Questions tagged [fourier-analysis]
The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.
1,021
questions
2
votes
0answers
55 views
Can we construct a computable sequence of trigonometric polynomials that converges pointwise to a given continuous function defined on the Torus?
Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric plynomials), with the band width (degree of the ...
0
votes
1answer
64 views
Can we construct a sequence of trigonometric polynomials that converges pointwise to a given continuous function on the torus?
Consider any continuous function $f$ on an $m$-dimensional Torus $\mathbb{T}^m$. Can we construct a sequence of band limited functions (trigonometric polynomials), with the band width (degree of the ...
2
votes
0answers
131 views
Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?
Consider the following formula which defines a piece-wise function which I believe corresponds to a series representation for the Dirac delta function $\delta(x)$. The parameter $f$ is the evaluation ...
2
votes
0answers
61 views
(Dis)continuity of periodic functions with non-summable Fourier series
Let $f : [0,2 \pi)^d \rightarrow \mathbb{R}$ be a square-integrable periodic function in $L^2( [0,2 \pi)^d )$ with $d \geq 1$.
We assume moreover that the square-summable Fourier coefficients of $f$, ...
2
votes
0answers
52 views
Radon transform range theorem and radial functions
(UPDATED for rapid decay considerations + new question)
In dimension 2, the Radon transform range theorem states that a rapidly decaying (Schwartz) function $g(t,\theta)$ can be represented as a ...
8
votes
0answers
180 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 $a\in (1,2)$.
2
votes
0answers
33 views
eigenvectors of a graph Laplacian VS Fourier basis
Could you please illustrate the following statement:
the eigenvectors of a
graph Laplacian behave similarly to a Fourier basis, motivating
the development of graph-based Fourier analysis theory.
3
votes
0answers
61 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 ...
5
votes
2answers
184 views
Analog of the Birkhoff's ergodic theorem for the sequence of squares
Consider a dynamical system $(X, \mathcal{B}(X), \mu, T)$ where $(X, \mathcal{B}(X), \mu)$ is a measure space and $T$ is a measure-preserving, invertible transformation.
Then by the classical ...
-1
votes
0answers
21 views
Unable to do question 3 in 7.3 from Folland's Fourier Analysis and its Application [migrated]
I'm unable to answer this question, where we were given $f(x)$:
$$f(x)=\begin{cases}
1, & \text{if }-1<x<1 \\
0, & \text{otherwise}\end{cases}$$
The questions asks me to compute $f*f$ ...
9
votes
2answers
602 views
Fourier series of $\log(a +b\cos(x))$?
By numerical computation it seems like, if $a_0 < a_1$:
$$
\begin{multline}
\log({a_0}^2 + {a_1}^2 + 2 a_0 a_1 \cos(\omega t)) = \log({a_0}^2 + {a_1}^2) \\
+ \frac{a_0}{a_1}\cos(\omega t)
- \frac{...
1
vote
0answers
184 views
Problem with completeness of an orthogonal system
For $\nu\in (-1, \infty)$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the sequence of positive zeros of the Bessel function $J_{\nu}$. The Fourier-Bessel "Laplacean" is given by
\begin{equation}...
3
votes
1answer
99 views
Example of a bounded function whose mean-zero mollification diverges at a point
For a Schwartz function $\psi(x)=xe^{-x^2}$ define $\varphi(x):=\psi'(x)$ and consider a family of $L^1$-dilations of $\varphi$ given by:
$$
\varphi_t(x)=\frac{1}{t}\varphi(x/t), \qquad t>0.
$$
$\...
2
votes
0answers
81 views
Proof that Littlewood-Paley vertical square function is NOT bounded on L^infinity
The classical heat semigroup on $\mathbb{R}$ is given by
$$
W_t f(x)=\frac{1}{t}\int_{\mathbb{R}}e^{-\pi (\frac{x-y}{t})^2}f(y)dy, \qquad t>0.
$$
Then the Littlewood-Paley vertical square ...
6
votes
1answer
161 views
the fractional integration method of the proof of Stein-Tomas theorem?
In Schalg's Classical multilinear and Harmonic analysis, he presented two methods of the proof of Stein-Tomas theorem, one of which is called the fractional integration method. As a matter of fact, in ...
1
vote
0answers
68 views
Strichartz estimate for the Schrödinger equation
Estimates of the extension operator can be seen as estimates of the initial value problem for the evolution Schrödinger equation. If $u(x,t)=e^{it\Delta}u_0$ is the solution to the IVP:
$$i\partial_t ...
3
votes
1answer
100 views
Strict inequality in decoupling inequality
I am working on the decoupling inequality developed by Bourgain and Demeter: https://arxiv.org/abs/1604.06032.
Is there an example where we have strict inequality in Theorem 1.1, say in the case $n=...
