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
vote
2answers
73 views
Delta-distribution composed with a function from the Fourier representation
A well known representation of Dirac's delta-distribution is via the Fourier transform of distributions:
\begin{equation}
\delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{...
4
votes
0answers
109 views
Weighted reverse Poincare inequality over a function class of neural networks
We consider a probability measure supported on the whole space $\mathbb{R}^n$, whose density is $p(x)$. We also consider a (one-layer) neural network function class $\mathcal{C}$, whose elements have ...
-4
votes
0answers
55 views
Fourier - integration over a period [closed]
I have hard to understand to solve a Fourier integration over a period.
Function $f$ is periodic with period 2 and for $0 < t < 2$ is $f(t)= t^2$. Calculate the integral $f(t)$ with boundaries ...
1
vote
0answers
27 views
Show a convolution of distributions ε-close to min-entropy k is ε^t-close to min-entropy k
Assume $X_1,...,X_t$ are independent distributions on $\mathbb{Z}_2^n$ s.t. each $X_i$ is $\epsilon$-close to min-entropy $k$; i.e. there exist distributions $Y_1,...,Y_t$ on $\mathbb{Z}_2^n$ s.t:
$$ \...
4
votes
0answers
111 views
When does a continuous function's “Fourier series” converge pointwise almost everywhere to the function?
Let $G$ be a compact topological group. By the Peter-Weyl theorem, the complex Hilbert space $L^2(G)$ is the Hilbert space direct sum of the spaces of matrix coefficients of all the irreducible ...
46
votes
5answers
3k views
Jean Bourgain's Relatively Lesser Known Significant Contributions
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions, such as the proof of Vinogradov's conjecture ...
5
votes
0answers
145 views
Divisor bound for $r_2$ off the origin
If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
5
votes
0answers
54 views
Are Stochastic Process Characterized by Their conditional Moments
Suppose that $X_t$ is a real-valued stochastic process. Then is $X_t$ characterized by it's conditional moments? In the sence that, if $Y_t$ is another process, such that
$$
\mathbb{E}\left[\int_s^T\...
0
votes
0answers
74 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 ...
8
votes
1answer
241 views
What's so special about the Orthonormal base $\{e_n\}$ of $L^2[0,1]$, where $e_n(x)=e^{2\pi i nx }$?
Let $f \in L^2([0,1])$ . Then Carleson's Theorem states that
$$\lim_{N\to \infty} \sum_{|n|<N} \langle f,e_n\rangle e_n(x)=f(x),\quad\text{a.e. } x\in[0,1],$$
where $\{e_n\}$ is the Orthonormal ...
5
votes
1answer
125 views
Bounds on the L^1 norm of a discrete Fourier spectrum
I am dealing with a function $f$ of the form
\begin{equation}
f(t):=\sum_{k=1}^Na_ke^{\mathrm{i}\phi_k t}
\end{equation}
and I have a promise that
\begin{equation}
0\leq f(t)\leq C\;\;\;\text{for all}...
1
vote
1answer
79 views
Multidimensional improper Riemann integrals with oscillatory kernels: Existence
I have asked this question three weeks ago here
https://math.stackexchange.com/questions/2998601/does-this-oscillatory-integral-exist/2998930#2998930
but received no relevant answers.
Let $n\geq 2$ ...
10
votes
0answers
283 views
Fourier transforms and nontrivial vector bundles
We know that in arithmetic, geometry and analysis, Fourier transforms of various forms show up. For example, we have the classical Fourier transform, Fourier-Mukai transforms in the setting of ...
3
votes
1answer
213 views
What are the almost periodic functions on the complex plane?
The almost periodic functions on the real line can be characterized as uniform limits of trigonometric functions. I was wondering whether a similar definition exists on the complex plane (a locally ...
6
votes
1answer
218 views
Is $\frac{\sin |\xi|}{|\xi|}$ in range of Fourier Transform for $n \ge 3$?
Does there exist $f \in L^1(\mathbb{R}^n)$ s.t., $\displaystyle \widehat{f}(\xi) = \frac{\sin |\xi|}{|\xi|}$ in case of dimension $n \ge 3$?
It is known that for $n = 2$, the function $\displaystyle ...
5
votes
1answer
149 views
Positive-definiteness of radial sinc function in three dimensions
In dimension one, it is well known that $\mathcal{F}\chi_{(-1,1)}=\frac{\sin{x}}{x}$. This implies, in particular, that $\frac{\sin{x}}{x}$ is a definite positive function. I wonder if a similar ...
11
votes
1answer
284 views
Poisson summation formula for number fields
Poisson summation formula is widely used in many parts of the litterature, its classical formulation for sums over integers as well as its adelic version. What is its corresponding form for more ...
