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0
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1answer
684 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
593 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 ...
0
votes
1answer
283 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
848 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
468 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
351 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
219 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
270 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
276 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
426 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 ...
5
votes
4answers
738 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
551 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
693 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
713 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
578 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 ...
7
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1answer
689 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
495 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
465 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
573 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
813 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
195 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
590 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
655 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
543 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 ...
0
votes
0answers
167 views

Fourier transform in n-dim Euclidean and Minkowski space

As far as I understood, the Fourier decomposition of a function $\boldsymbol{F}\colon\mathbb{R}^{n}\to\mathbb{R}^{m}$ where $\mathbb{R}^{n}$ is endowed with the Euclidean inner product ...
6
votes
1answer
446 views

Fourier transform

Does anyone know what the Fourier transform (in the sense of distributions) of $$ f(x) = (x^2 - 1)^{1/2}x, \quad |x|\ge 1, $$ and $f(x) = 0$ otherwise, is?
1
vote
1answer
195 views

Gausian distributions in the Frequency domain

I have read in many texts that the Fourier Transform of a Gaussian is yet another Gaussian, however how does the mean and standard deviation change? Also if we convolve a Gaussian with itself then ...
2
votes
0answers
78 views

Searching for inequalities relating a convolution-type integral of functions of modulus less than but close to one.

Suppose $f(x,y)$ and $g(x,y)$ are both measurable functions from $[0,1]\times[0,1]\to \mathbb{C}$ with $|f|,|g|<1$, and let $h(x,y)=\int_{0}^1 f(x,z)g(z,y) \ dz$. (So $|h(x,y)|<1$ also.) ...
2
votes
0answers
200 views

Continuous function sort

If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
6
votes
3answers
404 views

Simple closed curves and the coefficent of $\exp(i\theta)$ in the associated Fourier series

Given a continuous map $f:S^1\to \mathbb{C}$ from the unit circle to the complex numbers, one can form its Fourier series $\sum_{n=-\infty}^\infty a_n\exp(in\theta)$. I want to stick with those $f$ ...
3
votes
3answers
266 views

Asking for a Fourier inverse transform, which is related to stable laws

Dear friends, Denote the function $$ G_a(x)=\mathcal{F}^{-1}\left(e^{-|\xi|^a}\right)(x)= \frac{1}{2\pi}\int_R \exp\left(-i x \xi - |\xi|^a\right)d\xi\;. $$ It is well known that if $a\in ]0,2]$, ...
3
votes
2answers
2k views

How does the Laplace Transform work for circuit analysis? [closed]

I would like to understand how signals transformed from the time domain to the frequency domain for algebraic manipulation, can be transformed back to give solutions in the time domain. Knowing how to ...
2
votes
1answer
110 views

uniform bit generator

I'm trying this question for days but no luck, if someone can give me a lead or an article that solves this.. it would be great: A uniform bit generator is a function $f:\{0,1\}^n \times \{0,1\}^n ...
3
votes
4answers
766 views

Quick computation of the Pontryagin dual group of torus

I'm looking for a quick way to compute the Pontryagin dual group of the n-dimensional torus $\mathbb{T}^n$ (with $\mathbb{T} := \mathbb{R} / \mathbb{Z}$). The only way I know is from "Dikran Dikranjan ...
3
votes
3answers
601 views

Peter-Weyl theorem as proven in Cartier's Primer

I'm reading Pierre Cartier's A primer of Hopf algebras to educate myself. In its subsection 3.3 (which doesn't need any Hopf algebra theory), he sketches a proof why compact Lie groups are algebraic. ...