7
votes
2answers
320 views
Lower bound on exponential sums
Let $k\geq 2$. Consider the following norm of exponenetial sum:
$$
I(N,p,k)=\int_0^1\int_0^1 \left|\sum_{n=0}^N e^{2\pi i (n x+n^k y)}\right|^p dxdy.
$$
Bourgain mentioned on Page 118 of
https://...
0
votes
0answers
136 views
Reference request: Band limited interpolation of data
I have come up with an interpolation method for irregularly placed data points on a square domain. The method assumes the data points are discrete, that is they coincide with nodes of a uniform ...
2
votes
0answers
84 views
Inequality about exponential integrals
I am reading about Dirichlet polynomials in the book Analytic Number Theory by Iwaniec-Kowalski.
During the proof of Theorem 9.1 for any positive real numbers $T, N$ they define a piecewise linear and ...
0
votes
1answer
184 views
Poisson Summation Formula appears to fail when applied to Hermite Functions (why?) [duplicate]
I came across an odd circumstance where it appears as though the poisson summation formula fails to yield a correct answer (involving Hermite Functions), and I don't quite understand why this happens. ...
1
vote
0answers
92 views
A close formula for a Fourier transform
I would like to calculate "explicitly" the following integral, which is a Fourier transform: let $\alpha>0$ be a parameter, for $x\in \mathbb R$, we define
$$
I(\alpha, x)=\int_\mathbb R \cos(xt) e^...
2
votes
1answer
81 views
Weyl symbol of product
Are there explicit formulas for the Weyl symbol of $-f(x)D_x^2 $ where $D_x:=-i\partial_x $ and $\partial_x$ is the derivative and $f$ some sufficiently smooth function?
In the standard quantization ...
6
votes
1answer
230 views
Fourier transform on Minkowski space
Physicists Some people like to define the "Fourier transform" on Minkowski space as $\hat f(\xi) = \int e^{i \eta(x,\xi)} f(x) dx$, where $\eta(x,\xi)$ is the Minkowski form. I'm used to thinking of ...
4
votes
1answer
172 views
Idea behind Carleson's theorem modern proof “intitial reductions”
I'm having troubles to understand the philosophy behind the modern proof of Carleson's theorem. For convenience, let me state precisely what I am asking for.
For any $f \in L^2(\mathbb{R})$, let $\...
3
votes
0answers
45 views
Space contained in the Interpolation of $L^\infty$ and the Wiener Algebra $\mathcal{F}(L^1)$
Let $\ell^p$ be the space of sequences with power $p$ summable to $\ell^\infty$, $L^p = L^p(\mathbb{R^d})$ be the Lebesgue spaces and $\mathcal{F}$ be the Fourier $d$-dimensional Fourier transform.
...
2
votes
1answer
77 views
Evolution equation generated by Fourier multiplier
I am on the hunt for techniques regarding a field which I am not familiar with.
More precisely, I am considering equation of the form
$$ i \partial_{t} u(t,x) + p(D)u(t,x) = 0, \ \ u_{|_{t=0}}=u_0(x)...
7
votes
1answer
273 views
Compactly supported probability measure in high dimensions with fast Fourier decay?
For any sufficiently large $d\in\mathbb{N}$, does there exist a probability measure $\Psi$ supported on the Euclidean ball in $\mathbb{R}^d$ for which $|\widehat{\Psi}[\omega]|\le C\cdot \exp(-\|\...
1
vote
0answers
31 views
Higher-order inner products of an orthonormal basis
Let $\pi$ be a probability measure on some space $\mathcal{X}$, and let $\Phi = \{ \phi_k \}_{k \geqslant 0}$ be some (possibly complex-valued) orthonormal basis for $L^2 ( \pi )$, with $\phi_0 \equiv ...
6
votes
0answers
202 views
What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?
Here is the story as I see it.
Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
6
votes
1answer
102 views
Equivalence of antiderivative in L1 sense and in the usual sense
We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that:
$$
\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0
$$
where $\Vert \cdot \Vert_1$ is the $L_1$ ...
3
votes
0answers
104 views
On Pitt's inequality (weighted Fourier inequality)
One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$,
$$
\sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
0
votes
0answers
81 views
Wigner distribution
The Wigner distribution of $u\in L^2(\mathbb R)$ is defined as a function $W(u)$ on $\mathbb R^2$ given by
$$
W(u)(x,\xi)=\int_\mathbb R u\left(x+\tfrac z2\right) \overline{u\left(x-\tfrac z2\right)} ...
2
votes
0answers
68 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 ...
4
votes
2answers
380 views
Earliest use of deconvolution by Fourier transforms
From a previous discussion here Origin of the convolution theorem, it was shown that the property of convolution $y(t)$=$a$*$b$ becoming a multiplication after Fourier transform: $F$$(y(t))$= $F(a)F(b)...
