Skip to main content

Questions tagged [fourier-transform]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1 vote
0 answers
56 views

Operator norm of some type of discrete Fourier matrix

Let $N$ be a natural number and let $w$ be a complex number. We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows, $$ a_{k,l}=\begin{cases}1 & l=k+1\\ w &...
ABB's user avatar
  • 4,030
0 votes
1 answer
224 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
-2 votes
0 answers
91 views

Is the product of exponentiated elliptic curve basis elements invariant under FFT of scalars?

I am working with an elliptic curve defined over a finite field $\mathbb{F}_p$ and have a basis set of points ${g_0, g_1, \ldots, g_n}$. When I perform the FFT on these points, I obtain a new basis ${...
Nerses Asaturyan's user avatar
-2 votes
0 answers
124 views

Plancherel's Theorem in wider space

By Planchere's Theorem, we have for $f$, $g$ in $L^2(\mathbb{R}^n)$ we have $$\int_{\mathbb{R}^n}fg=\int_{\mathbb{R}^n}\hat{f}\hat{g}$$ Also, by distribution theorey, we can define the Fourier ...
Holden Lyu's user avatar
0 votes
1 answer
138 views

Generalisation of Paley–Wiener type results for unbounded sets

Do you know an unbounded open set $A\subset \mathbb{R}^d$, $d\geq 2$ with the following property: if some integrable function $f$ on $\mathbb{R}^d$ has its Fourier transform vanishing on $A^c$ and all ...
kaleidoscop's user avatar
  • 1,332
0 votes
0 answers
42 views

Function samplable from the past and Hardy spaces

What I am ultimately looking for is a $L^2$ function $f$ on the real line that can be sampled from the past, i.e. for each $x<0$ there are $L^2$ coefficients $c_n(x)$, $n\in \mathbb{N}$ such that, ...
kaleidoscop's user avatar
  • 1,332
0 votes
0 answers
20 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
121 views

Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?

Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere. Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$. $\hat{f}$ is the Fourier transform fora function f.
Edward's user avatar
  • 9
3 votes
0 answers
103 views

How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?

The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
Daron's user avatar
  • 1,915
1 vote
0 answers
41 views

Fourier transform relation for spherical convolution

Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$. The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as $$(f*g)(R) = \...
Goulifet's user avatar
  • 2,226
1 vote
0 answers
64 views

Is there an generalisation of convolution theorem to integral transforms

Basic convolutions can be computed efficiently by taking fourier transforms and applying the convolution theorem. Is there something analogous for a more general transform, where we have a varying ...
nathan pannifer's user avatar
0 votes
1 answer
114 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
2 votes
0 answers
95 views

Construction of an analytic function whose Fourier transformation has compact support [closed]

Is there a non-constant real analytic function $f$ on $\mathbb{R^2}$ satisfying the following properties? $f$ vanishes on $x$-axis and $y$-axis; the Fourier transformation $\hat{f}$ of $f$ has a ...
adobereader's user avatar
0 votes
1 answer
100 views

Spectral theory: a key to unlocking efficient insights in network datasets

In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as ...
ABB's user avatar
  • 4,030
2 votes
0 answers
193 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
10 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
223 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
0 votes
0 answers
95 views

Is there an inner product on $\mathbb{F}_p\left[S_n\right]$ for which $\langle x, x \rangle \ne 0$ for all $x$?

Let $\mathbb{F}_p\left[S_n\right]$ be the group algebra of the symmetric group $S_n$ over the finite field $\mathbb{F}_p$. One can define an "inner product" in the usual way: $$\langle x,y \...
Jackson Walters's user avatar
0 votes
0 answers
54 views

Is it possible that a system of differential equation has a solution in time domain but not in Fourier domain? If so, why does it happen?

I have to solve \begin{align} &\frac{\partial h'_{1,1m}(t,r)}{\partial t} + \frac{2}{r} h_{1,1m}(t,r) = 0 \label{beta_0_1}\\ &\frac{\partial^2 h_{1,1m}(t,r)}{\partial t^2} = 0. \label{...
AleNekro97's user avatar
0 votes
0 answers
75 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
4 votes
2 answers
184 views

Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function

Consider the function $$\phi_{a,b}(t)=\frac{|t|}{e^{a|t|}-e^{-b|t|}}, \ \ t\in\mathbb{R},$$ where $0<a<b$. Can $\phi_{a,b}$ be the Fourier transform of a positive function for some $a<b$?
Ribhu's user avatar
  • 361
2 votes
0 answers
140 views

Information about smoothness of function from its Fourier transform

We know that if the integrable function $f\in H^\alpha(\mathbb{R}), 0<\alpha<1$ (Hölder continuous), then its Fourier transform $\hat{f}$ has the asymptotic form $ O (1/x^\alpha)$ as $x\to\infty$...
eN.meshok's user avatar
1 vote
1 answer
101 views

Examining the Hilbert transform of functions over the positive real line

$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
Gabriel Palau's user avatar
2 votes
2 answers
222 views

Is there a compactly supported differentiable function whose Fourier transform is not in L1?

In my MSE answer here, I discussed the example of compactly supported continuous function $$g(x)= \begin{cases} \dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\ 0,&\text{otherwise} \end{cases}$$ ...
D.R.'s user avatar
  • 771
-2 votes
1 answer
297 views

Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT

In one line: Given an exponentially decaying sine wave $x(t)$, how can we predict the amplitude of the resulting peak in frequency spectrum using discrete Fourier transform. In nuclear magnetic ...
ACR's user avatar
  • 790
1 vote
0 answers
41 views

Looking at a frequency reassignment rule as a Möbius transform

Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed. I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...
mathim1881's user avatar
2 votes
1 answer
380 views

The Fourier transform of the Liouville function?

