All Questions
Tagged with fourier-transform harmonic-analysis
84 questions
0
votes
1
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105
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If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality
\begin{equation*}
\int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
- 39.1k
1
vote
0
answers
52
views
Sufficient conditions for boundedness of Fourier transform
This should be a well studied topic: I am looking for sufficient conditions on a function $u(x)$ on $\mathbb{R}$ ensuring that its Fourier transform is bounded. Of course one such condition is $u\in L^...
- 9,007
1
vote
2
answers
226
views
Show that the kernel $|x -y|^{-1}$ on $\mathbb{R}^3 \times \mathbb{R}^3$ is Hilbert Schmidt with respect to a weighted $L^2$ space
Let $\langle x \rangle := (1 + |x|^2)^{1/2}$, $x \in \mathbb{R}^3$. For $s > 1$, consider the weighted convolution operator
\begin{equation*}
T_s \varphi = \langle x \rangle^{-s} \int_{\mathbb{R}^3}...
- 6,035
6
votes
1
answer
343
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Integral convolution equation $\int_{B_n(R) } e^{- \| x - t\|} d\nu(t) = e^{- \|x \|^2/2}$ on $x \in B_n(R)$. Find measure $\nu$
Let $B_n(R)$ denote the $n$ ball centered at zero with radius $R$. We are interested in the following integral equation: given $R>0$ and $\lambda>0$, let
\begin{align}
\int_{B_n(R)} e^{- \...
- 17.2k
3
votes
0
answers
75
views
Non-vanishing of a "push-forward" Fourier–Harish-Chandra transform on a compact set
Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ ...
- 12.9k
23
votes
0
answers
1k
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Laplace Transform in the context of Gelfand/Pontryagin
Questions:
Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
- 1,124
2
votes
0
answers
80
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Prove uniqueness of Radon transform without using Fourier transform
The uniqueness of Radon transform can be expressed by the following claim (I assumed that the function has compact support for simplicity):
If a continuous function with compact support has zero ...
- 807
5
votes
1
answer
508
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Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)
I've been tackling the following problem for some time,
Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...
CommunityBot
- 1
1
vote
0
answers
72
views
Positive eigenfunctions of the discrete Fourier transform
Let $G$ be a finite cyclic group of order $n$ ($n$ need not be prime) and $\mathcal{F}$ the normalized discrete Fourier transform defined on $G$.
Is there a canonical way to construct an eigenfunction ...
- 549
0
votes
1
answer
255
views
Carleson's theorem: proof of a lemma
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
0
votes
0
answers
23
views
Is there a classification of 2D projective convolution kernels?
Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions
$$ \gamma\star\...
5
votes
0
answers
169
views
Fourier dimension of radial set
In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...
- 101
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votes
1
answer
126
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Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections
I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
- 7,817
0
votes
2
answers
227
views
Does this distribution exist?
Assume there is a distribution in two variables $\mathcal{W}\in\mathcal{S}'(\mathbb{R}^2)$ with Fourier transform $\hat{\mathcal{W}}(\alpha,\beta)\equiv \int_{-\infty}^\infty e^{i(\alpha x+\beta y)} \...
11
votes
3
answers
1k
views
What is the intuition behind applying the Mellin Transform to prime distribution?
I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem.
I understand that applying the Mellin Transform to the partial sum of the van ...
- 188k
2
votes
1
answer
168
views
Any references for generalised square functions?
In harmonic analysis, there is a big chunk of literature studying the square function $Sf=\|\{P_jf\}_{j=1}^\infty\|_{l^2}$, where $P_jf=(\psi_j\hat f)\check{}$ and $\{\psi_j\}$ is a partition of unity,...
- 938
1
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0
answers
73
views
$L^p$ norm of Fourier transform of function composed with a diffeomorphism
Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
- 265
2
votes
1
answer
254
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Fourier transforms of homogeneous functions [closed]
Compute Fourier transforms of homogeneous functions of the form,
$$
\frac{1}{|x|^{n+d}}P_d(x)
$$
where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.
- 16.2k
-1
votes
1
answer
213
views
Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
- 779
3
votes
0
answers
308
views
Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
- 870
1
vote
0
answers
108
views
Recovering phase function using Fourier decomposition
I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function
$$f = e^{i \phi(x)}. $$
I am interested in the following problem. If I know the function/distribution $...
- 63.9k
0
votes
1
answer
170
views
When some Fourier coefficients are fixed, can we control the extremals of the function?
Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f
:\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?
1- $-n\lneqq f_{\min}$ (where $f_{\...
- 3,808
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
9
votes
2
answers
628
views
How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?
In Bennett, Carbery and Wright's paper A non-linear generalisation of the Loomis–Whitney inequality and applications, Claim 5 was used to generalise the case from characteristic functions to simple ...
