Questions tagged [fourier-transform]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
2 answers
148 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
  • 311
2 votes
0 answers
77 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
83 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
159 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
  • 741
-2 votes
1 answer
210 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 ...
AChem's user avatar
  • 803
1 vote
0 answers
40 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
366 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
120 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
  • 289
1 vote
0 answers
58 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
  • 3,898
1 vote
1 answer
112 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 ...
AChem's user avatar
  • 803
33 votes
8 answers
3k 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
114 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
  • 15.1k
2 votes
1 answer
137 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
94 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
  • 15.1k
0 votes
0 answers
160 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
  • 1,061
1 vote
0 answers
64 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
153 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
86 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
42 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
166 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
  • 3,898
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
  • 1,033
1 vote
0 answers
86 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
65 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
103 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
77 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
332 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
274 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
161 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
58 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
127 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
  • 1,077
0 votes
0 answers
36 views

Conditions for an ODE with convolution term to have a probability distribution solution

Suppose we have a simple ODE like: $$ y''(x) + 2ay'(x) + by(x) = 0 $$ with the condition $y(0)=0$ for $x\leq 0$. Then the solution on $(0,\infty)$ will be of the form $Axe^{-ax}$ when $a^2=b$, $Ae^{-...
user1598's user avatar
  • 177
1 vote
1 answer
80 views

Characteristic exponent after Girsanov transformation

Let $B$ be a standard Brownian motion. Its characteristic exponent (or Fourier transform) is easily calculated to be $$ \mathbb E [e^{ixB_t}] = e^{-\frac 12 x^2 t}. $$ Now I want to apply a Girsanov ...
Benjamin's user avatar
  • 235
2 votes
1 answer
166 views

Is this integral solvable analytically?

I have this integral that comes from my research with some Fourier Transforms of spectrum functions: $$ G(\tau) = \int_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c_1 - c_2 e^{-c_3 x} ) } dx $$ where $...
CfourPiO's user avatar
  • 159
2 votes
1 answer
217 views

Fourier transforms of homogeneous functions [closed]

Compute Fourier transforms of homogeneous functions of the form, $$ \frac{1}{|x|^{n+d}}P_d(x) $$ where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.
user124297's user avatar
1 vote
0 answers
52 views

Comparison principle for porous medium equation in Fourier variables

Let $V:[0,\infty) \to[0,\infty)$ be convex, $C^2$ with $V(0)=0$. Define $F(u): = uV'(u)-V(u) $. Let $v\in L^1 (\Bbb R^d)$, $v\geq0$ so that $F\circ v\in L^1 (\Bbb R^d)$. For fixed $\varepsilon>0$, ...
Guy Fsone's user avatar
  • 1,033
1 vote
1 answer
152 views

Closed form of a Fourier transform

I apologize for not being able to motivate the question below; it would go into technicalities. Let $n=d+1\ge2$ be the space-time dimension, and $$H(y,t):=\left(\frac{t^2}{(t^2+|y|^2)^{1+d/2}}\right)^{...
Denis Serre's user avatar
  • 51.5k
2 votes
0 answers
55 views

Inequality for a weighted bilinear form in Fourier variables

Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$. Consider the ...
Guy Fsone's user avatar
  • 1,033
3 votes
0 answers
83 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
5 votes
0 answers
181 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
  • 1,033
4 votes
1 answer
367 views

Inequality for Fourier transform of a power exponential function

Let $$ f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }}, x \in \mathbb{R}, 0<\alpha<2, $$ where $\phi_1(\alpha)=\frac{\alpha}{2} \left\{{\{\Gamma(3/\alpha)\}^{1/...
Tanya Vladi's user avatar
0 votes
0 answers
103 views

Is this formula for 2D Fourier integral of diffraction kernel correct?

Well I have a function parametrized by $z$ $$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$ where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
VojtaK's user avatar
  • 151
-1 votes
1 answer
182 views

Building a smooth function from a rapidly decreasing sequence

Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function? More precisely: Let $\lbrace c_k\rbrace _{k \...
Peg Leg Jonathan's user avatar
3 votes
0 answers
288 views

Question on estimate in one of Jean Bourgain's 1992 papers

The paper in question is A Remark on Schrodinger Operators. The goal of the argument is to estimate the following integral: $$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
Dispersion's user avatar
6 votes
0 answers
165 views

(Finer) analogue between Fourier transform and (Fourier-)Mukai transform

Mukai transform gives a derived equivalence between the (bounded) derived category of coherent sheaves $D^b_{\mathrm{coh}}(A)$ of abelian variety $A$ and that of dual $A^\vee$, $D_{\mathrm{coh}}^{b}(A^...
Seewoo Lee's user avatar
  • 1,911
1 vote
0 answers
73 views

1D representation of 2D discrete Fourier transformation [closed]

I'm not too familiar with image processing, so I need a little help: In general, if we transform a discrete function $f$ with $n$-variables from the "spatial domain" using the Fourier ...
Albert's user avatar
  • 11
2 votes
1 answer
161 views

Existence of the special entire Hardy space function with infinitely many zeros in the strip

Question. Does there exist an entire function $h$ satisfying three following assertions: $h$ belongs to the $H^2$ Hardy space in every horizontal upper half-plane; $zh - 1$ belongs to $H^2(\mathbb{C}...
Pavel Gubkin's user avatar
4 votes
2 answers
204 views

Existence of nonzero entire function with restrictions of growth

Question. Is there an entire function $F$ satisfying first two or all three of the following assertions: $F(z)\neq 0$ for all $z\in \mathbb{C}$; $1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy ...
Pavel Gubkin's user avatar
1 vote
0 answers
107 views

Recovering phase function using Fourier decomposition

I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function $$f = e^{i \phi(x)}. $$ I am interested in the following problem. If I know the function/distribution $...
VojtaK's user avatar
  • 151
2 votes
1 answer
203 views

Fourier series of an arbitrary function of a cosine function

Is there a general expression for the Fourier series of the function $f(a\cos(\omega t))$ in terms of the derivatives of $f$? Obviously, the function can be expressed as a Maclaurin series $f(0)+af'(0)...
Jinyang Li's user avatar
0 votes
1 answer
72 views

Hamiltonian particle system and its frequency domain

I am interested in the following question. So let suppose we have finite number of point particles on plane $\mathbb{R}^2$. We can assume that every $j$ point is represented by Dirac delta function $\...
Dragomir's user avatar

1
2 3 4 5
10