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14
votes
2answers
713 views

A question in Fourier analysis

I recently came across this problem: Let $T=\mathbb R /2 \pi \mathbb Z$ the circle, with its proabability Haar measure $\mu$. Any integrable function $f : T \rightarrow \mathbb C$ has Fourier ...
5
votes
2answers
424 views

Fourier series representing a continuous function?

This is maybe not really research level, but I have not found anything in the literature, and asking on math.stackexchange wasn't successful either. Fourier series define an isometry $L^2(\mathbb{Z}) ...
10
votes
2answers
358 views

Uncertainty principle on finite groups

For a finite group $G$ with normal subgroup $H$, the induced representation $\text{Ind}_H^G(1)$ decomposes as a sum of irreducibles with the multiplicities equal to the dimensions, because it is is ...
2
votes
0answers
146 views

Fourier : from analyticity to exponential decay ; what prevents optimal decay ?

Hello everyone ! In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for simplicity) : if ...
41
votes
5answers
4k views

Quasicrystals and the Riemann Hypothesis

Let $0 < k_1 < k_2 < k_3 < \cdots $ be all the zeros of the Riemann zeta function on the critical line: $$ \zeta(\frac{1}{2} + i k_j) = 0 $$ Let $f$ be the Fourier transform of the sum ...
0
votes
1answer
122 views

upper bound on product of distances from points on a circle

Let C be a circle of radius 1 in the complex plane with n points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other n points. The goal ...
1
vote
0answers
184 views

An integral with Gamma functions (Part 2)

I was wondering if there is a generalization of the integral discussed here to a case like, \begin{equation}\int \frac{d^dq}{q^{\nu_1}\vert \vec{q} \pm \vec{k}_1\vert ^{\nu_2}\vert \vec{q} \pm ...
0
votes
1answer
245 views

Littlewood-Paley theory and norm estimation

In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1. It is claimed that Lemma 2 is ...
1
vote
0answers
223 views

A problem about Joint sine and cosine fourier transform

There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
2
votes
2answers
276 views

A sufficient condition for a probability measure to have compact support

Consider a probability measure $\mu$ on, let's say, $\mathbb R$. Is there a necessary and sufficient condition so that $\mu$ has compact support $Supp(\mu)$ ? I agree this question is too vague, ...
2
votes
0answers
158 views

On a differential inequality

The question has probabilistic origins, but it would take too long to elaborate. $\newcommand{\ii}{\boldsymbol{i}}$ $\newcommand{\bR}{\mathbb{R}}$ $\newcommand{\eS}{\mathscr{S}}$ Fix a nonnegative ...
1
vote
1answer
264 views

Pointwise convergence of double Fourier series

I'm looking for references/theorems that deal with the pointwise convergence of double Fourier series expansions for a particular function. Let $D \subseteq [-\pi, +\pi]^2$ be an arbitrary set of ...
2
votes
2answers
371 views

Schönhage–Strassen algorithm

After brief intro to Fourier series, CFT, DFT and their basic properties I enjoyed implementing forward and backward FFT algorithm in complex numbers. I was happy to, at least, have an idea how is it ...
17
votes
2answers
604 views

Image of L^1 under the Fourier Transform

The Fourier Transform $\mathcal{F}:L^1(\mathbb{R})\to C_0(\mathbb{R})$ is an injective, bounded linear map that isn't onto. It is known (if I remember correctly) that the range isn't closed, but is ...
1
vote
1answer
246 views

Is this set a Riesz Basis of $L^2(0,\pi)$

Let $A=\{\sqrt{2}\sin(\sqrt{n^2+a} \pi x)\} _{n=1}^\infty$, where $a$ is a positive real number. Is $A$ a Riesz Basis of $L^2(0,1)$?
3
votes
1answer
144 views

Using Fourier Transform to speed up calculation of forces following an inverse square law

Suppose I have $n$ electric point charges in, say, two dimensions. Is there any algorithm (and I have a hunch that it might be related to the Fourier transform) to compute the net forces that act on ...
1
vote
1answer
102 views

Almost Everywhere Convergence of Walsh Series of $L^2$ functions

I am currently reading the Hunt's papar (http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0655.0662.ocr.pdf), and am wondering if there is any notes which presents his argument more ...
12
votes
1answer
633 views

On the $L^1$-norm of certain exponential sums.

I am stuck with an elementary-looking problem, which does not belong to my usual field of research so I eventually decided to ask it on MO. Let $S$ be a finite set of integers. For $P$ a subset of ...
2
votes
1answer
232 views

About the boundedness of a multiplication operator.