- 776
6
votes
1
answer
491
views
Harmonic analysis for a beginner
I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
- 188k
4
votes
1
answer
662
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The decay of Fourier coefficients and the continuity of functions
Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not ...
- 24.2k
4
votes
2
answers
549
views
A proof of Bernstein's inequality
I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below:
"The function $\frac{\xi^\beta}{|\xi|...
- 6,035
0
votes
0
answers
75
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
2
votes
2
answers
917
views
Decay of the Fourier transform of a non-differentiable function
It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. ...
- 11
2
votes
1
answer
198
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$|\hat\mu(\xi)| \lesssim |\xi|^{-1/2}$ where $\mu$ is $f\mapsto \int_{\mathbb R} \psi(r) \int_{S^{1}} f(rx,r)\, d\sigma(x)\, dr $
I have questions about the proof of Theorem $2.1$ here. The proof is on Pg. $10$. I am trying to work out the $d = 2$ case in particular.
$$\mathcal C^d = \{(x_1, \ldots, x_{d+1}): |(x_1, \ldots, x_d)|...
- 165
1
vote
2
answers
152
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Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
- 188k
3
votes
1
answer
262
views
Low/high-frequency estimates in $\mathrm{L}^\infty$ for Lipschitz nonlinearities
Let $f \colon \mathbb{R} \to \mathbb{R}$ be a Lipschitz nonlinearity with $f(0) = 0$ and suppose $u \in \textrm{H}^s(\mathbb{R}) \cap \textrm{L}^\infty(\mathbb{R})$ for some $s \in [0, \tfrac{1}{2}]$. ...
CommunityBot
- 1
2
votes
0
answers
149
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An oscillatory integral
Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates?
\begin{...
- 351
3
votes
2
answers
413
views
A Sobolev embedding theorem for functions on spheres
$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
- 351
1
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0
answers
79
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A problem arising from Wiener-Levy theorem on the real line
Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
- 4,058
1
vote
0
answers
245
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On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)
Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
- 111
6
votes
1
answer
397
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Absolute values of two functions and absolute values of their Fourier transform coincides
Let $f, g \in L^2(\mathbb{R})$.
Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?
I am not able to prove it or ...
- 13k
2
votes
0
answers
191
views
Convergence in $S'(\mathbb R^d)$ of the paraproduct $\dot{T}_uv$
Let $B = B(0,4/3)$, $C = \{x \in \mathbb R^d : 3/4 \leq \|x\|_2 \leq 8/3\}$ and $\tilde{C} = \{x \in \mathbb R^d : 1/12 \leq \|x\|_2 \leq 10/3\}$.
For a fixed Littlewood-Paley decomposition $\chi \in \...
- 233
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1
answer
226
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Transformation of Fourier Transform
Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression ...
- 16.2k
25
votes
3
answers
13k
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Fourier transform of the unit sphere
The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu ...
- 14k
3
votes
0
answers
269
views
Finding (and saturating) a sharp Babenko-Beckner inequality for finite fields
My question is a follow-up to Abdelmalek Abdesselam's recent post
What makes Gaussian distributions special? Local field version?
asking about various characterizations of (real-valued) Gaussian ...
- 2,137
7
votes
0
answers
389
views
Certain Fourier transforms involving Whittaker function and Bessel functions
I recently meet the following two weird "Fourier transform" questions.
(I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ ...
- 960
2
votes
2
answers
331
views
Estimate for a simple oscillatory integral
If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that
$$
\Big|\int_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\...
- 870
26
votes
2
answers
3k
views
Image of L^1 under the Fourier Transform
The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
- 846
7
votes
1
answer
1k
views
Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
- 107
3
votes
1
answer
404
views
The sign of the tail of Fourier transform of a positive function/ characteristic function
I am interested in a specific density (positive function) and would like to prove that the tail of its characteristic function (Fourier transform) is positive ($>0$). Here is the density $f(x)=c_\...
- 17.2k
1
vote
1
answer
460
views
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 91.8k
0
votes
0
answers
166
views
Parseval-Plancherel identity involving absolute value
Let $\hat{f}$ be the fourier transform of $f$.
By Parseval-Plancherel identity, for suitable $f,g$, we have
$$\left\|\hat{f}*\hat{h}\right\|_{L^2_{\xi}}^2=\left\|f\cdot h\right\|_{L^2_{x}}^2.$$
Let ...
3
votes
0
answers
79
views
Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$
It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...
- 489
3
votes
2
answers
1k
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Composition of Riesz potentials
For $0<\alpha<n$ and $n\geq 2$ we define the Riesz potential by
$$
(I_\alpha f)(x) = \frac{1}{\gamma(\alpha)}
\int_{\mathbb{R}^n} \frac{f(y)}{|x-y|^{n-\alpha}}\, dy\, ,
\quad
\text{where}
\quad
\...
- 28k