Questions tagged [fourier-transform]
For questions about the Fourier transform
546 questions
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Fourier transform of compactly supported functions
Let $F:\mathbb C^n\longrightarrow \mathbb C$, be an entire function of exponential type, that is $F$ is holomorphic on $\mathbb C^n$ and there exist constants $C, A$ such that
$$
\vert F(z)\vert\le C ...
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$L^2$-functions orthogonal to their own Fourier transform
It is well-known that, besides the standard Gaussian $e^{-|x|^2/2}$, there are many interesting functions which are eigenfunctions of the Fourier transform, for example the Hermite functions.
Mainly ...
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Weak L2 norm in proof of Carleson's theorem
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. Under Lemma 2.18 on page 10 it says: "...
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Major arc approximations - partial summation and Gallagher's lemma
The motivation for my question is that I think I need to use Gallagher's lemma for exponential sums (``A large sieve density estimate near $\sigma =1$", Lemma 1, https://link.springer.com/article/...
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Calculating probabilities for the naive Fisher-Yates shuffling algorithm
The Fisher-Yates shuffle is the standard implementation for randomly permuting a finite list of $n$ elements. The algorithm has several incorrect implementations, one being that in each step permuting ...
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About bilinear arguments of the Fourier restriction
A bilinear argument of the Fourier restriction (not bilinear Fourier restriction) can be described follows:
This excerpt is from the book of C. Demeter, Fourier restriction, decoupling, applications. ...
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Proving that the ratio of the Z-transform of two anti-causal filters is also anti-causal
Let $f$ be some discrete series where $f[m]=0\space \forall m \geq 0$, and $S_{XX}^- (z)$, be the anti-causal part of the spectrum of a regular process (a process whose spectrum can be written as $\...
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Discrete inequality about sum of fractional parts function
I found a hard problem on an old olympiads handouts (without solution).
For an integer $n\geqslant 2$, and two positive integers $r_1$ and $r_2$ coprime to $n$, the following inequality holds: $$ \...
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Isomorphically narrow the domain of definition in the process of the inverse Fourier transform?
Let us have a function $f(x)$ defined on a finite interval $[a,b]$. Outside the interval it is identically zero. I want to obtain the Fourier transform of this function. Limits of integration $a,b$. ...
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Functions with compactly supported Fourier transform
The following is a repeat from the following question on MathStack Exchange. I am just hoping to have more success here.
This is a follow-up from that question. The question is this: I want to ...
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Laplace transform of a distribution determines its order
Suppose that $T\in\mathcal{D}'(\mathbb{R}^n)$ is a compactly supported distribution such that its Laplace transform (defined as $\widehat{T}(z) := \langle T(x), e^{i z x} \rangle $ for $z\in \mathbb{C}...
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What is known about decomposing time series into sums of excitated and damped oscillations
the subject of the question may have aspects of the discrete Fourier transform and the discrete Laplace transform, but I couldn't find any dedicated information on the subject.
Question:
what is ...
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Does replacing "finite relative extrema" with "finite strict extrema" in Dirichlet's convergence theorem maintain Fourier series convergence?
In Stein and Shakarchi's book: Fourier Analysis: An Introduction, page 128, the author said that ``Dirichlet's theorem states that the Fourier series of a real continuous periodic function $f$ which ...
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Is a continuous function with finitely many strict extrema on a closed interval necessarily a bounded variation function? convergence of Fourier
We know that if $f$ is a continuous function on $[a,b]$ and has finite number of relative maxima and minima on $[a,b]$, then $f\in \mathrm{BV}([a,b])$. In fact, under this condition, we can prove that ...
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Spectral representation for the Laplace operator
$\DeclareMathOperator\H{\mathcal{H}}$The spectral representation theorem for unbounded self-adjoint operators state that if $\H$ is a separable Hilbert space, $T:D(T) \to \H$ is a self-adjoint ...
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Fourier transform on basic affine space as a normalized intertwining operator
There is a Fourier transform $F_{\omega}$ acting on the basic affine space defined in the paper by Braverman and Kazhdan On the Schwartz space of the basic affine space as a generalization of the ...
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Reducing boundary artifacts in discrete‐Fourier (integer or fractional) derivatives
I am interested in calculating integer and fractional derivatives of a experimental data using discrete Fourier transform. There is a paper Calculating numerical derivatives using Fourier transform: ...
