# Questions tagged [fourier-transform]

The fourier-transform tag has no usage guidance.

385
questions

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### 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}(\...

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1
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### Fourier transform of a Radon measure

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 ...

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### 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 ...

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### 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-...

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### 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 ...

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### 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. ...

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2
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135
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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{...

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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 ...

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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 ...

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### 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 ...

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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{...

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185
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### 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}\!}...

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141
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### 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+...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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 ...

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604
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### 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 ...

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137
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### 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)=\...

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### 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 \...

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### 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 $\ell_1$ ...

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### 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}...

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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 $...

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### 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 ...

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91
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### 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=\...

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### 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(\...

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### 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 ...

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### 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}^\...

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votes

1
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### 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 $...

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### 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:
$$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$
$\...

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### 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 ...

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### 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 $g(x)$...

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### 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_{...

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### 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 ...

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### 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|<\...

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85
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### 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,...

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### 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 ...

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### 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 $...

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### 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<\...

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### 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:...

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### 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^...

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### 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 ...

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### 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 ...

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### Measure of set of singular points of a generalized discrete Fourier transform

Assume we are given a function that can be expressed in a Fourier-like expansion
$$
f\colon \mathcal{X} = [0, 2\pi]^d \to \mathbb{C}, \ \boldsymbol{x}\mapsto \sum_{i = 1}^K c_{i} e^{-i \boldsymbol{\...

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### 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 ...

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### 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 + &...

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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 ...

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### Discrete Fourier transform up to a diagonal matrix

Let $N$ be a natural number and $w=e^{\frac{2\pi i}{N}}$ and $u=e^{\frac{\pi i}{N}}$. Let's consider the diagonal matrix $D=\operatorname{diag}(1,u,u^2,\cdots,u^{N-1})$. It is well known that the set ...

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0
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### How to prove the Fourier transform of $e^{-x^p}$ is positive [duplicate]

I wonder how to prove that
$$\int_0^\infty\exp(-x^p)\cos(tx)\,dt\geq 0, \quad \frac{1}{2}<p<1.$$
This conclusion is used in the answer to another question here
Looking for sufficient conditions ...

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269
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### Fourier-positivity of a certain function

I am wondering how to prove the below Fourier transform is non-negative? I did much simulation and it seems to be non-negative.
$$\int_0^\inf (be^{-at^p}-ae^{-bt^p})\cos(tx)dt, 0<a<b, \frac{1}{2}...