Questions tagged [fourier-transform]

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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
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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
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2 votes
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51 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
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3 votes
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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
158 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
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4 votes
1 answer
335 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
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0 answers
93 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
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-1 votes
1 answer
117 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
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247 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}\...
Zachary's user avatar
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5 votes
0 answers
138 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
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1 vote
0 answers
55 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
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2 votes
1 answer
113 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
147 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
98 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
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0 votes
1 answer
123 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
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1 answer
55 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
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0 answers
18 views

How to calculate the power transformation of a spectral density function

There is a problem I have been trying to solve for a while. Let $X_t$ be a stationary (univariate) time series. The spectral density of the moving average process $$X_t=\sum^{\infty}_{j=-\infty}a_je_{...
Saïd Maanan's user avatar
1 vote
1 answer
75 views

Eigenvalues of a circulant: DFT or Inverse DFT Convention?

Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ...
AChem's user avatar
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1 vote
1 answer
106 views

Uniqueness of Fourier–Stieltjes transform for finite complex valued measures

Let $\mu$ be a finite complex valued measure on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform $$ \hat{\mu}(\omega)= \int_{\mathbb{R}} e^{it\omega} d \mu(t) $$ Question: Does $\...
Boby's user avatar
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1 answer
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When some Fourier coefficients are fixed, can we control the extremals of the function?

Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f :\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold? 1- $-n\lneqq f_{\min}$ (where $f_{\...
ABB's user avatar
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0 answers
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Fourier transform of an exponential function with radical argument divided by a radical

I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
Ft insat's user avatar
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0 answers
34 views

Are standing waves equivalent to travelling waves as a modelling tool to solve wave equation?

I am learning Fourier Analysis from Elias Stein's excellent textbook. He starts off by explaining difference between standing waves and travelling waves and then demonstrating how you can either to ...
nsimplex's user avatar
  • 101
3 votes
0 answers
157 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
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2 votes
1 answer
89 views

Is the Fourier multiplier $\mathcal F(G(-\hbar^2 \Delta)\psi)(p) = G(|p|^2)\hat \psi(p)$ justified for any real function $G$?

I am confused about a claim asserted in the paper "Higher Order Schrodinger Equations" published in IOP Science. The authors claim that a Fourier multiplier identity $$ \mathcal F(G(-\hbar^2 ...
Talmsmen's user avatar
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1 answer
177 views

The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$

Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
ABB's user avatar
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0 answers
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Question on possibility of uniquely defining the FRFT via certain properties

I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ...
Kanghun Kim's user avatar
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0 answers
104 views

Decay of the Fourier transform

I have read this post (and similar ones): Decay of the Fourier transform of a non-differentiable function and I have the following doubt: If $f\in C^{n}(\mathbb{R})$ $f^{(n+1)}$ is piecewise and of ...
Benigno's user avatar
2 votes
1 answer
208 views

Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space

Problem Statement Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
Jacob Helwig's user avatar
1 vote
0 answers
34 views

Does the constrained Wasserstein barycenter admit a blue noise property?

Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
0xbadf00d's user avatar
3 votes
0 answers
99 views

A new arranging of discrete sine transform

Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix $$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$ Let us denote by $s_{-,l}$ the $l^{\text{...
ABB's user avatar
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0 answers
103 views

Fourier transform and rotations in 3d

Let $f\in \mathcal{S}(\mathbb{R}^3)$ be a Schwartz function invariant under rotations in $\mathbb{R}^3$ and let $\hat f\in \mathcal{S}(\mathbb{R}^3)$ be its Fourier transform, i.e. $\hat f(p)=\int_{\...
user72829's user avatar
  • 508
9 votes
2 answers
587 views

How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?

In Bennett, Carbery and Wright's paper A non-linear generalisation of the Loomis–Whitney inequality and applications, Claim 5 was used to generalise the case from characteristic functions to simple ...
kathy4k's user avatar
  • 253
3 votes
0 answers
323 views

Is this FFT algorithm known?