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Absolute integrability and Fourier transform
In engineering we are mainly interested in linear time-invariant (LTI) systems which are bounded input, bounded output (BIBO). It's easy to prove that BIBO condition is equivalent to $$\int_{-\infty}^{...
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Example of Fourier transform of $L^1$ functions with certain regularity and growth condition
It is well-known that, vaguely speaking, for nice functions, the momentum operator corresponds to differentiation via the Fourier transform. However, I need some help in resolving the following ...
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Are there reasons to study the Fourier transform of a square root of a probability?
In probability theory, if a random variable has a probability density function, then modulo conventions one can equivalently work with the Fourier transform of that probability distribution (known as ...
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Decay of the discrete Fourier transform
Let $G$ be a finite abelian group of order $n$. Let $\hat{G}$ be the dual group of $G$, i.e., the group of characters on $G$. For $f:G\to \mathbb{C}$, we define its Fourier transform $\hat{f}:\hat{G}\...
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Fourier transform of functions which is compactly supported
Can one give me an explicit proof that if the Fourier transform of $f$ is compactly supported in a region, then $f$ is essentially constant on the dual region,i.e., $f \sim 1 $ on the dual region,i.e. ...
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Decomposition of a gauge-invariant 3-dimensional spin network state over the Wigner matrix elements
Below follows the passage of Rovelli and Vidotto's Covariant Loop Quantum Gravity that I do not understand. To give the context, let me clarify that a state $\psi$ is a function in $L^2[\text{SU}(2)^L]...
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Can Parseval's theorem be extended to the "nested" Fourier series representation of $f(x)$?
This is a cross-post of this question I posted on Math StackExchange a couple of months ago that has not yet received any answers or even comments (other than a single comment of my own).
Assuming ...
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Fourier transform of creation and annihilation operators of CAR algebra over $l^2(Z)$
Set-up
I want to consider the rigorous $C^*$-algebra statistical mechanics formulation (à la
Bratteli, Robinson) of fermions over a chain $\mathbb{Z}$, which is described by a CAR (canonical ...
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Paley-Littlewoord/paraproduct theory: Frequency of multiplication with respect to the japanese bracket
I'd like to prove (or disprove) the following bound for high frequencies $j$ of the dyadic decomposition of $\langle z \rangle^\alpha f$, for some $\alpha > 0$:
$$ | \triangle_j (\langle z \rangle^\...
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On optimum constant for upper density of zeros of an entire function
Let $f\in L^2(\mathbb R)$ be a not identically vanishing function that is compactly supported in the interval $(-\sigma,\sigma)$ for some fixed $\sigma>0$. For each $z\in \mathbb C$ let us define ...
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Operator powers
The Fourier transform $F$ is an operator on a suitable space of functions. The statement that $F^4$ is the identity operator is essentially the content of the Fourieer inversion Formula. This shows ...
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Discrete Fourier Transform using not the n-th roots of unity, but the 2n-th primitive roots of unity
I should start with stating that my question stems from the proof of lemma 6 in the following paper: Jung Hee Cheon, Hyeongmin Choe, Julien Devevey, Tim Güneysu, Dongyeon Hong,
Markus Krausz, Georg ...
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Prove or disprove: the Fourier transform of bounded $W^{1,1}$ functions is $L^1$
I am wondering whether the following statement is true. I don't think so, but I couldn't find a counterexample as well.
Let $f\in W^{1,1}(\mathbb{R})$, i.e. $\|f\|_{L^1},\|f'\|_{L^1}<\infty$. Then ...
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Perron-style formulae with compact support on a vertical line: references?
The following is a reference request.
Short version (for specialists):
when was it first clearly stated that you can have something like Perron's formula with no error term (thanks to smoothing) with ...
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Is the space $\mathcal{E}$ is suitable as testing function space for FRFT?
I know, that the Schwartz space is suitable as testing function space for Fractional Fourier transform (FRFT) because of it's properties like rapid decay & closure under Fourier transform.
My ...
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Does $\displaystyle \int_{-\infty}^\infty \frac{\zeta\left(\frac12+ix\right)}{\left(\frac12+ix\right)}\,\mathrm{d}x =-\pi$?
