Questions tagged [fourier-transform]

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Spectral family associated with the Laplacian operator in $L^{2}(\mathbb{R}^{n})$ [closed]

Let $\lambda>0$ be given. Define $$G_{\lambda}(\xi) = \chi_{_{\lbrace |\xi|^{2} \leq \lambda \rbrace }}. $$ and $$ E_{0}(\lambda)f = \mathcal{F}^{-1}[G_{\lambda}(|\xi|^{2})\mathcal{F}(f)], \ \ f \...
user253963's user avatar
2 votes
1 answer
191 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
7 votes
2 answers
1k views

What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?

What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"? I searched on Google but couldn't find a satisfiable answer. Thanks in advance :)
Yep's user avatar
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Discontinuity of the Fourier transform of $ x \mapsto (1+ x^2)^{- \gamma/2}$ for $\gamma \leq 1$

Fix $\gamma > 0$. Let $\mathcal{F}$ be the Fourier transform and consider the function $f(x) = (1+ x^2)^{- \gamma/2}$ for $x \in \mathbb{R}$. This function is in $\mathcal{S}'(\mathbb{R})$ and its ...
Goulifet's user avatar
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0 answers
111 views

Parseval identity extension?

I have stumbled upon the following three-dimensional series: $$\Lambda_p = \sum_{\underline{n}} \left(\frac{\left|n_1\right|}{\left|\left|\underline{n}\right|\right|_2}\right)^p \left|\hat{f}(\...
CNS's user avatar
  • 21
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1 answer
221 views

Fourier transform of a Radon measure [closed]

Let $\mu$ be a Radon measure on $\mathbb R^d$ with finite total mass: I guess that it is a tempered distribution on $\mathbb R^d$ and thus one may consider its Fourier transform. Now I guess that its ...
Bazin's user avatar
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0 answers
87 views

Cousin of Fourier transform for rescaling and translating functions

Is there a cousin of the Fourier transform which obeys the following property: if I rescale and translate a function, then its transformed form, at each frequency, only depends on the original ...
Kevin Ellis's user avatar
2 votes
1 answer
191 views

Non-Fourier complete orthogonal basis?

The Fourier Transform (FT) Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero Is invertible: info-preserving, has inverse function Is energy-...
OverLordGoldDragon's user avatar
1 vote
0 answers
113 views

Does the Ramanujan-Petersson condition correspond to a Fourier type property?

The Ramanujan-Petersson is one of the requirements used in Selberg's class of L-functions, and as such is a necessary condition for the Riemann Hypothesis to hold. The general converse conjecture ...
Sylvain JULIEN's user avatar
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1 answer
190 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
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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
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Discretizing a differential operator which is a function of the derivative operator

Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$. As an example, let $p(x)=x^{2}+2x$. Then ...
Mirar's user avatar
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Proof for $\Phi(t)$ is strictly decreasing for $t>0$ in Riemann's zeta function

I am looking for reference for proof that $\Phi(t)$ is strictly decreasing for $t>0$ and the first derivative of $\Phi(t)$ is negative for $t>0$ (see Page 5 in Conrey's article below) Conrey ...
John's user avatar
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1 answer
274 views

2-Wasserstein metric on convolution of probability distributions

I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
F J's user avatar
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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
332 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
1 answer
430 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
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1 vote
0 answers
70 views

Hyperplanes which equalize the Radon transforms of two distributions

Let $p_1$ and $p_2$ be "nice" probability densities on $\mathbb R^m$, for example the densities of a multivariate Gaussians $N(\mu_1,\Sigma)$ and $N(\mu_2,\Sigma)$ with common covariance ...
dohmatob's user avatar
  • 6,716
1 vote
1 answer
309 views

What is the 3-dimensional Fourier transform of $1/k^4$?

