Questions tagged [fourier-transform]
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516 questions
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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 $k$,
$$
K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
- 53
3
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0
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272
views
A generalization of Weierstrass transform
As stated in this article, the Weierstrass transform of $f(x)$ is defined as:
\begin{equation}
W[f](x)=\frac{1}{4\pi}\int_{-\infty}^{\infty}f(y)e^{-\frac{(x-y)^{2}}{4}}dy
\end{equation}
which can be ...
- 350
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212
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Two equivalent definitions of semisimplicity of group representations, proof by Zorn's lemma, a “counterexample” from the Fourier transform theory
Consider a representation $A$ of a group $G$ in a complex vector space ${\mathbb{V}}$:
$$
A:~~G~\longrightarrow~\operatorname{GL}({\mathbb{V}})~~,
$$
and let ${\mathbb{V}}$ be decomposable into a ...
- 603
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204
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The inversion formula for the square root of a positive function
Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\...
- 4,058
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answers
320
views
Does convolution by a Schwartz function preserve symbol classes?
I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ...
- 395
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270
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Finding (and saturating) a sharp Babenko-Beckner inequality for finite fields
My question is a follow-up to Abdelmalek Abdesselam's recent post
What makes Gaussian distributions special? Local field version?
asking about various characterizations of (real-valued) Gaussian ...
- 2,137
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309
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The $2\pi$ factor in the Fourier transform and dimensional analysis
I have been thinking about the $2\pi$ factor in the various conventions of the Fourier transform. For example, I was looking for a way to justify the following:
$(*)$ If we define $\hat f(\xi) = \int ...
- 613
3
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0
answers
79
views
Condition on a function to have a Fourier transform in $L^{2-\varepsilon}$
It is known that in general the Fourier transform of $L^p(\mathbb{R})$ functions for $p>2$ are not even function. However, for regular enough functions, the regularitytransfers into decay for $\hat ...
- 489
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140
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Decay of Laplace (or Mellin) transform beyond region of convergence?
Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
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155
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Why does division parallelize but not continued fractions and is there an analog of multiplication to continued fractions?
All the basic arithmetic operations $\times,+,/,-$ can be parallelized. However continued fraction representation of a rational number is not parallelized. The process of Euclid's algorithm looks ...
- 13.9k
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150
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Keeping $\max_{|t|\geq 1} |\widehat{\phi}(t)|$ small (uncertainty principle)
Let $\phi:\mathbb{R}\to \lbrack 0,\infty)$ be piecewise continuous, symmetric ($\phi(x)=\phi(-x)$) and with support on $(-1,1)$. Let $\Phi(x)=\int_{-\infty}^x \phi(u) du$; assume $\Phi(1)=1$.
What is ...
- 20.2k
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Does this definition of the Fourier intensity measure make sense?
Let $\epsilon_n$ be a sequence in $\{-1,1\}^{\mathbb Z_+}$.
For simplicity, assume that $\epsilon_n$ is just the Thue-Morse sequence with symbols $1$ and $-1$ (although the following definition is ...
- 785
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261
views
Extension of Paley-Wiener-Schwartz theorem to vector-valued distributions
Let $H_{j} := (H_{j}, \| \cdot \|_{H_{j}} ), j=0,1$ be a Hilbert space, and set
\begin{equation*}
{\mathscr S}'(\mathbb{R}^{n}, H_0; H_1) := {\mathscr L}( {\mathscr S}(\mathbb{R}^{n}, H_0), H_1)
\end{...
- 31
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Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
- 698
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Asymptotics of fast decaying oscillating integrals
Let $g(t):\mathbb{R}\rightarrow \mathbb{C}$ be a smooth even function such that for each positive integer $n$ the function $g^{(n)}(t)$ tends to 0 when $t$ goes to infinity and $\int_0^\infty |g^{(n)}(...
- 109k
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651
views
Decay of the Fourier transform of a surface area measure
Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...
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A question about Fourier transform of function of the type $Q(x)(1+P(x))^{z}$
For simplicity, consider in $\mathbb{R}^3$, and the Fourier transform of the following function
$$f=(x_1+x_2+x_3)(1+|x|^2+x_1^2(x_2^2+x_3^2)+x_2^2x_3^2)^{-t+is},~~ \frac12<t<1,~~s\in \mathbb{R}.$...
- 879
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225
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Green's Function for a Kernel with Symmetric Fourier Transform $\nabla^2-x^2$
I am trying to find the inverse of the following kernel in 3 dimensions
$$
\nabla^2-x^2,
$$
where,
$$
x^2=\vec{x}.\vec{x}
$$
It seems quit simple and one would think there should already be solutions ...
