All Questions
69 questions
2
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109
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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{...
- 20.2k
4
votes
1
answer
277
views
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:\...
- 20.2k
2
votes
0
answers
79
views
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 ...
- 20.2k
1
vote
1
answer
142
views
Operator norm of some type of discrete Fourier matrix
Let $N$ be a natural number and let $w$ be a complex number.
We define the $N\times N$ matrix $C_w=(a_{k,l})_{k,l=1}^N$ as follows,
$$
a_{k,l}=\begin{cases}1 & l=k+1\\
w &...
- 4,713
2
votes
0
answers
194
views
Functions such that the *integral* of the Fourier transform is non-negative?
Let $f:\mathbb{R}\to \mathbb{R}$ be in $L^1$, with its Fourier transform $\widehat{f}$ also in $L^1$. What is a necessary and sufficient condition on $f$ so that
$$\int_{-\infty}^x \widehat{f}(t) dt \...
- 20.2k
23
votes
0
answers
1k
views
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 ...
- 1,124
11
votes
2
answers
8k
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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\...
- 201
0
votes
1
answer
127
views
Why $\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w) < \infty$?
Let $f\in L^p(\Bbb R^n)$ and $S^{n-1}$ be the Unit sphere.
Why
$\int_{S^{n-1}} |\hat{f}(w)|^2d\sigma(w)<\infty$ when $1<p<2$.
$\hat{f}$ is the Fourier transform fora function f.
- 39.1k
0
votes
1
answer
126
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Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections
I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
- 7,817
6
votes
2
answers
458
views
Does the (distributional) support of the Fourier transform of an $L^p$-function with $p<\infty$ have positive measure?
Suppose that $f \in L^p(\mathbb R^n)$ such that $1\leq p < \infty$. Let $\hat f$ be the Fourier transform of $f$. Clearly, if $p=1$ or $p=2$ then the support of $\hat f$ has positive Lebesgue ...
4
votes
2
answers
1k
views
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
$$
\mathcal W\subset ...
5
votes
0
answers
194
views
When does the Fourier transform of a measure decay?
Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...
- 1,101
8
votes
2
answers
613
views
Pairs of elementary Fourier transforms in $L^2$
It is customary to teach Fourier transform on the real line by starting with functions from $L^1$, $L^2$ or the Schwartz space. It is not so easy to illustrate the theory by computing explicit pairs ...
- 18.7k
0
votes
1
answer
170
views
When some Fourier coefficients are fixed, can we control the extremals of the function?
Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f
:\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?
1- $-n\lneqq f_{\min}$ (where $f_{\...
- 3,808
3
votes
0
answers
162
views
The essential norm where some Fourier coefficients are fixed
Let us denote $C_{2\pi}$ by the set of all $2\pi$-periodic continuous functions $f:\mathbb{R}\to \mathbb{R}$.
Q. Let $\phi\in C_{2\pi}$. Is the following statement valid?
$$\|\phi\|_2=\inf_{g\in C_{2\...
- 4,058
2
votes
1
answer
272
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
4
votes
1
answer
255
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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
$$
H = \left\{f\in {\cal S}'(\mathbb{R}^d) \;\left| \; t\times\widetilde{f} \in {\cal L}^2(\mathbb{...
CommunityBot
- 1
6
votes
1
answer
491
views
Harmonic analysis for a beginner
I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
- 188k
1
vote
1
answer
230
views
Why we have $f=0$
Define the Fourier transform for a suitable function $f\in L^1(\Bbb R)$ by $\widehat{f}(\xi)=\int_{\Bbb R}f(x)e^{-ix\xi} dx$.
Assume the condition $$\int_{\Bbb R}\int_{\Bbb R}|\widehat{f}(\xi)f(x)|^...
- 91.8k
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
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
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 ...
- 188k
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. ...
- 4,729
1
vote
0
answers
79
views
A problem arising from Wiener-Levy theorem on the real line
Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
- 4,058
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:
$$\int_0^\infty f(t) e^{-itx} \, dt =\lambda \frac{1}{x} f\left(\frac{1}{x}\right)$$
$\...
- 1,422
3
votes
0
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
1
vote
0
answers
244
views
On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)
Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
- 111
1
vote
0
answers
62
views
Stable deconvolution of a band-limited function from its convolution with a Gaussian
Suppose that $f : \mathbb R \to \mathbb C$ is a band-limited function, i.e. its Fourier transform $\hat f$ has support in a compact interval $[-a,a]$. Let $\phi(t) = e^{-\frac{t^2}{2\sigma^2}}$ be a ...
- 437
1
vote
1
answer
439
views
Well-known conditions for the Fourier inversion formula
Let $f\in L^1(\mathbb{R})$.