In electrostatics, we often encounter the following 3-dimensional integral: \begin{equation} V=\int d^{3}\vec{k}\,\dfrac{e^{i\vec{k}.\vec{r}}}{|\vec{k}|^{2}} \end{equation} which yields the Coulomb ...
HoangNguyen's user avatar
1 vote
0 answers
150 views

Fourier transform of inverse of determinant of 1+ skew-symmetric matrix

I have asked the following question in math stackexchange(https://math.stackexchange.com/questions/4389626/fourier-transform-of-inverse-of-determinant-of-1-skew-symmetric-matrix), but did not receive ...
Zhan's user avatar
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1 answer
167 views

The semigroup of Laplace-Beltrami operator on 3-flat torus

I am studying a recent paper in which the author worked on the rectangular, flat 3 tori. It can be realized, the author explained, as $\mathbb{R}^3 \over (L_1 \mathbb Z \times L_2 \mathbb Z \times L_3 ...
Mr. Proof's user avatar
  • 159
8 votes
1 answer
987 views

Who introduced the discrete Fourier transform?

I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of ...
AChem's user avatar
  • 803
5 votes
1 answer
231 views

Infimum of Fourier transform of singular measure

Let $\mu$ be a finite non negative singular measure on $\mathbf{R}^d$. I would like to know if there exists some result on the infimum of the absolute value of its Fourier Transform $$\hat{\mu}(t)=\...
Tiblodocus's user avatar
2 votes
1 answer
267 views

Fourier transform of the indicator function of the semi-ball

I wonder if it is possible to derive an analytical form for the Fourier transform of the indicator function of a semi-ball: $$g_1(\underline{\xi}) = \int_{\mathcal{B}_+(R)} e^{i \underline{\xi} \cdot \...
CNS's user avatar
  • 21
3 votes
1 answer
437 views

Fourier transform of the unit ball in L1 metric

The Fourier transform of the (indicator function of the) unit ball is well known to be given by the Bessel functions (see Fourier transform of the unit sphere). What can be said about the $\ell_1$ ...
MCH's user avatar
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1 vote
0 answers
74 views

Calculation of a multi-dimensional Fourier transform

I am interested in the following multi-dimensional Fourier transform: $$ \int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}...
Y.Okuyama's user avatar
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1 vote
0 answers
73 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
  • 3,898
2 votes
1 answer
239 views

Unit singular value conjecture for discrete Fourier transform submatrix

This question was motivated by Singular value decomposition of truncated discrete Fourier transform matrix Consider for integers $1\leq k\leq N$, $1\leq n_0\leq N-k+1$ the $k\times k$ sub-unitary ...
Carlo Beenakker's user avatar
3 votes
0 answers
160 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
  • 3,898
4 votes
0 answers
160 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
1 vote
0 answers
235 views

Vector convolution?

I am working on a research problem which leads to the following optimization problem: \begin{equation} \hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
Mamal's user avatar
  • 273
23 votes
6 answers
4k views

Anti-delta function?

Did anyone ever consider a "function" or "distribution" $F(x)$ with the following property: its integral $\int_a^b F(x)\,dx=0$ for any finite interval $(a,b)$ but $\int_{-\infty}^\...
Anixx's user avatar
  • 9,306
6 votes
1 answer
1k views

Eigenvalues and eigenfunctions of the Laplace operator on entire plane

According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $...
UserA's user avatar
  • 587
8 votes
1 answer
454 views

Functional equation with Fourier transform and $\frac{1}{x} f(\frac{1}{x}) $

What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation: $$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$ $\...
Bertrand's user avatar
  • 1,121
4 votes
1 answer
472 views

The main topics (issues, problems) of the Fourier transform

To explain what we are looking for, let's have a quick review on some points in Fourier transform on periodic functions in both continuous and discrete cases. We emphasize that our attention is ...
ABB's user avatar
  • 3,898
1 vote
1 answer
161 views

Probability of two Boolean functions being equal expressed in terms of the maximum Fourier coefficient

This paper by Maslov et al. uses that the probability of two $n$-bit Boolean functions $l(x)$ and $g(x)$ being equal is bound in terms of $\hat{g}_\text{max}$, the largest Fourier coefficient of $g(x)$...
Sebastian's user avatar
5 votes
0 answers
289 views

Similarity in Navier-Stokes equation and convolution in finite abelian groups?