The Liouville function in number theory is defined as: $$\lambda(n) := (-1)^{\Omega(n)} \text{ where } \Omega(n) := \sum_{p|n} v_p(n)$$ Taking the discrete time Fourier transform and then taking the ...
mathoverflowUser's user avatar
1 vote
1 answer
129 views

Function with non Riemann-integrable Fourier transform

Does there exist a compactly supported continuous function $f$ on $\mathbb R$, such that $$ \lim_{n\to\infty}\int_{-n}^n\widehat{f}(x)\ dx $$ does not exist? Here $\widehat f$ is the Fourier transform ...
Nandor's user avatar
  • 183
1 vote
0 answers
65 views

The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components [closed]

Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$. It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity ...
ABB's user avatar
  • 4,030
1 vote
1 answer
141 views

Numerical partial differentiation of a convolution product with FFT

How can one numerically calculate the partial derivatives of a convolution function, particularly when the closed-form or analytical expressions of the derivatives are not readily available? I am ...
ACR's user avatar
  • 790
32 votes
8 answers
4k views

Motivation and physical interpretation of the Laplace transform

Concerning the one-sided Laplace transform, $$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$ what is a motivation to come up with that formula? I am particularly interested in "physical&...
AlpinistKitten's user avatar
2 votes
1 answer
120 views

Singular Integrals and $L^1$

Let us consider in one dimension the Fourier multiplier $\vert D\vert$ and the derivative $iD$. Both are well-defined on the Schwartz space $\mathscr S(\mathbb R)$ with the derivative sending $\...
Bazin's user avatar
  • 15.7k
2 votes
1 answer
157 views

Prove if the fractional Laplacian of a function is bounded

Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$. Here $(-\Delta)^s$ is the ...
Physics user's user avatar
2 votes
0 answers
107 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
  • 15.7k
0 votes
0 answers
185 views

Relationship between Fourier inversion theorem and convergence of "nested" Fourier series representations of $f(x)$

$\DeclareMathOperator\erf{erf}\DeclareMathOperator\sech{sech}\DeclareMathOperator\sgn{sgn}\DeclareMathOperator\sinc{sinc}$This is a cross-post of a question I posted on MSE a couple of weeks ago which ...
Steven Clark's user avatar
  • 1,091
1 vote
0 answers
115 views

Circulant matrix inverse in $GF(p)$

For a polynomial $C(x)=c_0+\dots+c_n x^n$, consider a circulant matrix $C$ such that $$ C= \begin{pmatrix} c_0 & c_{n-1} & \cdots & c_2 & c_1 \\ c_1 & c_0 &...
Oleksandr  Kulkov's user avatar
2 votes
1 answer
162 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
3 votes
0 answers
88 views

Efficient multiplication of Cayley-Dickson numbers

The question was already asked here, but doesn't have any meaningful answer, hence I'd like to re-post it. Assuming that we have an algebra with conjugation, we can use Cayley-Dickson construction to ...
Oleksandr  Kulkov's user avatar
1 vote
0 answers
44 views

Why do we need the concept of Fourier measurability with growth function $\mathcal F$?

I'm studying the book Higher Order Fourier Analysis by Terence Tao (https://terrytao.files.wordpress.com/2011/03/higher-book.pdf). There, it defines that a function $f:[N]\to\mathbb{C}$ has Fourier ...
El Tudey's user avatar
2 votes
2 answers
207 views

Theoretical/Practical Implications of DFT Eigenvectors

Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
ABB's user avatar
  • 4,030
2 votes
0 answers
143 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
  • 1,043
1 vote
0 answers
96 views

Poisson summation for solutions of the Burgers equation in the form 1/x

Long story short: I'm looking for a good way of showing that the Fourier transform of $1/x$ is a sign function. Motivation and why this has been a problem: I'm dealing with an equation similar to the ...
Rafael's user avatar
  • 93
2 votes
0 answers
69 views

Fourier Transform Of Fractional Laplacian [closed]

Fourier Transform Of Fractional Laplacian I dont know why we have the last 2 inequality and why it occurs the characteristics ball B1
Phan Trung Hiếu's user avatar
-4 votes
1 answer
106 views

An integral similar to the Delta function [closed]

I have an integral on the form $\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau$ that I would like to simplify (or basically solve). This indeed comes from a problem ...
owp's user avatar
  • 3
-3 votes
1 answer
81 views

What is the asymptotic behavior of the Levy distribution $P (x)$ when the independent variable $x$ approaches $0$ [closed]

What is the asymptotic behavior of the Levy distribution $$P(x)=\frac{1}{\pi}\int_{0}^\infty \exp(-\gamma q^\alpha)\cos qx\,dq$$ when the independent variable $x$ approaches $0$?
吴月红's user avatar
2 votes
3 answers
383 views

Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc

I am considering a PhD research topic. I only have a math Bachelor's degree with working experience mostly in teaching and I have been working on a paper. I have deep interest in Fourier Series and ...
2 votes
0 answers
310 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
0 votes
1 answer
188 views

A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$

Is it possible to express $$\exp\left[\mathrm{e}^{x^2/2}p\left(\frac{d}{dx}\right)\mathrm{e}^{-x^2/2}\right] f(x)$$ as an integral transform or something similar? $p(x)$ is a polynomial. $$\exp\left[\...
Mirar's user avatar
  • 350
1 vote
0 answers
66 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
0 votes
1 answer
131 views

A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$

Is there a probability distribution $\mu$ (with reasonably nice density $f$ on $\mathbb{R}$) such that the Fourier transform (aka. characteristic function) $\psi_\mu(t) = \int_{\mathbb{R}} e^{itx} \, ...
Tardis's user avatar
  • 1,253

1
2 3 4 5
10