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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 ...
Greg Zitelli's user avatar
  • 1,124
11 votes
0 answers
707 views

What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?

The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions: $\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...
Sara's user avatar
  • 111
6 votes
0 answers
158 views

Fourier-like transforms for a Day convolution?

The presheaf category on a monoidal category inherits the monoidal structure via the Day convolution. Moreover you can inherit (bi)closed monoidal structure. In the study of Fourier analysis we can ...
Steven Schaefer's user avatar
6 votes
0 answers
203 views

Uniform estimates of Fourier transform of tempered functions with parameters

Consider the following function in $\mathbb{R}^3$: $$ f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2}, $$ where $\frac{1}{2}<\alpha<1$, and ...
Tomas's user avatar
  • 879
5 votes
0 answers
194 views

When does the Fourier transform of a measure decay?

Let $\mu$ be a Borel measure on $\Bbb R^d$. It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies $$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$ However if ...
Guy Fsone's user avatar
  • 1,101
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
5 votes
0 answers
82 views

Are Stochastic Process Characterized by Their conditional Moments

Suppose that $X_t$ is a real-valued stochastic process. Then is $X_t$ characterized by it's conditional moments? In the sence that, if $Y_t$ is another process, such that $$ \mathbb{E}\left[\int_s^T\...
ABIM's user avatar
  • 5,405
5 votes
0 answers
210 views

Existence of $A\subset\Bbb{R}^n$ of finite measure and $\hat{1_A}\in\bigcap_{q>1}L^q$, but s.t. for some $1<p<\infty$, $1_A$ is no $L^p$-Fourier mult

I am interested in the following somewhat obscure question: Is there some $n \in \Bbb{N}$, and a set $A \subset \Bbb{R}^n$ of finite measure such that the Fourier transform $\widehat{1_A}$ of its ...
PhoemueX's user avatar
  • 734
5 votes
0 answers
124 views

Extension of function that minimizes function of Fourier transform

Suppose that $f$ is a given (smooth) function defined on $B\subset \mathbb R^n$. (For simplicity, take $B$ to be the unit ball but more generally we can take $B$ to be some other measurable set). How ...
Holden Lee's user avatar
5 votes
0 answers
326 views

Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?

The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ : $$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) = \frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
Bertrand's user avatar
  • 1,199
5 votes
0 answers
286 views

$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?

For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$ Let $1\leq p \leq ...
Inquisitive's user avatar
  • 1,051
4 votes
0 answers
188 views

Branch cuts, inverse Fourier transform and large time asymptotics

Let the Fourier transform of $f(t)$ be defined as $F(\omega) = \int_{-\infty}^\infty dt f(t) e^{i\omega t}$ for values of $\omega$ where the integral exists. What are the precise conditions on $F(\...
Fetchinson0234's user avatar
4 votes
0 answers
140 views

Given $\theta$, find $f$ such that $\int_{\mathbb{T}} \text{e}^{i\theta} \cos(h \cdot f) = 0,$ for all $h \in \mathbb{N}$

Let $\theta$ be a $C^{\infty}$ (resp. analytic) real-valued function on $\mathbb{T}=[0,2\pi]/\{0,2\pi\}$. When can one find $f \neq 0$, $C^{\infty}$ (resp. analytic) real-valued function on $\...
Leonardo's user avatar
  • 405
4 votes
0 answers
595 views

On smoothness of a function and decay of its Fourier transform

I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here. I am interested in a quantitative version of the principle that smoothness of ...
Durac's user avatar
  • 41
4 votes
0 answers
116 views

Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?

As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, ...
Robin Saunders's user avatar
4 votes
0 answers
171 views

Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions

I asked the question below on Math Stack Exchange, https://math.stackexchange.com/questions/2592555/convergence-of-integral-formula-for-fourier-inversion-and-hilbert-transform-fo, but [despite it ...
Julian Newman's user avatar
4 votes
0 answers
349 views

Is the Fourier transform of $\frac{1}{\mu+|\xi|^{2\alpha}}$($\mu>0$) a bounded function?