Let be $f$ a $2\pi-$periodic function and $\hat{f}(k)=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}dx$. Consider the operator: \begin{equation} Tf(x)=\sum_{k\in\mathbb{Z}}sign(k)\ \hat{f}(k)\ e^{ikx}. ...
6
votes
1answer
260 views

For which metric measure spaces is the Hardy-Littlewood maximal operator not of weak type (1,1)?

Let $(X,d,\mu)$ be a metric measure space, i.e. $\mu$ is a Borel measure on the metric space $(X,d)$. I'll denote the Hardy-Littlewood maximal operator - either centred or uncentred, I don't mind ...
3
votes
1answer
1k views

The Paley-Wiener theorem and exponential decay.

Consider a function whose Fourier transform is supported on a half-ray: $$ A(t)=\int_0^\infty \omega(E) e^{-iEt}d E, $$ where I can suppose $\omega(E)\geq 0$ and any suitable regularity conditions on ...
0
votes
1answer
328 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 ...
4
votes
1answer
351 views

Inequality on Trigonometric polynomials

My question comes from trying to understand a technical step in this paper by Bourgain. Let $R,L$ be positive integers and let $f(x)=\sum_{|n|\leq RL}a_ne^{2\pi inx}$ be a trigonometric polynomial. ...
5
votes
2answers
278 views

Efficient computation of “discrete infimal convolution”

This question arises from an application to graphical models in probability theory, but I have abstracted that part out so only algebra remains. Let $\mathbb{R}$ denote standard field of real numbers ...
8
votes
0answers
233 views

Uncertainty principle in Entropy terms

Math Questions: Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm $ ||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2}, $ and Fourier transform $ (F\psi)(\xi) = ...
6
votes
4answers
864 views

fourier analytic proofs

While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I ...
3
votes
2answers
285 views

Sufficient condition for $L^p$ multiplier on the torus

Hello, I am searching for a reference of an " easy " sufficient condition insuring that a bounded sequence $(b_{\mathbf{n}\in\mathbb{Z}^d})\in\ell^\infty$ defines a bounded operator from ...
6
votes
3answers
1k views

Fourier Coefficients and Hölder Continuity

Suppose we are given the Fourier coefficients of an $L^2$ function on the circle. Are there necessary and sufficient conditions on the coefficients that allow us to determine that $f$ is Hölder ...
4
votes
1answer
257 views

Is there a good way to estimate the Fourier transform of $\frac{1}{\lambda-iP(\xi)}$

Assume that P is a real valued strong elliptic polynomial, then what do we know about the following $$ K(\lambda,x)=\int{\frac{e^{ix\xi}}{\lambda-iP(\xi)}}d\xi,\quad \lambda\in \mathbb{R}\0 $$ The ...
0
votes
2answers
741 views

Derivative indicator function

I am wondering what is the derivative of the following function with respect to $x(t)$ in sense of distributions. $$ I\left(\int_0^t x(\tau)d\tau \leq c\right) $$ where $I$ is the indicator function ...
6
votes
1answer
302 views

faces in the discrete cube

This arose from a question Gil Kalai asked about a problem I posed involving the Fourier transform on the discrete cube. Maybe it is more tractable. I'm afraid I'm not sure how to do this kind of ...
30
votes
0answers
1k views

Fourier transform on the discrete cube

Notation: identify an element of $\{-1,1\}^n$ with the set $S \subseteq \{1, \ldots, n\}$ on which it takes the value $-1$. The following is an asymptotic question. "Close to one" means "more than ...
1
vote
1answer
98 views

Asymptotics of a one-parameter family of Schwartz functions

For $\tau > 0$ define $\theta_{\tau}(x) = e^{\tau(x-x^{2})}$. I am curious about the asymptotics of $\widehat{\theta}_{\tau}(\tau)$, that is $\int_{\mathbb{R}} e^{\tau(x - x^{2})}e^{-2\pi i ...
0
votes
0answers
287 views

Help with an irregular integral

I am looking for help with doing the following integral : $$\frac{1}{2\pi i}\int_{1}^{\infty}\ln\left(\frac{1-e^{-2\pi i x}}{1-e^{2\pi i x}} \right )\frac{dx}{x\left(\ln x+z\right)}\;\;\;\;z\in ...
2
votes
2answers
153 views

