Questions tagged [fourier-transform]
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516 questions
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Is there a practical application of natural integral or differintegral?
The following formulas give natural differintegral (that is one with naturally fixed integration constant):
$$f^{(s)}(x)=\sum_{m=0}^{\infty} \binom {s}m \sum_{k=0}^m\binom mk(-1)^{m-k}f^{(k)}(x)$$
$$f^...
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Are Stochastic Process Characterized by Their conditional Moments
Suppose that $X_t$ is a real-valued stochastic process. Then is $X_t$ characterized by it's conditional moments? In the sence that, if $Y_t$ is another process, such that
$$
\mathbb{E}\left[\int_s^T\...
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Existence of $A\subset\Bbb{R}^n$ of finite measure and $\hat{1_A}\in\bigcap_{q>1}L^q$, but s.t. for some $1<p<\infty$, $1_A$ is no $L^p$-Fourier mult
I am interested in the following somewhat obscure question:
Is there some $n \in \Bbb{N}$, and a set $A \subset \Bbb{R}^n$ of finite measure such that the Fourier transform $\widehat{1_A}$ of its ...
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Extension of function that minimizes function of Fourier transform
Suppose that $f$ is a given (smooth) function defined on $B\subset \mathbb R^n$. (For simplicity, take $B$ to be the unit ball but more generally we can take $B$ to be some other measurable set). How ...
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Can we extend the twisted Poisson Summation formula with functions having a singularity in zero?
The following "twisted" Poisson Summation formula for $\chi$ primitive of conductor $q$ :
$$ \sum_{n\in\mathbb{Z}}\chi(n)f\left(\frac{nx}{\sqrt{q}}\right) =
\frac{A}{x}\sum_{n\in\mathbb{Z}}\bar\chi(n)...
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Discrete versus Continuous Hilbert Transform
Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform $\...
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Szegő curve for partial sum of Taylor series of Riemann $\Xi(z)$ function
I am sorry that this is long post. But it might be of interest to you.
This post is related to zeros of partial sum of Taylor series of $e^x-1$.
Entire functions $e^z$, $\cos(z)$, and $\sin(z)$ can ...
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$f, \hat{f} \in L^{p}\cap L^{\infty} \implies f\in B(\mathbb R)$ (algebra of Fourier- Stieltjes transforms )?
For a bounded complex Borel measure $\mu$ on $\mathbb R$, we define, its Fourier-Stieltjes transform, $\hat{\mu}(y)= \int_{\mathbb R} e^{-2\pi ix\cdot y} d\mu(x); (y\in \mathbb R).$
Let $1\leq p \leq ...
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Computing a 2D Fourier transform on a disk
I am looking for a reliable and fast way of evaluating an integral like
$$
F(r, \phi)= \int_0^1 \int_0^{2\pi} f(\rho, \theta) e^{2\pi i \rho r \cos(\theta - \phi)}\rho\, d\theta\,d\rho,
$$
where $f$ ...
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Is there a compactly supported function that its Fourier transfrom vanishes at given n real points?
My question is as follows: Given ${{\lambda }_{1}},\,{{\lambda }_{2}},...,{{\lambda }_{n}}\in \mathbb{R}$ where $\underset{1\le j\le n-1}{\mathop{\min }}\,\left| {{\lambda }_{j+1}}-{{\lambda }_{j}} \...
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The Hubbard-Stratonovich transformation
is there a way to extend the Hubbard-Stratonovich transformation
$$e^{\frac{1}{2}Ks^2}=\left(\frac{K}{2\pi}\right)^{1/2}\int_{–\infty}^\infty e^{–Kx^2+Ksx}dx$$
to the case $e^{\frac{1}{2}Ks^p}$ for $...
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Space of Bandlimited Functions
I was asking myself about some necessary and/or sufficient conditions for a function to be bandlimited (i.e. its Fourier transform is zero t residing out of $[-B,B]$ for some $B>0$).
Of course, if ...
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Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function
Consider the function $$\phi_{a,b}(t)=\frac{|t|}{e^{a|t|}-e^{-b|t|}}, \ \ t\in\mathbb{R},$$ where $0<a<b$. Can $\phi_{a,b}$ be the Fourier transform of a positive function for some $a<b$?
