All Questions
Tagged with fourier-analysis fourier-transform
275 questions
2
votes
1
answer
273
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
8
votes
1
answer
1k
views
Who introduced the discrete Fourier transform?
I am trying to find the original reference which introduced the definition of discrete Fourier transform as used today. When did this modern formulation (which includes the indexing from n to N-1) of ...
- 879
3
votes
0
answers
86
views
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 ...
- 31
2
votes
2
answers
920
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
11
votes
2
answers
8k
views
About the Fourier transform of the logarithm function
I want to calculate / simplify:
$$\mathcal{F} (\ln(|x|)\mathcal{F(f)}(x))=\mathcal{F} (\ln(|x|)) \star f$$
where $\mathcal{F}$ is the Fourier transform ($\mathcal[f](\xi)=\int_{\mathbb R}f(x)e^{ix\...
- 1,199
1
vote
0
answers
73
views
$L^p$ norm of Fourier transform of function composed with a diffeomorphism
Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
- 265
5
votes
1
answer
325
views
The discrete Fourier transform's Gaussian-like eigenvector
I have the $N$x$N$ matrix below where $N$ is a power of 2 (usually 64 or 256) and $\omega = 2\pi/N$. What is its largest eigenvalue?
$\begin{bmatrix}
2 & 1 & 0 & 0 & \cdots & 0 &...
- 1,547
1
vote
1
answer
217
views
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 $\...
- 671
4
votes
1
answer
663
views
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 ...
- 1,219
-1
votes
1
answer
213
views
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 \...
- 581
6
votes
1
answer
491
views
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, ...
- 159
20
votes
1
answer
1k
views
Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$
Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
- 178
0
votes
0
answers
113
views
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 ...
- 151
1
vote
1
answer
230
views
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 $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
- 531
12
votes
3
answers
2k
views
Looking for sufficient conditions for positive Fourier transforms
I am looking for some sufficient conditions for an even, continuous, nonnegative, non-increasing, non-convex function to be non-negative definite. In other words
$$
\int_0^\infty f(x)\cos(x\omega) \, ...
- 178
4
votes
1
answer
245
views
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
$$
\begin{align}
F(x) := \sum_{n\in\mathbb{Z}} f(x + n).\tag{$...
- 595
5
votes
3
answers
2k
views
Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
- 595
1
vote
0
answers
108
views
Recovering phase function using Fourier decomposition
I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function
$$f = e^{i \phi(x)}. $$
I am interested in the following problem. If I know the function/distribution $...
- 151
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
0
votes
1
answer
170
views
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_{\...
- 4,058
5
votes
4
answers
953
views
Limit of an integral vs limit of the integrand
I have a simple Fourier transform problem, originating from mathematical physics (system of linear PDEs), which reduces to taking the integral
$$
I(\alpha)\equiv\int_{-\infty}^\infty e^{ikr} \cfrac{\...
1
vote
2
answers
2k
views
Fourier transform of a holomorphic function
Question: Is there a simple method for calculating the Fourier transform of a holomorphic complex function ${f{{\left({z}\right)}}}:\Omega\to{\mathbb{{{C}}}}$?
In order for my question to be well-...
- 547
4
votes
1
answer
256
views
Proof that elements of Beppo-Levi-like spaces are functions (and not just distributions)?
Context. I am trying to undestand the theory underlying "Beppo-Levi"-like spaces defined as
$$
H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
3
votes
0
answers
164
views
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 $k$,
$$
K_N(\xi_0 n)\left[1-\cos\bigg(\frac{2\pi\xi_0 n}{p}\...
- 53
15
votes
4
answers
2k
views
Positivity of certain Fourier transform
Is the Fourier transform of the function
$$ f(\xi) = e^{-t|\xi|^{2m}}$$
positive for $t>0$ and $m \in \mathbb{N}_0$?
- 9,853
3
votes
1
answer
2k
views
Deconvolution using the discrete Fourier transform
Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
- 879
2
votes
1
answer
141
views
The inequality $\int^\infty_0 \frac{\sin(rt)}{rt}\frac{r^4}{\sinh^2(r)} e^{-ar\coth(r)}dr\leq c \big(e^{-At}\big)$
Let $a>0$. How to prove the following inequality $$\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)...
- 531
2
votes
1
answer
495
views
Fourier transform of a function of bounded variation
I know if $f\in L^2(\mathbb R)$ is two times continuously differentiable, then we must have that the Fourier transform is integrable. Is there any more relaxed condition than this? For example if $f$ ...