10
votes
2answers
875 views
Differentiability of Fourier series
Consider the function defined by the Fourier series
$$ f(x;\alpha) = \sum_{n=1}^\infty \frac{1}{n^\alpha} \exp(i n^2 x ) , $$
where $\alpha >1 $.
For what values of $\alpha $ is $f$ ...
0
votes
0answers
66 views
Speed of divergence series
Given a positive sequence $\{a_k\}$ and an $s>0$ such that $\sum_{k=1}^\infty k^sa_k = \infty$, but $\sum_{k=1}^\infty k^s a_k/(\log k)^2 = 1$. For any $t\in [0,b]$, where $b$ is a fixed number, ...
8
votes
0answers
121 views
A basis of the Banach space $L^p(\mathbb T^\omega)$ consisting of characters
Problem: For $1<p<\infty$, $p\ne 2$, has the complex Banach space $L^p(\mathbb T^\omega)$ got a Schauder basis consisting of characters of the compact topological group $\mathbb T^\omega$?
(...
9
votes
1answer
202 views
The discrete Hardy-Littlewood-Sobolev inequality
Let $p>1$, $q>1$, $0<\lambda<1$ be such that
$\frac{1}{p}+\frac{1}{q}+\lambda=2$. Suppose that
$(a_{k})\in \ell^{p}(\mathbb{Z})$ and $(b_{k})\in \ell^{q}(\mathbb{Z})$.
It is known ([1,2,3]...
1
vote
1answer
86 views
Accurate estimate/lower bound for the l2 approximation error of trigonometric polynomial approximation
This is a restated version of my original very broad question.
Let $P$ be probability a measure on an interval $[a,b]$ ($-\infty<a<b<\infty$) that's dominated by Lebesgue measure. Let $\...
2
votes
1answer
134 views
Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames?
I have found some applications of the Frame Theory in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection.
As we ...
2
votes
1answer
107 views
$L^2(X) \cong L^2(X',\xi)$
Recently, I read a notes about Sakellaridis and Venkatesh conjecture. It mentions a technique called "unfolding" and gives an example:
Let X=A\G, X'=N\G, where G=PGL(2), A={
$\left[\begin{array}{...
1
vote
1answer
68 views
Maximum Magnitude Deviation between DFT and DTFT
This is a cross-post from signal processing forum as it was not conclusive.
Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then
$$
X(\omega) = \sum_{n = 0}^...
12
votes
3answers
510 views
Which bounded sequence can be realized as the Fourier Series of a probability measure on the circle?
Given a finite Borel measure $\mu$ on $\mathbb{S}^1 = \mathbb{R}/\mathbb{Z}$, define its Fourier coefficients by
$$ \hat\mu(n) = \int e^{2i\pi nx} d\mu(x) \qquad\forall n\in \mathbb{Z}.$$
Clearly, $(...
0
votes
1answer
95 views
A question about the convolution theorem
I have the following "argument" about Fourier series, which I know is wrong because it yields a ridiculous conclusion. However, I don't know where the mistake is, and need to know which step is the ...
3
votes
1answer
119 views
Does zero Fourier dimension imply there is no Rajchman measure?
Let $K\subset R^d$ be a compact set. It is well known that its Fourier dimension is defined by
$$\dim_F K=\sup\{s\ge 0: \exists \mu \in M_1(K) s.t. \hat{\mu}(x)=O(|x|^{-s/2})\}(|x|\to\infty),$$
where $...
9
votes
1answer
342 views
Fast convolution of sparse functions
Let $F:\mathbb{R}\to \mathbb{Z}$ be a step function with at most $k$ discontinuities, at given rationals $a_1<a_2<\dotsc<a_k$. Let $g:\mathbb{R}\to \mathbb{Z}$ be given as a linear ...
6
votes
3answers
349 views
Connections between martingales and Fourier analysis
I have had this strange feeling recently that somehow, the theory of martingales we study in probability, and the theory of Fourier analysis are very alike. But I am not able to formalize my thoughts.
...
4
votes
0answers
81 views
On smoothness of a function and decay of its Fourier transform
I am not sure that this question is research level, but it was not answered at MSE for several days, so I place it here.
I am interested in a quantitative version of the principle that smoothness of ...
3
votes
1answer
430 views
Functions belong to $L^{\frac{2n}{n+1}}$ whose Fourier transforms are infinite on $S^{n-1}$
I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?
This seems to be a well-known result, but I can ...
1
vote
0answers
73 views
Is the ratio of two distinct zeros of a Bessel function of the first kind an irrational number?
Let $J_1$ be the Bessel function of the first kind with parameter $\alpha=1$. Namely, $J_1$ satisfies the differential equation for $y$ given by $x^2y''+xy'+(x^2-1)y=0$ and $J_1(0)=0$.
