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7
votes
1answer
680 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
268 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
78 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
139 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
237 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
267 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
3answers
508 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
319 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
963 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
712 views

Frequency calculation using fourier transform [closed]

How to calculate the frequency of an audio file using Fourier Transform
3
votes
1answer
323 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
595 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
290 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
878 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
245 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
344 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
86 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
473 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
352 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
221 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
275 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
119 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 ...
51
votes
4answers
2k 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
431 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
873 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 ...
0
votes
2answers
562 views

Understanding the inverse Laplace transform of a function with essential singularities

I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in ...
9
votes
5answers
697 views

Eigenvalues of a sum of Hermitian positive definite circulant matrix and a positive diagonal matrix

Suppose I have a real $n\times n$ matrix $\mathbf{C}$ that is Hermitian, positive-definite, and circulant. We know that its eigenvalues $\{\lambda_0,\ldots,\lambda_{n-1}\}$ are extraordinarily nice ...
11
votes
4answers
732 views

Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
6
votes
2answers
583 views

Number of integers coprime to l

A long time ago I've seen a paper considering, given $\ell$ fixed, estimates for $$ \sum_{n \leq x, (n, \ell) = 1} 1 $$ Of course, this is easy to estimate with a trivial error term of ...
8
votes
1answer
697 views

Steinmetz, Laplace and Fourier Transforms

I am looking for references on Steinmetz Transform and its relation with Laplace and Fourier Transforms. There is an Italian Wikipedia page about this topic but with no references.
4
votes
1answer
514 views

Max of Fourier transform?

Let f be a real-valued function (or distribution) on $\mathbb{R}$. (You can assume it is nice in one way or another.) What would be some practical ways to bound $\max_{\alpha \in \mathbb{R}} ...
1
vote
2answers
1k views

A question about normalization in the Fourier transform

Is there a reason why most people use the normalization $$ \hat{f}(x) = \int_{-\infty}^{\infty} e^{- 2 \pi i t x} \cdot f(t) d t $$ instead of $$ \hat{f}(x) = \int_{-\infty}^{\infty} e^{2 \pi i t x} ...
0
votes
1answer
479 views

how to prove the range of a closed linear operator is closed ?

The closed range theorem tells us that given two banach spaces X,Y,and a closed densely defined linear operator T:$X \to Y$. We have the following equivalence $R(T)$ is closed in $Y \iff R(T^{*})$ is ...
0
votes
1answer
276 views

higher fourier analysis: splitting periodic parts and 'nilsequence' noise?

In Fourier Analysis notes by Terence Tao (and "algebraic" version by Balasz Szegedy) there seem to be two versions of Fourier analysis floating around: To decompose periodic functions $L^2(S^1)$ ...
3
votes
1answer
578 views

Fourier and Bessel

Oliver Heaviside, on page 387 of Electrical Papers, Vol. I, Macmillan and Co., 1892, available here, writes $$v = 1 - \frac{n^2r^2}{2^2} + \frac{n^4r^4}{2^2 4^2} - \frac{n^6r^6}{2^24^26^2} + \ldots ...
4
votes
3answers
826 views

History of the Sampling Theorem

In January, 1949, Shannon publishes the paper Communication in the Presence of Noise, Proc. IRE, Vol. 37, no. 1, pp. 10-21, available here, which establishes the Information Theory. In this paper, the ...
2
votes
1answer
179 views

Function with Fourier coefficient of order $o(n^{-m})$

Let $(a_n)$, $(b_n)$ be the Fourier coefficients of a periodic, locally integrable function $f: \mathbb{R} \rightarrow \mathbb{R}$. Assume that $n^m a_n, n^m b_n \rightarrow 0$ when $n \rightarrow ...
3
votes
0answers
198 views

The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwinig operator $V_k$ on $C(\mathbb{R}^d)$ is defined by: $$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$ where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...
3
votes
1answer
593 views

karhunen-Loeve expansion of Poisson process

Let $X_t, t\geq 0$ be a Poisson process with rate parameter $\lambda$. Compute the karhunen-loeve expansion of $X$ in interval [0, T]. How about the KL expansion of the centered process $X_t-\lambda ...
7
votes
2answers
660 views

Analogues of the Riemann-Roch Theorem

In his thesis, Tate derives a Poisson formula on the adeles and a theorem which he calls the "Riemann-Roch Theorem". More specifically, given a continuous, $L^1$ function $f$ on the adeles such that ...
1
vote
3answers
1k views

General Procedure for Inverse of an Integral Transform

Is there a general inversion formula or procedure for an integral of the form (where f is the function being transformed and g depends on the type of transform) $\int^{a}_{b} f(x) g(x,\xi) dx $ ? ...
0
votes
1answer
176 views

Decay of Sobolev fractional norms outside a ball

Suppose you have a function $u$ that lies in the (fractional) Sobolev space $H^s$, with $0<s<1$. Take a smooth cut-off $\varphi$ such that $\varphi(x)=0$ if $|x|>R+1$. What can be said ...
6
votes
1answer
549 views

Fastest decay of Fourier Transform for Generalized Functions of compact support

What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential ...
2
votes
1answer
91 views

Characterizing Identification of a Deconvoluted Density Without Independence

Hey, I'm new here and this is my first attempt at a question. My background is in econometrics, so I apologize in advance if I use unfamiliar notation or display ignorance of important work. Suppose ...