- 31
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151
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Meaning of fractional Fourier transform with imaginary iteration count
(I'm reposting this from math.stackexchange as I didn't get an answer there and thought it might be "advanced" enough for this site.)
As one may know, the Fourier Transform
$$F[f](\nu) = \int_{-\...
- 1,687
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1
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77
views
Level sums, displacements: how to determine them efficiently?
Let $R =\mathbb{Z}/N \mathbb{Z}$. Let $f:R\to \mathbb{R}$,
$\rho:R\to \lbrack 0,1\rbrack$. We assume that it takes trivial time to compute any given value $f(m)$ or $\rho(m)$.
Define $$S(\delta,m) = ...
- 20.2k
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votes
2
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272
views
The inequality $\int^\infty_0 (\sin(rt)r^3/\sinh^2(r)) dr\leq cte^{-At}$
How to prove the following inequality $$\forall t>0,\quad\int^\infty_0 \sin(rt)\frac{r^3}{\sinh^2(r)} dr\leq c \big(te^{-At}\big)$$
for some constants $A>0,c>0$
- 521
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1
answer
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views
nonnegative Fourier Transform
Let $\widehat{f}(\xi)$ be Fourier transform of $f$ given by
\begin{align}
\widehat{f}(\xi)=\int_{\mathbb{R}^n} e^{-ix\cdot\xi}f(x)dx.
\end{align}
Suppose that $\widehat{f}(\xi)$ is nonnegative and ...
user9663
2
votes
2
answers
1k
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Decay estimate of Fourier transform of a compactly supported function
Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate
$$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$
for some $\...
- 903
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2
answers
331
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Estimate for a simple oscillatory integral
If $\varphi$ is a smooth function on $\mathbb{R}$, then integration by parts implies that there exists a constant $C>0$ such that
$$
\Big|\int_0^1 \varphi(x)\, e^{i \lambda x}\, dx\Big|<\frac{C}\...
- 421
2
votes
1
answer
279
views
Integral involving Bessel function
I have to work out the integral:
$$
\int_0^{\infty} dq \frac{J_0(q \xi)}{q+1}
$$
where $ J_0(z) $ is the Bessel function of the first type and order zero, $\xi \in \mathbb{R}$, $\xi \ge 0 $. So far, ...
2
votes
2
answers
858
views
Proof of Biot-Savart theorem in 3D using distributions and fourier transform
While studying the book Vorticity and Incompressible Flow by A. Majda and A. Bertozzi I came across the following Biot-Savart theorem in 3D:
The solution to:
$$div \ v=0$$
$$curl(v)=\omega$$
is
$$v=\...
- 489
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votes
2
answers
235
views
Theoretical/Practical Implications of DFT Eigenvectors
Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
- 4,058
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votes
3
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457
views
Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
I am considering a PhD research topic. I only have a math Bachelor's degree with working experience mostly in teaching and I have been working on a paper. I have deep interest in Fourier Series and ...
2
votes
2
answers
365
views
Is there a compactly supported differentiable function whose Fourier transform is not in L1?
In my MSE answer here, I discussed the example of compactly supported continuous function
$$g(x)=
\begin{cases}
\dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\
0,&\text{otherwise}
\end{cases}$$
...
- 833
2
votes
1
answer
404
views
The Fourier transform of the Liouville function?
The Liouville function in number theory is defined as:
$$\lambda(n) := (-1)^{\Omega(n)} \text{ where } \Omega(n) := \sum_{p|n} v_p(n)$$
Taking the discrete time Fourier transform and then taking the ...
- 3,064
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votes
1
answer
168
views
Any references for generalised square functions?
In harmonic analysis, there is a big chunk of literature studying the square function $Sf=\|\{P_jf\}_{j=1}^\infty\|_{l^2}$, where $P_jf=(\psi_j\hat f)\check{}$ and $\{\psi_j\}$ is a partition of unity,...
- 275
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2
answers
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views
Decay of the Fourier transform of a non-differentiable function
It is well known that if $\varphi$ is a Schwartz function on $\mathbb{R}$ (i.e. smooth and decaying at infinity faster than polynomials), then its Fourier transform decays faster than polynomials. ...
- 421
2
votes
1
answer
250
views
Can a Fourier transform be performed on irregularly sampled data with timestamps?
Normally, when I think of performing a Fourier transform, I imagine that my samples are spaced regularly in time (or space).
If I have a set of samples that are spaced irregularly, but have accurate ...
