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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,...
JZS's user avatar
  • 481
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^...
Piero D'Ancona's user avatar
1 vote
2 answers
225 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}...
JZS's user avatar
  • 481
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 $G$ ...
Sentem's user avatar
  • 81
6 votes
1 answer
343 views

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^{- \...
Boby's user avatar
  • 671
2 votes
0 answers
80 views

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 ...
Zhang Yuhan's user avatar
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 ...
Itay's user avatar
  • 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 ...
Alexander's user avatar
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\...
Nicolas Medina Sanchez's user avatar
0 votes
1 answer
126 views

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 ...
Mohammad A's user avatar
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 ...
onionbread's user avatar
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)} \...
Nicolas Medina Sanchez's user avatar
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,...
enihcamemit's user avatar
5 votes
1 answer
508 views

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 ...
Daniel Fonseca's user avatar
1 vote
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 ...
Simplyorange's user avatar
2 votes
1 answer
254 views

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.
user124297's user avatar
-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 \...
Peg Leg Jonathan's user avatar
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}\...
Dispersion's user avatar
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 $...
VojtaK's user avatar
  • 151
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_{\...
ABB's user avatar
  • 4,058
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\...
ABB's user avatar
  • 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 ...
enihcamemit's user avatar
4 votes
1 answer
661 views

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 ...
Luis Yanka Annalisc's user avatar
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, ...
CfourPiO's user avatar
  • 159
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|...
Jiawen Zhang's user avatar
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 \...
Mr. Proof's user avatar
  • 159
2 votes
2 answers
916 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. ...
Tony419's user avatar
  • 421
2 votes
1 answer
198 views

$|\hat\mu(\xi)| \lesssim |\xi|^{-1/2}$ 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)|...
stoic-santiago's user avatar
1 vote
2 answers
152 views

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{...
Johnny T.'s user avatar
  • 3,625
2 votes
0 answers
149 views

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{...
Dapao Zhang's user avatar
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}\!}...
Dapao Zhang's user avatar
1 vote
0 answers
79 views

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 $...
ABB's user avatar
  • 4,058
1 vote
0 answers
244 views

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 ...
Dionel Jaime's user avatar
6 votes
1 answer
397 views

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 ...
J.Mayol's user avatar
  • 489
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 \...
Desura's user avatar
  • 233
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 ...
Jeanne Scott's user avatar
  • 2,137
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}\...
Tony419's user avatar
  • 421
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?
ABB's user avatar
  • 4,058
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_\...
Tanya Vladi's user avatar
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 ...
YT_learning_math's user avatar
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 ...
J.Mayol's user avatar
  • 489
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 ...
Manlio's user avatar
  • 342
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 $\mathbb{R}$ ...
Q-Zh's user avatar
  • 960
3 votes
1 answer
328 views

Large Fourier submatrices with small operator norm

Consider a finite abelian group $G$ (I'm mostly interested in $\mathbb{Z}_2^n$). For two subsets $A$ and $B$ of $G$, one can form a submatrix of the Fourier transform matrix on $G$ by keeping only ...
alesia's user avatar
  • 2,772
4 votes
1 answer
783 views

Fourier transform derivation from Laurent series

Using Laurent Series of a function $f(z)$ around a point $a \in \mathbb{C}$ $$f(z) = \sum^{\infty}_{n=-\infty} c_n(z-a)^n \ \ \ \ (1)$$ where $$c_n = \frac{1} {2\pi i}\int\limits_{\gamma}\frac {f(z)} {...
user740171's user avatar
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 $s \in [0, \tfrac{1}{2}]$. ...
F. H.'s user avatar
  • 63
1 vote
0 answers
668 views

Asymptotics of a function from its Fourier transform

My question is: given a Fourier transform $\hat f$ of a function $f$, is it possible to estimate its asymptotic behaviour without performing the inverse transform? Let me give a concrete example. ...
jonathan wolf's user avatar
1 vote
1 answer
385 views

Interchanging Integration Order involving Fourier Transform

$$f(\omega,u):=\frac1{\omega+iu}$$ where $i$ is the imaginary unit number. We see that the integral of a Fourier transform $$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
Hans's user avatar
  • 2,239
0 votes
1 answer
226 views

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 ...
ABIM's user avatar
  • 5,405
6 votes
1 answer
419 views

Positive-definiteness of radial sinc function in three dimensions

In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar ...
Capublanca's user avatar