2
votes
0answers
72 views
Sobolev convergence of Fourier series
Consider $f\in H^{\sigma}(S^1)=W^{\sigma, 2}$ (the usual Sobolev space on the circle) and let $S_Nf$ be its truncated Fourier series $S_Nf = \sum_{|n|\leq N} \hat{f}(n)e^{2\pi i n x}$. I am looking ...
0
votes
0answers
74 views
Existence of the inverse Fourier transform, Carr Madan
I have a function $C_T(k)$ that is not $L_1$, because its limit in negative infinity is a constant.
So I dampened it by $ e^{\alpha k} $. Let's call the transformed function (of the dampened function) ...
1
vote
0answers
33 views
Example of periodic semidifferentiable function without absolutely convergent Fourier series
Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but
with a non-absolutely convergent Fourier ...
1
vote
0answers
129 views
Logarithm of the Fourier transform?
I've found this paper on the logarithm of the discrete fourier transform which proves that
$$
log F = 1/4 i \pi (I - (1 +i)F + F^2 - (1 - i)F^3)
$$
where $F$ is the unitary discrete Fourier ...
9
votes
1answer
245 views
Bounding a Fourier coefficient of a non-negative periodic function in terms of its $L^2$-norm
This question is motivated by the earlier MO question: Show that $(\sum_{k=1}^{n}x_{k}\cos{k})^2+(\sum_{k=1}^{n}x_{k}\sin{k})^2\le (2+\frac{n}{4})\sum_{k=1}^{n}x^2_{k}$ .
It is a cleaned up ...
1
vote
0answers
71 views
Reconstructing a sine wave using square waves and Möbius inversion: L² convergence?
Let $s$ be the (“square wave”) $1$-periodic real function such that $s(x) = 1$ if $0<x<\frac{1}{2}$ and $s(x) = -1$ if $\frac{1}{2}<x<1$ (and maybe $s(0)=s(\frac{1}{2})=0$ for the sake of ...
0
votes
0answers
47 views
Fourier coefficients of a variation in Teichmuller theory
Prove that for $\dot w[\mu](\zeta)=-\frac{(\zeta-1)(\zeta+1)(\zeta+i)}{\pi}\left\{\iint_{\Delta} \frac{\mu(z) d x d y}{(z-1)(z+1)(z+i)(\zeta-z)}+\iint_{\Delta} \frac{i \overline{\mu(z)} d x d y}{(\bar{...
2
votes
0answers
66 views
Point wise convergence of Laplace transform and convergence of functions
Assume that functions $f_n(t), f(t)\in C_b(R_+)$. For every $\lambda >0$, we have
$$
\bigg|\int_0^\infty e^{-\lambda t}f_n(t)d t-\int_0^\infty e^{-\lambda t}f(t)d t\bigg|\leq C_\lambda n^{-1},
$$
...
4
votes
1answer
112 views
Stationary phase method for $\varphi''(x_0)= 0$
Stationary phase method (in the usual setup) gives asymptotic for
$$
I(\lambda)=\int_{a}^{b} f(t) e^{i \lambda \varphi(t)} d t,
$$
when at any stationary point $x_0$ ($\varphi'(x_0)=0$) second ...
26
votes
1answer
691 views
Linear combination of sine and cosine
I was explaining to my students the other day why $\cos(2x)$ is not a linear combination of $\sin(x)$ and $\cos(x)$ over $\mathbb{R}$. Besides the canonical method of using special values of sine and ...
4
votes
0answers
196 views
Simultaneous Hahn-Banach theorem
Let $C(\mathbb{T})$ be the Banach algebra of continuous functions on the unit circle. Let $n \in \mathbb{N}$ and let $P_n(\mathbb{T})$ be the subspace of trigonometric polynomials of degree at most $n$...
2
votes
1answer
121 views
Explicit construction of Kakeya sets using Perron tree
I have found many excellent notes online that illustrate how to construct a Kakeya needle set (with measure $<\varepsilon$.) Yet none of them gives full argument about the construction of a Kakeya ...
4
votes
1answer
199 views
Earliest reference on the calculation of derivatives by Fourier transform
I was looking for an earliest reference or the name of the mathematician who showed calculating the derivatives is possible in the Fourier domain?
The Fourier transform of the derivative is (...
1
vote
1answer
87 views
Duality relation of Lorentz space $L^{p,1}$
want to prove the duality relation:
$$||f||_{L^{p,1}} =C_{p} \cdot \sup\{\int_X fg d\mu: \text{ for any } ||g||_{L^{p',\infty}}\le 1 \} $$
where $\frac{1}{p}+\frac{1}{p'}=1, p>1, \mu$ is $\sigma$-...
1
vote
0answers
100 views
Fourier Inversion Theorem
I have a reference request concerning the Fourier Inversion Theorem.
Let $G$ be a compact hausdorff abelian group. $L_2(G)$ has a unique Haar integral $\int_G - d \mu$ arising from the completion of ...