Consider $m(\xi)=\frac{1}{\mu+|\xi|^{2\alpha}}$, where $\xi\in\mathbb{R}^n$, $\mu, \alpha>0$, I want to know that if $m(\xi)$ is a multiplier of $\mathcal{M_{1}^{\infty}}$,i.e., whether the ...
Tomas's user avatar
  • 879
3 votes
0 answers
136 views

Possible gaps for a function and its Fourier transform

This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish. A nice remark by Christian ...
kaleidoscop's user avatar
  • 1,352
3 votes
0 answers
144 views

Minimizing vertical integral of a Mellin transform

Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say). Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small. What is the choice of $...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
86 views

Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$

Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial , so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that $g(|x|^\gamma)$ is positive ...
Alexey S's user avatar
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
3 votes
0 answers
164 views

Green-Tao's "Polylogarithmic bound for $r_4(N)$"

On P.23 of https://arxiv.org/pdf/1705.01703.pdf, they seemed to suggest that by the non-negativity of $\psi\big(\frac{k}{N}\big)$ for all $k$, $$ K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
Jonathan Lam's user avatar
3 votes
0 answers
204 views

The inversion formula for the square root of a positive function

Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\...
ABB's user avatar
  • 4,058
3 votes
0 answers
320 views

Does convolution by a Schwartz function preserve symbol classes?

I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
Ervin's user avatar
  • 395
3 votes
0 answers
309 views

The $2\pi$ factor in the Fourier transform and dimensional analysis

I have been thinking about the $2\pi$ factor in the various conventions of the Fourier transform. For example, I was looking for a way to justify the following: $(*)$ If we define $\hat f(\xi) = \int ...
Alan C's user avatar
  • 613
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
3 votes
0 answers
140 views

Decay of Laplace (or Mellin) transform beyond region of convergence?

Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
150 views

Keeping $\max_{|t|\geq 1} |\widehat{\phi}(t)|$ small (uncertainty principle)

Let $\phi:\mathbb{R}\to \lbrack 0,\infty)$ be piecewise continuous, symmetric ($\phi(x)=\phi(-x)$) and with support on $(-1,1)$. Let $\Phi(x)=\int_{-\infty}^x \phi(u) du$; assume $\Phi(1)=1$. What is ...
H A Helfgott's user avatar
  • 20.2k
3 votes
0 answers
66 views

Does this definition of the Fourier intensity measure make sense?

Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$. For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is ...
Darren Ong's user avatar
3 votes
0 answers
261 views

Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions

Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set \begin{equation*} {\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1) \end{...
Alex Pereira's user avatar
3 votes
0 answers
214 views

Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?

For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
Rajesh D's user avatar
  • 698
3 votes
0 answers
211 views

A question about Fourier transform of function of the type $Q(x)(1+P(x))^{z}$

For simplicity, consider in $\mathbb{R}^3$, and the Fourier transform of the following function $$f=(x_1+x_2+x_3)(1+|x|^2+x_1^2(x_2^2+x_3^2)+x_2^2x_3^2)^{-t+is},~~ \frac12<t<1,~~s\in \mathbb{R}.$...
Tomas's user avatar
  • 879
2 votes
0 answers
111 views

Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support

This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction? Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
79 views

Function that is (essentially) a self-convolution but not a multiple of a self-convolution

Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
194 views

Functions such that the *integral* of the Fourier transform is non-negative?

Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that $$\int_{-\infty}^x \widehat{f}(t) dt \...
H A Helfgott's user avatar
  • 20.2k
2 votes
0 answers
187 views

Three optimization problems of uncertainty principle/Paley-Wiener type

Let $\phi:\mathbb{R}\to\mathbb{R}$ be an even function with support on $[-1,1]$. Assume that it is in $L^1\cap L^2$ and that its Fourier transform is also in $L^1\cap L^2$. Assume as well that $|\phi|...
H A Helfgott's user avatar
  • 20.2k
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
2 votes
0 answers
204 views

Finding specific coefficients of product of high-dimensional Fourier series faster than FFT

I need a fast algorithm to perform a specific Fourier-type computation in my physics research. Suppose I have the following two Fourier series in three dimensions $$ a(t_1,t_2,t_3)=\sum_{j=-n}^{n}\...
groupoid's user avatar
  • 620
2 votes
0 answers
156 views

Fourier multiplier on $L^1$

On the Wikipedia page, one can read that an iff condition for L1 boundedness of the Fourier multiplier m(D) is that $$ \hat m\quad\text{ is a Borel measure with finite total mass. } $$ There is no ...
Bazin's user avatar
  • 16.2k
2 votes
0 answers
144 views

Positivity of the Fourier transform: prove or disprove that $\operatorname{Re}(\overline{\widehat{u}}(\xi) \widehat{F\circ u}(\xi))\geq0$

Let $F:[0,\infty) \to[0,\infty)$ be increasing, $C^1$ and $L-$Lipschitz with $F(0)=0$. Let $u\in L^1 (\Bbb R^d)$, $u\geq0$ so that $F\circ u\in L^1 (\Bbb R^d)$ I would like to prove (or disprove) ...
Guy Fsone's user avatar
  • 1,101
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
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
2 votes
0 answers
172 views

What are the necessary/sufficient conditions for a Fourier transform to have at least $k$ roots?

Let $f(x)$ be a symmetric function from $\mathbb{R}\to \mathbb{R}$, and $\hat f(k)$ be it's Fourier transform. What are the necessary and sufficient conditions for $\hat f(k)$ to have at least $n$ ...
SarthakC's user avatar
  • 139
2 votes
0 answers
127 views

Failure of Strichartz estimates for the wave equation: elaboration of a counter-example

One can read in Oh - Probabilistic perspectives in nonlinear dispersive PDEs (Proposition 64, p. 60) that there exists a function $F \in L^2_tL^{1}_x (\mathbb{R}_t\times \mathbb{R}^3_x)$ which is ...
J.Mayol's user avatar
  • 489
2 votes
0 answers
91 views

(Dis)continuity of periodic functions with non-summable Fourier series

Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$. We assume moreover that the square-summable Fourier coefficients of $f$, ...
Goulifet's user avatar
  • 2,306
2 votes
0 answers
127 views

eigenvectors of a graph Laplacian VS Fourier basis

Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory.
ABB's user avatar
  • 4,058
2 votes
0 answers
811 views

Existence of unbounded $M \subset \Bbb{R}$ of finite measure s.t. $1_M$ is $L^p$-Fourier multiplier

I would like to know if there is a measurable set $M \subset \Bbb{R}$ such that $M$ has finite Lebesgue measure $0 < \lambda(M) < \infty$, $M$ is unbounded in the sense that $\lambda(M \...
PhoemueX's user avatar
  • 734
2 votes
0 answers
143 views

Need to show bounded behavior of a particular Fourier transform

First let me be briefly state the relevant information to my problem: $\beta(s) \in C_0^{\infty}([-1,1])$, and $\beta \equiv 1$ around $s=0$. The $\beta$ I'm using is an even function, but it doesn't ...
Patch's user avatar
  • 377
2 votes
0 answers
70 views

Reading off the Fourier coefficients whether a function is everywhere locally bounded except for finitely many points

Suppose we consider an $L^2$-function $f:[0,1]\rightarrow \mathbb{R}_{\ge 0}$. How does the property "$f$ is a.e. bounded by a rational function" translate in terms of the Fourier coefficients? I ...
Olli's user avatar
  • 21
2 votes
0 answers
224 views

On uniform or simple convergence of Poisson Summation formula

Under good conditions on an even function $f(x)$ we have the Poisson Summation formula ($x>0$): $$f(0) + 2 \sum\limits_{n =1}^{\infty} f(nx)= \frac{1}{x} \left( \hat{f}(0) + 2 \sum\limits_{n =1}^{\...
Bertrand's user avatar
  • 1,199