
660
views
Fractional Fourier transform [closed]
Let $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ be the Fourier transform. Is there any reasonable definition of fractional Fourier transform (i.e. operator $A$ such that $A^{\alpha}=T$ for $\...
- 3,323
4
votes
2
answers
1k
views
Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
- 1,839
4
votes
5
answers
3k
views
Generalize Fourier transform to other basis than trigonometric function
The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
In how far can this ...
- 5,327
5
votes
3
answers
753
views
Regularity of sparse Fourier transforms
Suppose $F$ has discrete Fourier transform $(a_n)$ where $a_n=0$ unless $n=2^k$ for some $k > 0$, in which case $a_n=1/k$ (or $a_n=1/k^2$ if you want: I'm happy with anything polynomial). What ...
- 18.7k
21
votes
2
answers
1k
views
Is there an L^p tauberian theorem?
From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...
- 13k
2
votes
3
answers
3k
views
What function has fourier series the harmonic series?
I know that this is on the boundaries of what's allowed, but hopefully someone'll answer before it gets closed!
What (periodic) function has Fourier series the harmonic series? I really want the ...
- 26.8k
2
votes
2
answers
1k
views
Decoupling lemma for the Lambda(p) problem
I'm attempting to work through Bourgain's paper "Bounded orthogonal systems and the $\Lambda(p)$-set problem". There is a step in the proof of the decoupling lemma that I am stuck on, and thought ...
- 13k
10
votes
5
answers
1k
views
What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...
- 2,745