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Fourier transform of tempered distribution
I'm wondering whether anyone knows a reference or proof for finding the Fourier transform of $f(t):=(t+1)^{1/2}t_+^{1/2}$? (Here $t_+=\max (t,0)$.)
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Is there a measure on the sphere with positive Fourier transform?
Is it possible to have an even probability measure $\mu$ (that is $\mu(A)=\mu(-A)$ for any set $A\subset \mathbb{R}^d$) supported on the unit sphere $S^{d-1}$ such that its Fourier Transform
$$
\...
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Why does Fourier transform of a function give the frequency spectrum?
Coefficients of the complex Fourier series give spikes at discrete frequencies. I'd like to understand why F.T. gives us the continuous frequency spectrum?
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How to unperiodise a function
We know that given a sufficiently regular function $f: \mathbb{R} \to \mathbb{R}$, then its periodisation (say to period $1$) is given by
$$
\begin{align}
F(x) := \sum_{n\in\mathbb{Z}} f(x + n).\tag{$...
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Existence of nonzero entire function with restrictions of growth
Question. Is there an entire function $F$ satisfying first two or all three of the following assertions:
$F(z)\neq 0$ for all $z\in \mathbb{C}$;
$1/F - 1\in H^2(\mathbb{C}_+)$ -- the classical Hardy ...
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A proof of Bernstein's inequality
I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below:
"The function $\frac{\xi^\beta}{|\xi|...
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Fourier transform of a Lorentz invariant generalized function
Consider on $\mathbb{R}^{n+1}$ the indefinite quadratic form defining the Minkowski metric
$$B(p)=(p^0)^2-(p^1)^2-\dots-(p^n)^2.$$
Let $\mu$ be a generalized function on $\mathbb{R}^{n+1}$ which is ...
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Is there an uncertainty principle for Fourier pairs everywhere dominated by $t^{-A}$?
Hardy's uncertainty principle states that a real function $f$ and its Fourier transform $\widehat{f}$ may not both decay faster at infinity than the standard Gaussian $e^{-\pi t^2}$, unless $f = 0$. ...
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The decay of Fourier coefficients and the continuity of functions
Let $ f $ be a function on $ \mathbb{T}=[0,1] $ ($ 1 $-periodic) with bounded variation. Prove that if $ \widehat{f}(k)=\int_0^1f(x)e^{-2\pi ikx}dx=o(1/|k|) $, then $ f\in C(\mathbb{T}) $. I do not ...
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2-Wasserstein metric on convolution of probability distributions
I have two related questions. Let $\mu$ and $\nu$ be two distinct probability measures on $\mathbb{R}^n$ with finite second moments, and $W_2(\cdot,\cdot)$ be the $2$-Wasserstein metric. The question ...
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Hörmander-Mikhlin theorem on the torus
Let me first recall a particular case of the classical Hörmander-Mikhlin multiplier theorem: Let $m$ be a bounded function on $\mathbb {R} ^{n}$ which is smooth except possibly at the origin, and ...
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Characterizations of Wiener algebra
The Wiener algebra $\mathcal W$ is defined as $\text{Fourier}(L^1(\mathbb R))$, i.e. the image by the Fourier transform of $L^1(\mathbb R)$. Riemann-Lebesgue's lemma ensures that
$$
\mathcal W\subset ...
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Eigenvalue of a convolution and a restriction?
Let $\epsilon>0$ be small. Let $\eta(t) = \frac{2\epsilon}{\epsilon^2+(2\pi t)^2}$ (the Fourier transform of $x\mapsto e^{-\epsilon |x|}$). Let $V$ be the space of integrable, bounded functions $f:\...
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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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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.
$$\int_0^\inf (be^{-at^p}-ae^{-bt^p})\cos(tx)dt, 0<a<b, \frac{1}{2}...
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Vanishing of the product of a function and its own Fourier transform
I have found the following question to be surprisingly hard:
Is there a non-zero $f\in L^1(\mathbb R)$ or $f\in L^2(\mathbb R)$ such that
$$
f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere},
$$
...
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Fourier transform on finite groups in characteristic $p>0$
Is there a Fourier theory for finite groups in characteristic $p>0$? Assume that $p$ divides the order $|G|$ of finite groups (or just work with $p$-groups), i.e., in a modular representation-...
