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Fourier decay implies what kind of regularity

We consider a function $f:\mathbb R^2 \to \mathbb C$ that is compactly supported and bounded. In addition, we know that $$\lim_{\vert x\vert \to \infty} \vert x \vert^2 \vert \hat{f}(x)\vert =0,$$ ...
Yizheng Yuan's user avatar
0 votes
1 answer
115 views

Fourier transform of exponential over torus

I found the following formula for the Fourier transform on a flat 2-torus, but I don't quite know how to derive it. We have a variable $q=(q_x,q_y) \in [0,2\pi)^2$ and by considering it in polar ...
António Borges Santos's user avatar
9 votes
1 answer
429 views

A curious norm related to the L¹ norm

If $f \in C^0([0,1])$, one can define $$\Vert f \Vert_? = \sup_{J \subset [0,1]} \left\lvert \int_J f \right\rvert,$$ where $J$ runs among all subintervals of $[0,1]$. This is a norm on $C^0([0,1])$ (...
PseudoNeo's user avatar
  • 575
3 votes
1 answer
111 views

Sobolev inequalities and Wiener algebra

It follows from the Gagliardo-Nirenberg inequality that for a locally integrable function $f$ defined on $\mathbb R^d$ (we assume $d\ge 3$) such that $\nabla f$ belongs to $L^2(\mathbb R^d)$ and $$ \...
Bazin's user avatar
  • 16.2k
1 vote
1 answer
136 views

$\ell^2 \to L^\infty$-inequality for almost periodic functions

Suppose $u : \mathbb R \to \mathbb C$ is a smooth, (Bohr) almost periodic function. Formally, such a function admits a Fourier series expansion $$u(x) = \sum_{\lambda \in \Lambda} \widehat u(k) e^{i \...
Jason Zhao's user avatar
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
4 votes
1 answer
255 views

Asymptotic behavior and of an integral on a d-dimensional torus

I am trying to evaluate the asymptotic behavior of the following integral as $t \to \infty$: $$ I(t; \mathbf{v}) = \int_{[-\pi, \pi]^d} \frac{\sin(t f(\mathbf{k}))}{\sin(f(\mathbf{k}))} e^{i t \mathbf{...
Ko Hey's user avatar
  • 81
4 votes
3 answers
507 views

Fourier transform in $L^1$?

Let $f \in L^1 \cap L^2$. Are there any natural conditions on $f$ that ensure that the Fourier transform $\hat f$ is in $L^1?$ I don't want to have anything as restrictive as Schwartz. I am rather ...
António Borges Santos's user avatar
4 votes
1 answer
222 views

estimate a singular integral using a dyadic decomposition

Let $0<\alpha_{j}<1$, $j=1,\dots,d+1$. I am trying to estimate the following singular integral: $$I(y_{1},\dots,y_{d},z):=\int_{\substack{ x\in[0,1]^{d}\\ 1/2<|x|<1}} \frac{d x_{1} \dots d ...
Medo's user avatar
  • 852
1 vote
1 answer
111 views

How to show such result for generalized $ O(|x|^{-1/2}) $ function?

Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
Luis Yanka Annalisc's user avatar
4 votes
1 answer
441 views

Reference or proof of a theorem of L. Fejér on summability of Fourier series

In the article "Ensembles exceptionnels" by A. Beurling, the author cites the following theorem of Fejér: Suppose that a $2\pi$ periodic function $ f $, Lebesgue integrable in $(0,2\pi)$ ...
an_ordinary_mathematician's user avatar
1 vote
1 answer
112 views

A bilinear estimate with a simple one-dimensional oscillatory integral kernel

Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$. I am trying to show that $$\int_{\mathbb{R}}\int_{\mathbb{R}} \,K(y,z)\, \frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
Medo's user avatar
  • 852
0 votes
0 answers
124 views

Counterexamples in Laplace transforms

Of all the examples I know of (bilateral) Laplace transforms $F$ defined on their maximal vertical strips $V_{a,b}=\{ z \in \mathbf{C} : a < \operatorname{Re}(z) < b \}$ with $-\infty < a <...
proofromthebook's user avatar
0 votes
0 answers
88 views

Convolution of $\mathscr{F}\{ \log \}(x) * \mu$ with compactly supported measure $\mu$

As I read in this post the Fourier transform of $\psi(\lambda) = \log{|\lambda|}$ must be interpreted in distributional sense and it is given by: $$\mathscr{F}\{\psi\}(x)=-2\pi \gamma \delta(x)-\pi \...
Grandes Jorasses's user avatar
0 votes
0 answers
112 views

Fourier integral operators and parametrix

Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary. Question: Is there an expression for the ...
0x11111's user avatar
  • 593
2 votes
1 answer
231 views

Fourier coefficients of the logarithm of a given function

Let $f$ be a $1$-periodic real function that I know is bounded away from zero: $$ f(x) = \sum_{n = -\infty}^\infty c_n e^{2\pi i n x} $$ Let me also assume that $f$ is analytic with Fourier ...
Romain Gicquaud's user avatar
5 votes
0 answers
243 views

Is there a way to solve this integral on the sphere explicitly?

Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that $k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral $$f(y):=\int_{\...
Medo's user avatar
  • 852
11 votes
0 answers
3k views

Eric T. Sawyer's proof of Fourier restriction conjecture

Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
a curious fellow's user avatar
1 vote
0 answers
98 views

Periodicity in one Fourier variable

Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument. We can thus write $f$ as $$ f(x) = \sum_{n \in \...
António Borges Santos's user avatar
0 votes
1 answer
624 views

Does this dyadic sum converge?

Let $a\in (0,1)$ and define $$J(j):=\int_{0}^{\infty} e^{- 2^{j} s} \frac{s^{a}}{1+s^{2a}} ds,\quad j\in \mathbb{Z}.$$ Note that rescaling $2^{j} s\mapsto s$ shows that $$J(j)\leq 2^{-j(1+a)}\int_{0}^...
Medo's user avatar
  • 852
6 votes
1 answer
310 views

Surjectivity of a class of integrals in dimensions two

Let $\Omega \subset \mathbb{R}^2$ be an open set and $G(x,\theta): \Omega \times [0,2\pi]\rightarrow \mathbb{R}$ be a positive continuous function. Assume $F:\Omega \rightarrow \mathbb{R}^2$ defined ...
MathLearner's user avatar
3 votes
1 answer
929 views

What is the Fourier series of $\sin(1/x)$ in $[-\pi,\pi]$?

What is the Fourier series of $\sin(1/x)$ (or $x^k\sin(1/x)$, where $k$ is a positive integer) in $[-\pi,\pi]$? This function evidently does not satisfy Dirichlet's conditions. However, Dirichlet's ...
user37022'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
215 views

Asymptotics for oscillatory integral

Consider the following integral for $f \in C_c^{\infty}(\mathbb R^n)$, $x_0$ fixed (possibly zero), and $n \ge 3$ $$F(\lambda) = \int_{\mathbb R^n} e^{i\lambda \vert x-x_0 \vert^2} \frac{f(x)}{\vert x ...
António Borges Santos's user avatar
1 vote
0 answers
67 views

Estimating commutator of Fourier integral

Let $f(x)= \log(\vert x\vert)$ on $\mathbb R^2$ and define $s_n:H^2 \to L^2$ where $H^2$ is the second Sobolev space by $$ s_n(g)(x) = \frac{nf(x)}{4\pi i} \int_{\mathbb R^2} e^{\frac{in\vert x-y\...
António Borges Santos's user avatar
8 votes
2 answers
670 views

Asymptotic behavior of a certain oscillatory integral

Let $x>0$ and consider the integral $$I(x):=\int_0^\infty \frac{e^{i r}}{r^{\frac{1}{2}}} \int_0^\infty \frac{e^{-s}}{s^{\frac{1}{2}}} \frac{r}{sx+\sqrt{sxr}+r} \, ds \, dr.$$ I am trying to ...
Medo's user avatar
  • 852
0 votes
1 answer
245 views

Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e

The Riemann-Liouville integral is defined by $$ I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t $$ where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
Grandes Jorasses's user avatar
0 votes
0 answers
317 views

What is the "best" good kernel?

A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied: $\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$, $...
Dr. Pi's user avatar
  • 3,062
3 votes
0 answers
141 views

Direct analytic proof of positive definiteness of stable characteristic functions

Is there a direct analytic proof that the function $$ f ( t ) = \exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right), \qquad \lambda > 0, \quad |\theta| < \...
tsnao's user avatar
  • 620
0 votes
1 answer
150 views

The asymptotic behaviour of a singular integral

Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$. I am trying to determine the asymptotic behaviour of $$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\...
Medo's user avatar
  • 852
2 votes
1 answer
236 views

Approximation of Hölder functions by Fourier series

Let $Q$ be a bounded domain in $\mathbf R^N$ with smooth boundary. Let $f\in C^a(\overline{Q})$, $0<a<1$. Denote $\psi_k(x)$ normalized eigenfunctions and $\lambda_k$ eigenvalues ($k=0,1,2\...
user500030's user avatar
6 votes
3 answers
267 views

Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?

Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let $$ F_a(t) = \sum_{k \in \mathbb Z} f(t+ak) $$ be the ...
user975628's user avatar
1 vote
1 answer
148 views

Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?

