Questions tagged [fourier-transform]
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208 questions with no upvoted or accepted answers
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Laplace Transform in the context of Gelfand/Pontryagin
Questions:
Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
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707
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What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?
The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions:
$\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...
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"Small" zero divisors in $\mathbb C[\mathbb Z/p\mathbb Z]$
If $p$ is a prime, and $a,b$ are non-zero elements of the group algebra $\mathbb C[\mathbb Z/p\mathbb Z]$ satisfying $a\ast b=0$, then $$|{\rm supp}\ a|+|{\rm supp}\ b|\ge p+2.$$ This is easy to prove ...
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Fourier Transforms of Convolutions
Straightforward computations lead to the following standard property of Fourier transformation: it transforms convolutions into products, i.e. for functions $f$ and $g$ Schwartz class we have
$$\...
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Determining convexity of a polygon from its Fourier coefficients
Consider an $n$-sided polygonal curve in the plane, represented by an ordered set of points $(x_0, x_1, \ldots, x_{n-1})$; line segments connect consecutive points and also $x_{n-1}$ to $x_0$. It is ...
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512
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Montgomery's conjecture and lower bound on certain Fourier transform.
Recently I have come across the following question, while meditating about Matt Young's answer to this question of mine, explaining the heuristic (or at least, one possible heuristic) behind ...
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Certain Fourier transforms involving Whittaker function and Bessel functions
I recently meet the following two weird "Fourier transform" questions.
(I), Suppose that $F$ is a $p$-adic field (the same question can be asked over any local field, including $\mathbb{R}$ ...
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Are these identities Newton series?
Newton series is the following expansion of a function:
$$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$
Now ...
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6
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Fourier-like transforms for a Day convolution?
The presheaf category on a monoidal category inherits the monoidal structure via the Day convolution. Moreover you can inherit (bi)closed monoidal structure.
In the study of Fourier analysis we can ...
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(Finer) analogue between Fourier transform and (Fourier-)Mukai transform
Mukai transform gives a derived equivalence between the (bounded) derived category of coherent sheaves $D^b_{\mathrm{coh}}(A)$ of abelian variety $A$ and that of dual $A^\vee$, $D_{\mathrm{coh}}^{b}(A^...
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Fourier transformation of a distribution
We have no idea how to tackle the following Fourier transformation of a distribution:
$$
\lim_{\epsilon\to0^+}\int_{-\infty}^{\infty}\mathrm{d} t\int_{\mathbb{R}^{d-1}} \mathrm{d}^{d-1}\vec{r} e^{-\...
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Uniform estimates of Fourier transform of tempered functions with parameters
Consider the following function in $\mathbb{R}^3$:
$$
f_t(x)=(1+|x|^2)^{-\alpha}e^{-g(x)t},\,\,\,\,\, \text{where}\,\, g(x)=\frac{x^2_1\cdot x^2_2}{1+|x|^2},
$$
where $\frac{1}{2}<\alpha<1$, and ...
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Real interpolation space between the Wiener algebra and $L^2$
The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for $\...
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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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Similarity in Navier-Stokes equation and convolution in finite abelian groups?
Let $G$ be a finite abelian group, $X = (x_g)_{g \in G}$ be a vector of variables.
Set for $g \in G$:
$$\tau_g(X) := \frac{1}{|G|} \sum_{\rho \text{ irred. }} \chi_{\rho}(-g) \exp(\sum_{s \in G} \chi_{...
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Fourier dimension of radial set
In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...
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Properties of microlocalization
Let $i: M\hookrightarrow X$ be the inclusion of a closed submanifold in a smooth manifold $X$. I denote by $T_MX$ the normal bundle to $M$ in $X$, by $T^{\ast}_MX$ its dual bundle, and by $D^b(X)$ the ...
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Is there a practical application of natural integral or differintegral?
The following formulas give natural differintegral (that is one with naturally fixed integration constant):
$$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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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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Any references on infinite-dimensional Fourier-Plancherel theory?
Let $M$ be a measure on an infinite-dimensional topological vector space (in fact, only the measure type matters), such that $M$ is quasi-invariant under a dense subspace $S$ of shifts (let's assume ...
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Is the Fourier transform of $\frac{1}{\mu+|\xi|^{2\alpha}}$($\mu>0$) a bounded function?
Consider $m(\xi)=\frac{1}{\mu+|\xi|^{2\alpha}}$, where $\xi\in\mathbb{R}^n$, $\mu, \alpha>0$, I want to know that if $m(\xi)$ is a multiplier of $\mathcal{M_{1}^{\infty}}$,i.e., whether the ...
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Inverse Fourier Transform involving a Bessel Function, Exponential, and Power
I'm interested in this integral as a function of $r$ for various spectral densities $S(s)$:
$\frac{2 \pi}{r^{p/2}-1} \int_{0}^{\infty} S(s) J_{p/2-1}(2 \pi r s) s^{p/2} ds $, where $J_{p/2-1}$ is a ...
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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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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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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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How wild is the maximal ideal space of the Fourier-Stieltjes algebra of the real line?
The Fourier-Stieltjes algebra of $\mathbb R$ is the set of all sufficiently nice measures on $\mathbb R$. The vector product is convolution of measures. By identifying each measure with its Fourier ...
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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 ...
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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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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:
$$K_1(x,y)=\int_{\mathbb{R}^2} e^{i(x-y)\cdot\xi+i(t(x)-t(y))|\xi|^2}\...
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The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
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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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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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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 $k$,
$$
K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
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A generalization of Weierstrass transform
As stated in this article, the Weierstrass transform of $f(x)$ is defined as:
\begin{equation}
W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy
\end{equation}
which can be ...
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Two equivalent definitions of semisimplicity of group representations, proof by Zorn's lemma, a “counterexample” from the Fourier transform theory
Consider a representation $A$ of a group $G$ in a complex vector space ${\mathbb{V}}$:
$$
A:~~G~\longrightarrow~\operatorname{GL}({\mathbb{V}})~~,
$$
and let ${\mathbb{V}}$ be decomposable into a ...
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