In his beautiful paper "An electromechanical investigation of the Riemann zeta function in the critical strip", Bull. Am. Math. Soc. 53, 976-981 (1947), MR22712, Zbl 0032.21602, Balthasar ...
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Uncertainty principle: minimize $\int_{-\infty}^\infty |t| |\widehat{f}(t)|^2 dt$ for $f$ of compact support
This is a question of uncertainty-principle type stemming from Eigenvalue of a convolution and a restriction?
Let $f:\mathbb{R}\to \mathbb{R}$ be even, absolutely continuous and supported in $[-\frac{...
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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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If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality
\begin{equation*}
\int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
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Function that is (essentially) a self-convolution but not a multiple of a self-convolution
Call a function $F:\mathbb{R}\to C$ nice if it is of the form $F = f\ast \tilde{f}$, where $\tilde{f}(x) = \overline{f(-x)}$. (Of course nice functions are precisely those whose Fourier transform is ...
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Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
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Three optimization problems of uncertainty principle/Paley-Wiener type
Let $\phi:\mathbb{R}\to\mathbb{R}$ be an even function with support on $[-1,1]$. Assume that it is in $L^1\cap L^2$ and that its Fourier transform is also in $L^1\cap L^2$. Assume as well that $|\phi|...
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Possible gaps for a function and its Fourier transform
This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.
A nice remark by Christian ...
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Sufficient conditions for boundedness of Fourier transform
This should be a well studied topic: I am looking for sufficient conditions on a function $u(x)$ on $\mathbb{R}$ ensuring that its Fourier transform is bounded. Of course one such condition is $u\in L^...
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Minimizing vertical integral of a Mellin transform
Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say).
Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small.
What is the choice of $...
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Fourier-like transforms for a Day convolution?
The presheaf category on a monoidal category inherits the monoidal structure via the Day convolution. Moreover you can inherit (bi)closed monoidal structure.
In the study of Fourier analysis we can ...
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Asymptotic expansion inverse discrete Fourier transform
Let $\ell^1(\mathbb{Z})$ be the space of biinfinite sequences $f = (f(n))_{n \in \mathbb{Z}} \subset \mathbb{C}$ such that it is absolutely summable. The discrete Fourier transform or Fourier series ...
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Show that the kernel $|x -y|^{-1}$ on $\mathbb{R}^3 \times \mathbb{R}^3$ is Hilbert-Schmidt with respect to a weighted $L^2$ space
Let $\langle x \rangle := \left(1 + |x|^2\right)^{1/2}$, $x \in \mathbb{R}^3$. For $s > 1$, consider the weighted convolution operator
\begin{equation*}
T_s \varphi = \langle x \rangle^{-s} \int_{\...
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Non-vanishing of a "push-forward" Fourier–Harish-Chandra transform on a compact set
Let $G \subset \operatorname{GL}_d(\mathbb{R})$ be a non-compact semi-simple Lie group and $K \subset G$ a maximal compact subgroup. Let $\mathfrak{g}$ (resp. $\mathfrak{k}$) be the Lie algebra of $G$ ...
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Help on integral regarding analytical Fourier transform
To explain my problem I start with two functions to be sine transformed. This question is a problem of current research in the field of electrolyte transport theory. The first Function is given by:
$$...
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Integral convolution equation $\int_{B_n(R) } e^{- \| x - t\|} d\nu(t) = e^{- \|x \|^2/2}$ on $x \in B_n(R)$. Find measure $\nu$
Let $B_n(R)$ denote the $n$ ball centered at zero with radius $R$. We are interested in the following integral equation: given $R>0$ and $\lambda>0$, let
\begin{align}
\int_{B_n(R)} e^{- \...
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Inversion formula for discrete sine and cosine transforms
$\newcommand{\wh}[1]{{\widehat{#1}}}
\newcommand{\R}{{\mathbb{R}}}
$I am looking for a proof of the inversion formulas for the discrete sine and cosine transforms, i.e. a proof of the fact that these ...
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How can discrete Fourier transform approximation prove the completeness of complex exponentials in $L^2(T)$?
I have a question about the completeness of complex exponentials in function spaces.
For the discrete set $ S = \{1, 2, \ldots, n\} $, it is clear and intuitive that $ e^{2\pi ikx/n} $ for $ k = 0, 1, ...
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