Taylor expansion convergence relation to power-spectrum

Is there some connection between the power-spectrum of a real function $f:\mathbb{R}\to\mathbb{R}$ (that is, its Fourier transform) and the convergence radius of its Taylor expansion around arbitraty ...
2
votes
1answer
158 views

Approximation power of wavelets

The Wikipedia article on Wavelet Transform states that: Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, ...
1
vote
3answers
524 views

Fourier Transform, for entire function

On THIS site, Alexandre used Fourier transform to solve the problem. If we use Fourier transform, how to define it to ensure any entire function has a FT? Classical FT is defined by $$ ...
3
votes
3answers
1k views

Integral kernel for the resolvent of the laplace operator

Consider the Laplace operator defined in the biggest possible subset of $L^2(\mathbb{R}^2)$ and let $z \in \mathbb{C}\backslash\mathbb{R}$. Therefore $z \notin \sigma (\Delta)$ the spectrum of ...
7
votes
1answer
476 views

Questions about the Fourier expansion of $e^{iz\cot(x)}$

Referring to a question I posted on MS, I post it here, as I didn't get an answer: By analogy with the Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form : ...
2
votes
1answer
224 views

Fourier transform of a particular function

In order to estimate the fundamental solution of some particular types of differential operators,I need estimates on some kind of oscillatory integrals.For simplicity, consider the Fourier transform ...
0
votes
2answers
161 views

Convolutional Equation

Hi, Given C>0. Let $f,g,h$ be $L^2$ functions such that $f,g,h$ have a compact and finite-measure support, and $f*f(x)=g*g(x)=h*h(x)=f*g(x)=g*h(x)=f*h(x)=0$ (where $*$ is the convolution) for all ...
5
votes
1answer
327 views

Weakest assumption for pointwise convergence of Fourier series

This should be a quick one, but so far books, my brain, and the internet have not produced a clear answer. Or maybe it's subtle and exposes a weakness in my understanding of FS! Suppose ...
6
votes
3answers
770 views

Property/Relations using Fourier series/transform, which give complete information about all the jump singularities of a function.

Consider a function which has only jump singularities of the form of the function itself or one of its derivatives jumping. Now let $\hat{f}(k)$ be its Fourier transform/series. We know the decay of ...
3
votes
1answer
406 views

Uncertainty principle (really for Mellin, but never mind that!)

Is there a smooth funtion $f:\mathbb{R}\to \mathbb{C}$ such that (a) $f(x)$ decreases faster than $e^{-e^x}$ when $x\to \infty$, (b) $\widehat{f}(t)$ decreases faster than $e^{-|t|}$ when $t\to ...
2
votes
0answers
328 views

How well can you approximate a function by a band-restricted function?

Say I have a compactly supported $C^1$ function $f:\mathbb{R} \to \mathbb{R}$. Let $R>0$. Let $\nu$ be some reasonable measure on $\mathbb{R}$ -- take, for instance, (a) $d\nu(t)=dt$ or (b) ...
2
votes
2answers
394 views

Does the Fourier series of an $L^1$ function converge to the function *weakly* in $L^1$?

Let $f$ be a periodic $L^1$ function, and $S_n[f]$ the $n$-th partial sum of its Fourier series. I am aware that $S_n[f]$ might not converge toward $f$ in $L^1$ (i.e., in norm). However, does it at ...
1
vote
2answers
208 views

Berry Esseen inequality for multidimensional distributions

The classical Berry-Esseen theorem asserts that if $f$ and $g$ are the characteristic functions of two distribution functions $F(t)$ and $G(t)$ respectively then with $T$ arbitrary $$ \sup_{t \in ...
2
votes
2answers
361 views

Understanding Discrete Cosine Transformation

I'm currently working on some software and a key component is 2D DCT. But my question is more general, as I'm trying to understand the DCT in general, let's say from engineers point of view. For ...
0
votes
2answers
280 views

Estimates for Fourier transform

Let $f(x)$ be a smooth function from $\mathbb{R}$ to $\mathbb{R}$. Let $\mu$ be a compactly supported Borel measure (not necessarily positive) on $\mathbb{R}^n$. Define $$ \tilde{\mu}(\xi) = \int ...
6
votes
1answer
233 views

Minimizing |FT(X)|_{\infty} by permutation of X_i - question on Fourier transform related to engineering problem (peak factor of OFDM system)

Consider vector X =( X_1 ... X_N), consider the discrete Fourier transform $Y=F(X)$. I am interested to minimize $|Y|_{\infty}$, by permutation of numbers X_i, how to do it ? Here $|Y|_{\infty}$ is ...