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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
290 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
157 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
164 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
563 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
554 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
228 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
163 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
334 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
791 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
408 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
410 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
231 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
406 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
286 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
234 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 ...
7
votes
1answer
708 views

Norm concentration of trigonometric polynomials - Uncertainty principle

Hi all, I am interested in the following question (which is quite similar to one I posed a long while ago): Let $P_{N}(t)=\underset{k=-N}{\overset{N}{\sum}}c_{k}e^{ikt}$ be a unit norm trigonometric ...
5
votes
1answer
273 views

What properties should a transform have to deserve the descriptor Fourier?

Two MO questions, "Heuristic behind the Fourier-Mukai transform" and "Explaining Mukai-Fourier transforms physically," compel me to ask these two related questions: 1) What properties do you feel are ...
1
vote
0answers
80 views

Decay rate of Discrete Prolate Spheroidal Sequences in frequency

What is the decay rate of DPSS sequences in frequency? Consider an interval $T\subset\mathbb{Z}$ of length N in time. Consider another interval $[-W,W]$ in frequency with $W<1/2$. Let $\phi_0$ ...
0
votes
1answer
141 views

On a limit at the boundary of $\mathbb{D}$ related to complex and harmonic analysis

Let $p(z,t)=\frac{1}{2\pi}.\frac{1-|z|^2}{|z-t|^2}$ be the Poisson kernel on the open unit disk $\mathbb{D}$, fix $0<\alpha<1$ . Let $a\in \partial\mathbb{D}=S^1$ be fixed. Then my question is : ...
6
votes
1answer
238 views

A fourier series related to spin Chern numbers almost commuting matrices

Let $$ f(x)=\sin(x)\sqrt{1+\cos^{2}(x)+\cos^{4}(x)}. $$ In my study of almost commuting unitary matrices, $U$ and $V$, I have need for a bound like $$ \left\Vert ...
2
votes
1answer
208 views

Bound on trigonometric sum

I want to show that there is some $\gamma(n)=o(n^{-1})$ and some $C(n) \to -\infty$ such that for $\gamma \leq \theta \leq \pi$ we have $\sum_{k=1}^n -1+\cos(k\theta) \leq C(n)$. If we rewrite this ...
28
votes
7answers
2k views

How should an analytic number theorist look at Bessel functions?

(And a related question: Where should an analytic number theorist learn about Bessel functions?) Bessel functions occur quite frequently in analytic number theory. One example, Corollary 4.7 of ...
1
vote
2answers
281 views

Characteristic Function of a Non-negative Random Variable Evaluated at a Complex Value

Suppose we have a non-negative random variable $X$ with density $p(x)$,and its characteristic function, evaluated at a complex number $z$, being $\phi(z)=E[e^{z X}]=\int_{0}^{\infty}e^{zx}p(x)dx$. It ...
1
vote
1answer
532 views

Estimating parameters of a mixture of normal distributions.

I want to estimate the parameters $\mu_i$ and $\sigma^2_i $ of a countable mixture of Gaussians with assumed equal weights, variance and identically spaced means. I intially thought that the Fourier ...
5
votes
1answer
327 views

Kahn-Kalai-Linial for intersecting upsets

Is there any known improvement on the Kahn-Kalai-Linial inequality (on the influences of boolean functions) in the special case in which $f$ is the indicator function of an intersecting monotonic set ...
3
votes
1answer
1k views

History of the Fourier transform

Does anyone know a good book or article on the History of the Fourier transform? It's first appearance (of the transform) and use in particular? Or at least some source with some historical ...
0
votes
1answer
765 views

Frequency calculation using fourier transform [closed]

How to calculate the frequency of an audio file using Fourier Transform
3
votes
1answer
333 views

Does anybody know an estimation of L4 norm of fejer kernel ?

Hi, I need an estimation or an exact closed form expression for the following integral $\int_{0}^{2\pi} K_N^4(s) ds $ where $K_N(s)= \frac{1}{N2\pi} (\frac{sin(Ns/2)}{sin(s/2)})^2$, the Fejer ...
6
votes
2answers
606 views

What is the simplest oscillatory integral for which sharp bounds are unknown?