Recently I've been thinking about alternatives to the usual Cooley-Tukey FFT for multiplying polynomials. I think I've come up with a pretty nifty algorithm for multiplying polynomials. So my question ...
Görre Mörre's user avatar
5 votes
1 answer
470 views

Fastest decay of Fourier transform of function of (one-sided or two-sided) exponential decay

Let $f:\mathbb{R}\to \mathbb{R}$ be a function in $L^2$ satisfying $|f(x)|\ll e^{-a_1 x}$, $a_1>0$, for $x\to \infty$. (Variant: assume as well that $|f(x)|\ll e^{a_2 x}$, $a_2>0$, for $x\to -\...
H A Helfgott's user avatar
  • 18.7k
9 votes
2 answers
434 views

Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?

(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem) Let $f:\mathbb{R}\to [0,\infty)$ be such that (a) $\int_{\mathbb{R}} f(x) dx = 1$, (b) $\...
H A Helfgott's user avatar
  • 18.7k
6 votes
1 answer
615 views

Fourier optimization problem related to the Prime Number Theorem

Let $\kappa>0$ be given. What is the function $f:\mathbb{R}\to [0,\infty)$ with $\int_\mathbb{R} f(x) dx = 1$ such that $$\int_\mathbb{R} |x| f(x) dx + \kappa \int_{|t|\geq T}\left| \frac{\widehat{...
H A Helfgott's user avatar
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6 votes
2 answers
208 views

Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?

Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
J. Swail's user avatar
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3 votes
0 answers
152 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
4 votes
1 answer
284 views

When does the sine transform result in a positive function?

For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$ In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...
Bobby Ocean's user avatar
2 votes
0 answers
74 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
  • 18.7k
5 votes
2 answers
229 views

An optimization problem: $\Phi(0)$, $\widehat{\Phi}(0)$, $\Phi$ a majorant

(This is a problem that arose from my own answer to Mean value theorem for Dirichlet series - optimize? ) Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such ...
H A Helfgott's user avatar
  • 18.7k
2 votes
0 answers
99 views

Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform

Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
Mirar's user avatar
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1 vote
0 answers
75 views

Kernel representation of a power of (pseudo-)differential operator

Let $\mathcal{T}$ be a (pseudo-)differential operator that admits the following kernel representation: \begin{equation} \mathcal{T}f(x) = \int_{-\infty}^{\infty} K(x,t)f(t)dt. \end{equation} What can ...
Mirar's user avatar
  • 308
4 votes
1 answer
392 views

The decay of Fourier coefficients and the continuity of functions

Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not ...
Luis Yanka Annalisc's user avatar
6 votes
1 answer
354 views

Harmonic analysis for a beginner

I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
CfourPiO's user avatar
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3 votes
0 answers
193 views

A generalization of Weierstrass transform

As stated in this article, the Weierstrass transform of $f(x)$ is defined as: \begin{equation} W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy \end{equation} which can be ...
Mirar's user avatar
  • 308
0 votes
1 answer
159 views

Find an integral kernel for the solution of a partial differential equation: an initial value problem

Consider the following partial differential equation with an initial condition $u(x,0)=f(x)$: \begin{equation} \frac{\partial}{\partial t} u(x,t)=g_{1}(x)\frac{\partial u}{\partial x}+g_{2}(x)\frac{\...
Mirar's user avatar
  • 308
1 vote
1 answer
219 views

Why we have $f=0$

Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$. Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
zoran  Vicovic's user avatar
4 votes
2 answers
436 views

A proof of Bernstein's inequality

I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below: "The function $\frac{\xi^\beta}{|\xi|...
Jiawen Zhang's user avatar
0 votes
0 answers
75 views

To show a function is zero, assuming some integral conditions on its Fourier transform

Let $f\in L^1(\mathbb{R})$ such that $$\int_0^\infty e^{-yt}e^{ixt}\widehat{f}(t)dt=0,……(i)$$ $$\int_{-\infty}^0 e^{yt}e^{ixt}\widehat{f}(t)dt=0, ……(ii)$$ for all $x\in \mathbb{R}, y>0.$ Questions: ...
user483450's user avatar

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