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Fourier transform derivation from Laurent series
Using Laurent Series of a function $f(z)$ around a point $a \in \mathbb{C}$
$$f(z) = \sum^{\infty}_{n=-\infty} c_n(z-a)^n \ \ \ \ (1)$$
where
$$c_n = \frac{1} {2\pi i}\int\limits_{\gamma}\frac {f(z)} {...
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If $f$ is non-prime, can we say $|f|$ is also a non-prime; in convolution algebra?
By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that,
$$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < \...
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When does the sine transform result in a positive function?
For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$
In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...
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Fourier coeffients of Cantor measure
For $0<\theta<\frac{1}{2}$, denote by $\mu_\theta$ the uniform Cantor measure with dissection ratio $\theta$. It is not hard to show that the Fourier–Stieltjes transform of $\mu_\theta$ is
$$
\...
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Fourier transform in terms of special function?
I have a Fourier integral
$$\int\limits_{-\infty}^{\infty}\mathrm{d}t\,\frac{1}{t^2}\exp\left({\mathrm{i}\frac{t^3}{3}+\mathrm{i}Yt+\frac{\mathrm{i}\lambda^2}{4t}}\right),$$
where $Y$ and $\lambda$ ...
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Integral solving request
Dear all,
please help me solve the following integral.
I need to solve this integral for one of my problems.
$$(\frac{1}{2\pi})^2\int_0^\infty\int_{-\infty}^\infty \frac{J_0(\rho R_0)J_0(\rho r)L*...
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Inequality for Fourier transform of a power exponential function
Let
$$
f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }},
x \in \mathbb{R}, 0<\alpha<2,
$$
where
$\phi_1(\alpha)=\frac{\alpha}{2}
\left\{{\{\Gamma(3/\alpha)\}^{1/...
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Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?
Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as
$$
H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
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Distribution boundary value of analytic function and wave front sets
Assume $f(z)$ is analytic in the tube domain $\mathbb R^n\oplus iC$, where $C\subset \mathbb R^n$ is a convex cone. Under the assumption $|f(x+iy)|\leq 1/|y|^k$, we know by a Theorem of Martineau (see ...
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Advantage of fractional Fourier transform over multiscale wavelet
What is the best argument of fractional Fourier transform over multiscale wavelet in data analysis purpose.
Optimization of the good time-frequency domain parameter ? "Good" will be, find the %time-%...
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Does the Fourier transform preserve the separation property?
The space of Schwartz functions on the plane is denoted by $\mathcal{S}$.
The usual multiplication and the convolution multiplication on $\mathcal{S}$ are denoted by $m_1$ and $m_2$, respectively.
...
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Solution to the fractional differential equation
What is the solution of the fractional differential equation
$$
f^{(\alpha-1)}(t) = tf(t)
$$
where $(\alpha)$ denotes the fractional derivative of order $\alpha$
EDIT: Background behind this ...
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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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Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
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How related are Fourier transforms on finite groups and Fourier transforms on graphs?
Here are two generalizations of the notion of a Fourier transform. I am also aware of the Pontryagin Duality generalization for locally compact abelian groups, though I am personally more concerned ...
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Given $\theta$, find $f$ such that $\int_{\mathbb{T}} \text{e}^{i\theta} \cos(h \cdot f) = 0,$ for all $h \in \mathbb{N}$
Let $\theta$ be a $C^{\infty}$ (resp. analytic) real-valued function on $\mathbb{T}=[0,2\pi]/\{0,2\pi\}$.
When can one find $f \neq 0$, $C^{\infty}$ (resp. analytic) real-valued function on $\...
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On smoothness of a function and decay of its Fourier transform
I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here.
I am interested in a quantitative version of the principle that smoothness of ...
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Is there a categorical foundation for manifolds of bounded geometry and bandlimited functions?
As an outsider to both, manifolds of bounded geometry and bandlimited functions appear rather connected: for example, bounded geometry is defined in terms of bounds on curvature and its derivatives, ...
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Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions
I asked the question below on Math Stack Exchange, https://math.stackexchange.com/questions/2592555/convergence-of-integral-formula-for-fourier-inversion-and-hilbert-transform-fo, but [despite it ...
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Isbell duality in Joyal and Street's Introduction to Tannaka Duality
In Sec. 3 of Joyal and Street's Introduction to Tannaka Duality and Quantum Groups, the authors give a commutative triangle of isomorphisms of compact topological groups (Corollary 8). This diagram ...
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