- 1,879
8
votes
1
answer
640
views
Rate of decrease of the Fourier transform of standard mollifiers
What is the the rate of decrease of $|\widehat{f_p}(t)|$ (as $t\to\infty$), where $p\in(0,\infty)$,
$$\widehat{f_p}(t):=\int_{\mathbb R} e^{itx}f_p(x)\,dx,$$
and
$$f_p(x):=e^{-1/(1-x^2/p)^p}1(|x|<\...
- 128k
7
votes
1
answer
1k
views
Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
- 4,058
6
votes
1
answer
397
views
Absolute values of two functions and absolute values of their Fourier transform coincides
Let $f, g \in L^2(\mathbb{R})$.
Is it true that if both $|f|=|g|$ and $|\hat f|=|\hat g|$ hold, then there exists $\theta \in \mathbb{R}$ such that $f=ge^{i\theta}$?
I am not able to prove it or ...
- 489
-3
votes
1
answer
101
views
Asking for reference about a relation related to Fourier transform [closed]
Sorry for the not-perfect question. I am asking for a reference for the following relation:
$$\int f . g. h ...= \int_{\xi_1 +\xi_2 +...=0} \hat{f}(\xi_1) \hat{g}(\xi_2)... d\xi_1 d\xi_2...$$
Could ...
- 159
8
votes
1
answer
491
views
Functional equation with Fourier transform and $\frac{1}{x} f(\frac{1}{x}) $
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$, they satisfy following functional equation:
$$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$
$\...
- 1,199
0
votes
1
answer
268
views
Constant in Amrein-Berthier uncertainty principle
Let $S,\Sigma$ in $\mathbb{R}^d$ be finite measure set. The Amrein-Berthier uncertainty principle states that there exists $C=C(S,\Sigma)>0$ such that for all $f\in L^2(\mathbb{R}^d)$, $\int_{\...
- 321
1
vote
1
answer
203
views
Explanation of a step in a work by C. E. Kenig and A.D. Ionescu
I am studying the work
Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
- 159
24
votes
3
answers
1k
views
Is there a 'certainty' principle?
Heisenberg's uncertainty principle is a restriction on which probability distributions can describe the position and momentum of a quantum particle.
In mathematical terms it says that if $\psi\in L^2$ ...
- 3,137
2
votes
0
answers
79
views
For $\Phi$ a majorant of $1_{[-1/2,1/2]}$, how small can the total variation of $\widehat\Phi$ be?
Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such that $\Phi(t)\geq 1$ for $|t|\leq 1/2$. Assume furthermore that $\Phi$ and $\widehat\Phi$ are both in $L^1\...
- 20.2k
4
votes
1
answer
520
views
The main topics (issues, problems) of the Fourier transform
To explain what we are looking for, let's have a quick review on some points in Fourier transform on periodic functions in both continuous and discrete cases. We emphasize that our attention is ...
- 4,058
0
votes
1
answer
130
views
Riesz transform after linear transformation
I am encountering the term $\partial_x \mathcal{R}_x(f(x,y))$. I needed to do the following linear transformation
$$x' = a x+ by,\,\,\,\,\, y'=ax-by,\,\,\, and \,\,f(x,y)=g(x',y') $$
I ended up with ...
- 159
1
vote
0
answers
180
views
A potential wrong proof of a Lemma
Consider the following lemma: Let $g \in H^s_{x,y}(S)$ where $S = \mathbb{R}^2$ or $S = \mathbb{T}^2$, and $\eta \in C^\infty(\mathbb{R})$, $\operatorname{Supp}(\eta) \subset [-2,2]$, and $\eta \equiv ...
- 159
3
votes
2
answers
413
views
A Sobolev embedding theorem for functions on spheres
$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds:
$$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...
- 351
1
vote
2
answers
152
views
Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
- 3,625
2
votes
1
answer
260
views
Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
1
vote
1
answer
509
views
Fourier transform of the fractional Poisson kernel
Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows:
$$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
- 537
0
votes
0
answers
150
views
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:
...
- 73
5
votes
0
answers
169
views
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 ...
- 342
0
votes
0
answers
75
views
$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
- 159
3
votes
2
answers
590
views
On the Fourier inversion formula
For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
...
- 4,058
4
votes
1
answer
295
views
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
$$
\...
- 675
2
votes
0
answers
149
views
An oscillatory integral
Let $s>0, v\in \mathbb{R}^d, w\in \mathbb{R}, |w|\leq 1$. Pick a cut-off function $B(0,1)\prec \eta \prec B(0,2)$ and a large real number $N$. Do we have the following type of estimates?
\begin{...
- 351