I have either heard or read that sharp asymptotics and bounds for oscillatory integrals of the form $ \int e^{i \lambda \Phi(x)} \psi(x) dx \quad \lambda \to \infty $ are unknown when the critical ...
1
vote
1answer
311 views

Generalized Friedrichs Lemma

Taylor's PUP book on pseudodifferential operators in II.7 has an extension of the pseudodifferential version of Friedrichs' lemma to generalized Friedrichs' mollifiers $J_\epsilon$ on a compact ...
0
votes
1answer
916 views

In Fourier Transforms: Positive Definite Functions, Bochner's Theorem, and Derivatives

I've been reading about Bochner's Theorem lately, but when I apply it to the derivative of a function, I seem to get a contradiction with the theorem. "Bochner's theorem states that a positive ...
0
votes
2answers
250 views

Convergence of Fourier series for $C^p$ functions

Let $f \in C^p[0,2\pi]$ and periodic. Denote $\omega_p$ as the moduli of continuous of $f^{(p)}$. Then $ |f - S_Nf| \le K \frac{\log{N}}{N^p}\omega_p(2\pi/N), $ where $S_N$ is the Fourier partial sum ...
2
votes
0answers
358 views

L^1-convergence of convolution exponential

Consider a differential equation \begin{eqnarray*} \frac{d}{d\tau}q^{\tau}\left(x\right)=Aq^{\tau}\left(x\right)+h\star q^{\tau}=Aq^{\tau}\left(x\right)+\int ...
0
votes
0answers
87 views

Convergence rate of fourier partial sum

Hi there, I am looking for the proofs of convergence rate in Section 4 of the following artical http://en.wikipedia.org/wiki/Convergence_of_Fourier_series in wikipedia. Could someone please point ...
10
votes
1answer
1k views

A Fourier-analytic inequality used by Jean Bourgain

I am currently reading Jean Bourgain's 1986 paper A Szemerédi type theorem for sets of positive density in $R^k$ and would appreciate some help in understanding a Fourier-analytic estimate used in ...
1
vote
1answer
486 views

A bijective correspondence induced by Fourier Transform

Let $G$ be a discrete Abelian group and denote by $\widehat G$ the (compact) Pontryagin-Van Kampen dual of $G$. I was reading in a paper of Justin Peters that Fourier Transform induces a bijection ...
3
votes
1answer
359 views

Ask for theory about the weighted L^2(R^d) space.

Dear MOs, I am now considering the following norm: $$ ||f||_{H}^2 := \iint f(x) H(x,y) f(y) d x d y\:. $$ where the integral is over the whole space $R^{2d}$ and $H(x,y)$ is some non-negative ...
2
votes
0answers
228 views

Continuous positive-definite function with prescribed support

Consider a (locally) compact Abelian group G and a compact neighborhood $C$ of the identity ($0$) of $G$. Is it possible to find a non-trivial (i.e., $\phi(0)\neq 0$) continuous, positive (with real ...
0
votes
1answer
297 views

Fourier coefficients of a rational function

Any ideas how to compute or to approximate integral $$\int_{0}^{1}\frac{(x+a)^{2q}(x+b)^{2q}}{(x-1)^{4q}+x^{4q}}\exp({-2\pi i x y})dx$$ where $q \in \mathbb{N}$ and $a,b =-2,-1,0,1$, $y \in (0,1)$
1
vote
2answers
277 views

Is it possible that the intersection of two nest algebras contains only scalars?

Dear all, I really want to know the answer of the following question. I would appreciate any help. Assume H is a separable Hilbert space, is it possible to find two nests N1, N2 such that the ...
0
votes
0answers
121 views

Finding peaks and determining noise

Hello , Im having one matrix which is product of two FFT transforms of one fits image ( astronomical image ). In that matrix you could find 3 peaks. One largest in center, and two around central ...
52
votes
4answers
3k views

How Does My Radio Work?

Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical. I live in an area with $n$ AM radio stations and $m$ FM radio stations. AM station ...
2
votes
2answers
443 views

Positive Fourier coefficients

Hi all, Is there any general way to construct a smooth 2pi periodic function which vanishes on an interval and has positive Fourier coefficients? And if that was too specific I can make this more ...
7
votes
6answers
908 views

Non-commutative Fourier transform

What is a good reference for introducing non-commutative fourier transform for Electrical Engineers and Theoretical Computer Scientists in an explicit way?
2
votes
1answer
2k views

FFT and Butterfly Diagram

Wikipedia presents butterfly as "a portion of the computation that combines the results of smaller discrete Fourier transforms (DFTs) into a larger DFT, or vice versa (breaking a larger DFT up into ...
4
votes
2answers
1k views

How the Fast Fourier Transform got its name

In 1971, T.S. Huang published a paper in IEEE Computer, May-June, pp.15, called How the Fast Fourier Transform Got its Name, available here. At the bottom of the paper, he wrotes: "The Chinese ...