Is it true ...
-2
votes
1answer
120 views
Relationship between “Radial” Fourier transform and Fourier transform, especially at infinity
Let $\phi:\mathbb{R}^n \to \mathbb{R}$ be a $C^{\infty}$ function with compact support.
What is the relationship between
$$
\widehat{\phi}(k) = \int e^{-2\pi i x \cdot k} \phi(x) dx, \quad k \in \...
4
votes
0answers
38 views
Multi-parameter stationary phase asymptotic expansion
I am looking for an asymptotic expansion of the oscillatory integral of the form
$$\int_{\mathbb{R}^n}f(x)\exp(i(\lambda_1\phi_1(x)+\dots+\lambda_k\phi_k(x))dx,$$
as $\lambda_i\to \infty$ ...
6
votes
0answers
143 views
Extremal polynomial majorants of $\log{|f|}$: a multivariate extension of a theorem of Carneiro and Vaaler
Carneiro and Vaaler have proved, as an application of their work on Beurling-Selberg extremal majorants, that for any non-zero complex polynomial $f(z) \in \mathbb{C}[z]$, the infimum value of the ...
3
votes
1answer
256 views
Detecting if a series represents a rational function
A lemma by Kronecker states that the series $f(z):= \sum_{i=0}^{\infty} c_i z^i$ represents a rational function if and only if for every $m \gg 0$ the determinant of the following matrix is equal to ...
1
vote
1answer
106 views
Uniqueness of solution to system of integral equations
Given the following system of integral equations for an integrable function $f(x)$:
For all integers $k \ge 1$ holds
$\int_{0}^{2\pi} [f(x)]^k e^{(ikx)} dx = 0$.
If $f(x)$ is real-valued and non-...
0
votes
0answers
78 views
Initial value theorem for Fourier transform
Initial value theorem states that for a bounded function $f(t) = O(e^{ct})$ and an existing initial value, one-sided Laplace transform $F(s) = \int\limits^\infty_{0^-}f(\tau)e^{-s\tau}\mathrm{d\tau}$ ...
1
vote
2answers
173 views
Fourier transform of a generalized function on the plane
Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
$$\frac{1}{x+y^2+i0}?$$
Remark. Equivalent question: consider the Schroedinger equation one the ...
5
votes
0answers
105 views
Convolution theorem on a non-abelian Lie group
Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the ...
11
votes
1answer
390 views
Nonlinear Schrödinger equation with discrete Laplacian
In the paper "Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on $\mathbb{R}^3$" by Colliander, Keel, Staffilani, Takaoka and Tao it is argued in the beginning ...
1
vote
1answer
193 views
Calculus of variation with discontinuous solutions?
I'm thinking of the following question:
Consider a function $f: U\rightarrow\mathbb{R}$ where $U=[0,L_1)\cup(L_1,L]$, and an energy functional $$F=\int_{U}\Big (\frac{\mathrm{d}f}{\mathrm{d}x}\Big)^2\...
7
votes
2answers
170 views
Characterizing pseudo-differential operators as a subalgebra of continuous endomorphisms of tempered distributions
I'm aware that the following question is at best a refined version of at least 2 questions which are already on this site. I think it is justified however in that it is more precise and has some new ...
0
votes
1answer
70 views
The minimum of the maximum of a sequence of sinc functions
I apologise if this is trivial or well known to be impossible:
Can one find a finite set of integers $2\leq a_1<a_2<\ldots<a_m<\infty$
such that for the function defined as
$$
f_{a_1,\...
2
votes
0answers
52 views
Star-convex curve and Fourier series
Let x(t) be a periodic function on [0, 2$\pi$].
I am interested in finding criteria in terms of the Fourier coefficients of x(t), such that the parametric curve $\left\{ x\left( t\right) ,\dot{x}\left(...
1
vote
0answers
72 views
Conditions for Poisson summation (for discontinuous functions)
Let $G$ be an locally compact abelian group with $\Gamma$ a discrete cocompact subgroup. I'm looking for precise conditions by which Poisson summation formula holds. That is, for some function $f$ on $...
13
votes
1answer
437 views
A question on the sine function
The Fejer-Jackson-Gronwall inequality involving the sine function is as follows:
$$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$
Here I ask the ...
4
votes
1answer
171 views
$L^1$ norm of Littlewood polynomials on the unit circle
A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$.
I'm interested in a "good" lower bound on ...
10
votes
0answers
114 views
A combinatorial proof of the Harrow--Kolla--Schulman theorem
Let $Q^n := \{0,1\}^n$ be the Hamming cube with the Hamming metric. (Recall that the Hamming is defined by the distance $d(x,y) := \# \{ i : x_i \neq y_i \}$.
For integers $0 \leq k \leq n$, define a ...