- 121
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1
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views
Is this integral solvable analytically?
I have this integral that comes from my research with some Fourier Transforms of spectrum functions:
$$ G(\tau) = \int_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c_1 - c_2 e^{-c_3 x} ) } dx $$
where $...
- 159
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1
answer
141
views
The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$
Let $a>0$. How to prove the following inequality $$\exists c>0,\exists A>0,\forall t>0:\quad\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)...
- 521
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votes
1
answer
334
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Fourier transform of the indicator function of the semi-ball
I wonder if it is possible to derive an analytical form for the Fourier transform of the indicator function of a semi-ball:
$$g_1(\underline{\xi}) = \int_{\mathcal{B}_+(R)} e^{i \underline{\xi} \cdot \...
- 21
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votes
1
answer
279
views
Unit singular value conjecture for discrete Fourier transform submatrix
This question was motivated by Singular value decomposition of truncated discrete Fourier transform matrix
Consider for integers $1\leq k\leq N$, $1\leq n_0\leq N-k+1$ the $k\times k$ sub-unitary ...
- 188k
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votes
2
answers
432
views
What is multidimensional Fourier transform of $e^{-\mathrm{abs}(x)}$?
What is multidimensional Fourier transform of $e^{-\mathrm{abs}(x)}$?
I know the answer for 1-dimension case. However, I cannot do integration for higher dimensional case.
I used spherical coordinate ...
- 21
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votes
1
answer
668
views
Does Bochner's Theorem apply to Fourier coefficients?
Let $f $ be a periodic function and denote by $c_n$, for $n \in \mathbb{N}$, its Fourier coefficients, i.e.
$$
c_n := \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{inx}\ dx.
$$
It is well known that Bochner's ...
- 595
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2
answers
251
views
Two classic problems concerning Fourier transform of an integrable function
I am looking for the following questions:
(1) True or false? for every $p<q$, one may find a function $f\in L^1(\mathbb{R})$ such that $\hat{f}\in L^q (\mathbb{R})$ but $\hat{f}\notin L^p (\...
- 4,058
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votes
1
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190
views
Half Poisson summation
Suppose $f$ is a Schwartz function on $\mathbb{R}$. Is there a closed formula for $$\sum_0^\infty \hat{f}(n)$$ where $\hat{f}$ is the $n$-th Fourier coefficient of $f$?
- 587
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views
Singular Integrals and $L^1$
Let us consider in one dimension the Fourier multiplier $\vert D\vert$ and the derivative $iD$. Both are well-defined on the Schwartz space $\mathscr S(\mathbb R)$ with the derivative sending $\...
- 16.2k
2
votes
1
answer
184
views
Prove if the fractional Laplacian of a function is bounded
Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$.
Here $(-\Delta)^s$ is the ...
- 123
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votes
1
answer
254
views
Fourier transforms of homogeneous functions [closed]
Compute Fourier transforms of homogeneous functions of the form,
$$
\frac{1}{|x|^{n+d}}P_d(x)
$$
where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.
- 343
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votes
1
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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 ...
- 53
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votes
1
answer
260
views
Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
2
votes
1
answer
129
views
Gronwall estimate with a Fourier transform
Suppose I have the following equality
$$\hat{u}_\epsilon(t,k) = \alpha(t,k) + \int_0^t\int_{\mathbb{R}^n} e^{ik\cdot(x +\epsilon \phi(s,x))}u_\epsilon(s,x)dxds$$
Where $\alpha(t,k) \geq 0$ and $\alpha(...
- 131
2
votes
1
answer
115
views
Fractional Schrödinger equation
Let $\mathcal{F}$ denotes the Fourier transform.
It is known that $\mathcal{F}(e^{-4\pi^2 i t |x|^2})(\xi)= e^{i |\xi|^2/4t}{(4\pi i t)^{-n/2}} \ (x, \xi\in \mathbb R^n).$
My question is: what is ...
- 305
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votes
1
answer
286
views
CTRW: solve a renewal equation
Let X(t) be a continuous time random walk, with exponentially distributed waiting times of pdf $f_T(t)= k e^{-k t}\; t\geq 0$ and a jump sizes pdf $f_J(x)$. Suppose the initial distribution $\rho_{X(0)...
- 634
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votes
1
answer
1k
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Decay of the Fourier transform of a compactly supported continuous BV function
If $f$ is a compactly supported bounded variation function on the real line $\mathbb R$, its Fourier transform $\widehat f$ can be estimated as $|\widehat f(\xi)| = O(|\xi|^{-1})$, $|\xi|\to\infty$. ...
- 1,329