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 $\varepsilon>0$, ...
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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
$$H(y,t):=\left(\frac{t^2}{(t^2+|y|^2)^{1+d/2}}\right)^{...
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Inequality for a weighted bilinear form in Fourier variables
Let $\phi:\Bbb R^d\setminus\{0\}\to [0,\infty)$ be a continuous and symmetric, i.e., $\phi(-\xi)=\phi(\xi)$. Let $F:\Bbb R\to[0,\infty)$ be increasing and $L-$Lipschitz with $F(0)=0$.
Consider the ...
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Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$
Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial ,
so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that
$g(|x|^\gamma)$ is positive ...
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When does the Fourier transform of a measure decay?
Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...
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Inequality for Fourier transform of a power exponential function
Let
$$
f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }},
x \in \mathbb{R}, 0<\alpha<2,
$$
where
$\phi_1(\alpha)=\frac{\alpha}{2}
\left\{{\{\Gamma(3/\alpha)\}^{1/...
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Is this formula for 2D Fourier integral of diffraction kernel correct?
Well I have a function parametrized by $z$
$$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$
where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
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Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
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Question on estimate in one of Jean Bourgain's 1992 papers
The paper in question is A Remark on Schrodinger Operators.
The goal of the argument is to estimate the following integral:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
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(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^...
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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 ...
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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}...
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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 ...
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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 $...
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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)...
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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 $\...
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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 $$X_t=\sum^{\infty}_{j=-\infty}a_je_{...
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Eigenvalues of a circulant: DFT or Inverse DFT Convention?
Currently, most engineering texts (and webpages including Wikipedia) define forward discrete Fourier transform with a negative sign on the exponential. This is a convention and the inverse discrete ...
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Uniqueness of Fourier–Stieltjes transform for finite complex valued measures
Let $\mu$ be a finite complex valued measure on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform
$$
\hat{\mu}(\omega)= \int_{\mathbb{R}} e^{it\omega} d \mu(t)
$$
Question: Does $\...
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When some Fourier coefficients are fixed, can we control the extremals of the function?
Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f
:\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?
1- $-n\lneqq f_{\min}$ (where $f_{\...
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Fourier transform of an exponential function with radical argument divided by a radical
I have $f(t)=\dfrac{e^{-i\sqrt{(t-t_0)^2+A^2}}}{\sqrt{(t-t_0)^2+A^2}}$ where $t_0$ and $A$ are constant. I need to take the Fourier transform of $f(t)$. I made few substitutions to take it to a form ...
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Are standing waves equivalent to travelling waves as a modelling tool to solve wave equation?
I am learning Fourier Analysis from Elias Stein's excellent textbook. He starts off by explaining difference between standing waves and travelling waves and then demonstrating how you can either to ...
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The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
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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
$$
\mathcal F(G(-\hbar^2 ...
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The proof of the invertibility of $\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$
Suppose that $n$ is even. Any suggestion/appraoch to prove that $S=\Big( \sin\frac{8kl\pi}{2n+1} \Big)_{k,l=1}^\frac{n}{2}$ is invertible?
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Question on possibility of uniquely defining the FRFT via certain properties
I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ...
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Decay of the Fourier transform
I have read this post (and similar ones):
Decay of the Fourier transform of a non-differentiable function
and I have the following doubt: If
$f\in C^{n}(\mathbb{R})$
$f^{(n+1)}$ is piecewise and of ...
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Proof of covariant convolution for a kernel function that is rotation symmetric in Fourier space
Problem Statement
Let $g:\mathbb R^{d}\to \mathbb R,d\in\{2,3\}$ be an integrable function (assumption I1). Suppose $\mathcal T$ is a rotation, and $f:\mathbb R^d\to\mathbb C$ (assumption C) is an ...
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Does the constrained Wasserstein barycenter admit a blue noise property?
Let $(E,d)$ be a metric space and $\nu$ be a probability measure on $\mathcal B(E)$. In this paper, it is mentioned that sampling from $\mu$ can be described as choosing $n\in\mathbb N$, $x_1,\ldots,...
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A new arranging of discrete sine transform
Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix
$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$
Let us denote by $s_{-,l}$ the $l^{\text{...
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Fourier transform and rotations in 3d
Let $f\in \mathcal{S}(\mathbb{R}^3)$ be a Schwartz function invariant under rotations in $\mathbb{R}^3$ and let $\hat f\in \mathcal{S}(\mathbb{R}^3)$ be its Fourier transform, i.e. $\hat f(p)=\int_{\...
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How was Claim 5 in "A non-linear generalisation of the Loomis–Whitney inequality and applications" thought up?
In Bennett, Carbery and Wright's paper A non-linear generalisation of the Loomis–Whitney inequality and applications, Claim 5 was used to generalise the case from characteristic functions to simple ...
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Is this FFT algorithm known?
Recently I've been thinking about alternatives to the usual Cooley-Tukey FFT for multiplying polynomials. I think I've come up with a pretty nifty algorithm for multiplying polynomials. So my question ...
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Fastest decay of Fourier transform of function of (one-sided or two-sided) exponential decay
Let $f:\mathbb{R}\to \mathbb{R}$ be a function in $L^2$ satisfying $|f(x)|\ll e^{-a_1 x}$, $a_1>0$, for $x\to \infty$. (Variant: assume as well that $|f(x)|\ll e^{a_2 x}$, $a_2>0$, for $x\to -\...
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Distribution $f$ such that (a) $\widehat{f}$ has compact support, (b) $\mathbb{E}(|X|)$ is minimal?
(What follows is motivated by an answer to Fourier optimization problem related to the Prime Number Theorem)
Let $f:\mathbb{R}\to [0,\infty)$ be such that
(a) $\int_{\mathbb{R}} f(x) dx = 1$,
(b) $\...
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Fourier optimization problem related to the Prime Number Theorem
Let $\kappa>0$ be given. What is the function
$f:\mathbb{R}\to [0,\infty)$ with $\int_\mathbb{R} f(x) dx = 1$ such that
$$\int_\mathbb{R} |x| f(x) dx + \kappa \int_{|t|\geq T}\left| \frac{\widehat{...
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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 ...
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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 $k$,
$$
K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
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When does the sine transform result in a positive function?
For each $x>0$ is, $$\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 ...
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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\...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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{\...
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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)|^...
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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|...
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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:
...
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