I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator $...
Medo's user avatar
  • 852
0 votes
0 answers
88 views

Closed formula for iterated Fourier series

I'm trying to obtain a closed formula for the following integral. \begin{align} I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt] & {}+ b_{1,...
Marco's user avatar
  • 293
4 votes
1 answer
198 views

How to compute the asymptotics of this oscillatory integral?

I posted this on Stackexchange but got no responses or comments. Consider the following integral, for $\epsilon\ne 0:$ $$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[...
Josh Lackman's user avatar
  • 1,198
7 votes
2 answers
407 views

$L^p-L^q$ boundedness of this simple singular oscillatory integral operator

Let $0<\alpha<1$ and define $$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$ The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of $Hf(x):=\int \frac{...
Medo's user avatar
  • 852
5 votes
1 answer
594 views

Exponential sum vs. exponential integral via Poisson summation

When we want to estimate an exponential sum $$ \sum_{M<m\le M'}e(f(m)) \quad\text{with}\quad 1\le M\le M'\le 2M \quad\text{and}\quad e(x):=\exp(2\pi ix) $$ where $e(x):=\exp(2\pi ix)$ and the phase ...
snufkin26's user avatar
  • 363
1 vote
0 answers
205 views

Fourier transform of functions mapping manifolds, is there a definition?

$\DeclareMathOperator\SO{SO}$I have a problem which boils down to the analysis of functions of the form $$ f : \mathbb{R} \to \SO(3)^n $$ Since $\SO(3)$ is a compact group so is $\SO(3)^n$. Now if ...
user8469759's user avatar
10 votes
1 answer
474 views

A basic estimate of exponential sums

Demeter in his book "Fourier Restriction, Decoupling, and Applications" (P287) used the following estimate: \begin{equation} \sup_{0\leq n\leq q}\bigg|\sum_{m=0}^n e^{2\pi i\frac{a}{q}m^2}\...
Dapao Zhang's user avatar
5 votes
2 answers
484 views

Optimizing a smoothing function with the Prime Number Theorem in mind

Let $f:[0,\infty)\to \mathbb{R}$ be a function with $f(x)=1$ for $0\leq x\leq 1$. Write $Mf$ for the Mellin transform of $f$. Let $c>0$, $T>10^6$ be constants. We are interested in minimizing ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
79 views

For $\Phi$ a majorant of $1_{[-1/2,1/2]}$, how small can the total variation of $\widehat\Phi$ be?

Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such that $\Phi(t)\geq 1$ for $|t|\leq 1/2$. Assume furthermore that $\Phi$ and $\widehat\Phi$ are both in $L^1\...
H A Helfgott's user avatar
  • 20.2k
1 vote
0 answers
47 views

Functional inequality for fractional Laplacian

Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
Matt Rosenzweig's user avatar
10 votes
1 answer
283 views

A function is of bounded variation if and only if the errors of its best approximation by trigonometric polynomials satisfy $\sum\frac{e_n}n<\infty$?

Let $\mathcal P_n$ be the set of trigonometric polynomials of degree less than or equal to $n$ and let $\lVert\cdot\rVert_\infty$ be the supremum norm. The error of the best approximation of $f$ of ...
Derivative's user avatar
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
3 votes
1 answer
251 views

Asymptotic behavior of a double oscillatory integral

Let $0<\theta_1,\theta_2<\pi/2$. Suppose $\psi$ is a smooth real-valued function with compact support. Consider the oscillatory integral $$I(t):=\int_{0}^{1}\frac{1}{(y-e^{\dot{\imath}\theta_1}) ...
Medo's user avatar
  • 852
2 votes
0 answers
216 views

Fourier transform of Dirac delta distribution

Let $f,g$ be Schwartz functions on $\mathbb R^4$, we denote them as $\mathcal S(\mathbb R^4)$, one can then define the transform $V$ mapping $f,g$ to a Schwartz function $\mathcal S(\mathbb R^8)$ $$ V(...
Guido Li's user avatar
6 votes
0 answers
213 views

Equivalent forms of Fourier restriction conjecture

this question is posted in mathstackexchange, but it seems that no one answers it. Sorry to the administrator if this question is not appropriate on Mathoverflow. I'm reading Pertti Maattila's book ...
Tutukeainie's user avatar
3 votes
1 answer
256 views

A sharp estimate for an oscillatory integral with a simple phase

Let $\alpha>1$ not necessarily an integer, and let $\psi:\mathbb{R}\rightarrow \mathbb{R}$ be a smooth function with compact support. Consider the oscillatory integral $$I(\lambda):=\int_{0}^{\...
Medo's user avatar
  • 852
1 vote
1 answer
203 views

Explanation of a step in a work by C. E. Kenig and A.D. Ionescu

I am studying the work Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
Mr. Proof's user avatar
  • 159
1 vote
1 answer
506 views

Fourier transform of the fractional Poisson kernel

Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows: $$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
Student's user avatar
  • 537

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