Let $G$ be a finite abelian group, $X = (x_g)_{g \in G}$ be a vector of variables. Set for $g \in G$: $$\tau_g(X) := \frac{1}{|G|} \sum_{\rho \text{ irred. }} \chi_{\rho}(-g) \exp(\sum_{s \in G} \chi_{...
mathoverflowUser's user avatar
0 votes
0 answers
99 views

How to construct non-abelian functions?

I have found some functions $t_g, g \in G$ for cyclic groups $G=C_n$ which seem to satisfy the following convolution identity: $$t_g(x+y) = \sum_{h \in G} t_{gh^{-1}}(x) t_h(y)$$ Example of such ...
mathoverflowUser's user avatar
6 votes
1 answer
495 views

Rate of decrease of the Fourier transform of standard mollifiers

What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$, $$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$ and $$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\...
Iosif Pinelis's user avatar
0 votes
0 answers
136 views

Binary polynomial evaluation

Let $p$ be a prime number and let $P(x) = a_0 + a_1x + a_2x^2 + \dots + a_{n-1}x^{n-1}$ be a polynomial in $\mathbb{Z}_p[x]$ with binary coefficients, i.e., such that $a_i \in \{0,1\}$ for all $i = 0,...
Bean Guy's user avatar
3 votes
0 answers
275 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
  • 393
1 vote
0 answers
131 views

Polynomial interpolation of binary vectors

Let $\mathbb{F}$ be a finite field and let $\boldsymbol{x} = (x_1, x_2, \dots, x_n)$ be $n$ pairwise distinct points in $\mathbb{F}$. Given the vector $\boldsymbol{y} = (y_1, y_2, \dots, y_n)$, with $...
Bean Guy's user avatar
0 votes
0 answers
252 views

Precise decay of density through Fourier transform

Suppose $f(x)$ is a probability density on $\mathbb{R}$. Let $\varphi(t)=\int e^{itx}f(x)dx$ denote the Fourier transform (characteristic function). It is well-known that if $\int |x|^p f(x)dx<\...
Uchiha's user avatar
  • 87
1 vote
1 answer
190 views

2-dimensional Fourier transform

Let $k=(k_1,k_2) \in \mathbb{Z}^2$. Let $\lambda=(\lambda_1,\lambda_2)\in [0,2\pi]^2$ and $F(\lambda)$ be a bounded real function of $\lambda\in [0,2\pi]^2$. I am interested in the following equation:...
HyyFly's user avatar
  • 187
0 votes
0 answers
160 views

Critical exponent in Sobolev scale and singular continuous measure

Recently I started the following discussion about Sobolev spaces where I was interested in finding the maximal $s \in \mathbb{R}$ such that given function/distribution belongs to the Sobolev space $H^...
truebaran's user avatar
  • 9,140
0 votes
1 answer
202 views

Laplace transform injectivity for different values of $p$

Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty ).$ Assume that there exist $p_{0},p_{1}\in %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion ,$ $p_{0}\neq ...
Gustave's user avatar
  • 545
2 votes
2 answers
383 views

What is multidimensional Fourier transform of $e^{-\mathrm{abs}(x)}$?

What is multidimensional Fourier transform of $e^{-\mathrm{abs}(x)}$? I know the answer for 1-dimension case. However, I cannot do integration for higher dimensional case. I used spherical coordinate ...
user409452's user avatar
1 vote
0 answers
36 views

Solving an equation containing Laplace transform

Consider the equation \begin{equation} \frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}% (y)(s_{2})=\mathcal{L(}y)\mathbf{(}p), \end{equation} where $\mathcal{L}$ is the ...
Goga's user avatar
  • 47
1 vote
0 answers
99 views

The Pushforward of the Liouville measure

Let's consider a Hamiltonian action of a torus $T$ on a symplectic manifold $(M, \Omega)$. We denote by $\mu: M \rightarrow \mathfrak{t}^*$ the moment map and by $\Omega_\mathfrak{t}(X) := \Omega + &...
Mira's user avatar
  • 129
1 vote
0 answers
213 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

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