One may easily check that
$$(*)~~~f', f''\in L^1(\mathbb{R})\Rightarrow \int_\mathbb{R}|\hat{f}| ~\text{is finite} \Rightarrow \int_\mathbb{R}\hat{f}(s)e^{2\pi is x}ds ~\...
CommunityBot
- 1
2
votes
2
answers
1k
views
Decay estimate of Fourier transform of a compactly supported function
Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate
$$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$
for some $\...
- 903
0
votes
1
answer
226
views
Transformation of Fourier Transform
Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression ...
- 16.2k
1
vote
0
answers
139
views
Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?
Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form:
$$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$
where the ...
- 435
3
votes
2
answers
589
views
On the Fourier inversion formula
For a given function $f\in L^1(\mathbb{R})$, suppose that the
$$\check{f}(x)=\int_\mathbb{R} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$$
almost every where converges in $\mathbb{R}$. Then, can we say that
...
- 17.2k
1
vote
0
answers
119
views
Integrable functions that may not satisfy the inversion Fourier formula
Let $f\in L^1(\mathbb{R})$. We define $\phi_f(x)=\int_{\mathbb{R}} \hat{f}(\zeta)e^{2\pi i\zeta x}d\zeta$ if the improper Riemann integral is finite otherwise, $\phi_f(x)=\infty$.
Does there exist ...
- 4,058
2
votes
2
answers
251
views
Two classic problems concerning Fourier transform of an integrable function
I am looking for the following questions:
(1) True or false? for every $p<q$, one may find a function $f\in L^1(\mathbb{R})$ such that $\hat{f}\in L^q (\mathbb{R})$ but $\hat{f}\notin L^p (\...
- 39.1k
5
votes
3
answers
2k
views
Extension of Poisson Summation formula
Under the condition f continuous, integrable and:
$|f(t)| + |\hat{f}(t)| \le C (1+|t|)^{-1-a}$ (with a>0)
we have the twisted Poisson formula that holds (where $\chi(n)$ is a primitive Dirichlet ...
- 113
1
vote
1
answer
487
views
Fourier Transform of an even function
Let $S^n$ be an $n$-dimentional unit sphere.
Consider $f: S^n \longrightarrow R_+$, where $f$ is an even continuous function.
Denote
$$
F(f):=\int_0^{\infty}\int_{S^n}f(y)g\left(\frac{|xy|}{t}\...
- 34.7k
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 $\psi\in L^2$ ...
- 2,369
7
votes
1
answer
1k
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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?
- 107
3
votes
1
answer
304
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Existence of probability measure on the circle with given Fourier coefficients
We say that a Hermitian symmetric (i.e., $f_{-n} = f_n^*$ for any $n \in \mathbb{Z})$ sequence $(f_n)_{n\in \mathbb{Z}}$ is positive-definite if, for any $N \geq 0$ and any $z_0 , \ldots, z_N \in \...
- 63.9k
11
votes
0
answers
707
views
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(;\...
- 111
1
vote
0
answers
353
views
Eigenvalues of convolution matrices
Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
- 315
1
vote
1
answer
460
views
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
I am reading a book "Fourier Series and Integrals" by Dym & McKean.
There is an exercise (Page 106):
Exercise: Check that if $f$ is a real, even, summable function and
if $f(0+)$ and $f(0-)$...
- 91.8k
3
votes
1
answer
423
views
Is there (fast) fourier transform for vector convolution?
Given a list of variables $u_1,\dots,u_m\in\mathbb R$ and $v_1,\dots,v_n\in\mathbb R$ the standard convolution is defined
$$U*V(t)={\sum_{i}} u_iv_{t-i}.$$
Given a list of vectors $u_1,\dots,u_m\in\...
CommunityBot
- 1
1
vote
0
answers
146
views
Functional equation with Fourier transform
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
- 1,199
1
vote
0
answers
50
views
Comparison of (square) of a function and its Fourier transform in an integral
I am completely stuck on a comparison between $f(t)^2$ and $\hat{f}(t)^2$ in an integral.
Considering $f(t)$ of rapid decrease at infinity such that near zero: $f(t) \sim_0 t^{-\frac{1}{2}- \alpha}+o(...
- 1,199
0
votes
0
answers
60
views
Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
4
votes
0
answers
116
views
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, ...
- 3,612
11
votes
1
answer
691
views
Reference request: Fourier transform on the multiplicative group of real numbers
Let us consider the three groups $(\mathbb{R},+)$, $(\mathbb{Z}/2\mathbb{Z},+)$ and $(\mathbb{R}^\times,\cdot)$ (where $\mathbb{R}^\times := \mathbb{R} \setminus \{0\}$). We endow $\mathbb{R}$ with ...
- 31.5k
1
vote
0
answers
141
views
Characterisation of functions for which the Fourier transform commutes with a particular operator
Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...
CommunityBot
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