Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Are DFT(x) and IDFT(x) complex conjugate?

I was playing a little with an FFT program I downloaded from the web, taking its source code as a basis for some experimentation. After reading a few texts on DFT/FFT, I was a little confused as to ...
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274 views

How to show integral of different order Hankel transformed functions are equal?

Say I have a function $p_v(r) \in L^2(\mathbb{R})$ given by $$p_v(r) = \int_0^\infty P(k) J_v(rk)\,k\,dk$$ From mucking around in MATLAB it seems the following is true: $$\int_{r=0}^\infty ...
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How do we know the complex exponentials “span” the set of all real functions?

So, we know if $L^2 (0,2\pi)$ is the space of all $2\pi$ periodic square-integrable functions, ie all functions that have finite energy: $$ \int_0^{2\pi} |f(x)|^2dx < \infty $$ Then those ...
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252 views

Is there an autocorrelation function with a constant integral whose absolute value integral diverges?

Suppose a function $g:\mathbb{R}\rightarrow\mathbb{R}$ such that: $|g(x)|\leq g(0)$; $g(x)=g(-x)$, i.e. $g(x)$ is even; $\int_{-\infty}^{\infty}g(x)dx=C$; There exists a Fourier transform of ...
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A simple application of Hölders inequality (I think)

I'm reading a paper where the following inequality appears. $$ \| \widehat{f} \|^2_{L^2(d\mu)} \leq \| f \ast \widehat{\mu} \|_p \| f \|_{p^\prime} $$ where $f$ is a real-valued measurable function on ...
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155 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 ...
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98 views

Independence of Random Variables (kernel ICA)

In the paper Bach, F. R., & Jordan, M. I. (2002). Kernel Independent Component Analysis. Journal of Machine Learning Research, 3(1), 1-48. doi:10.1162/153244303768966085 I stumpled upon ...
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328 views

Using Bohr sets to locate arithmetic progressions

I've just started to read about additive combinatorics and I'd like to know how I can use Bohr sets to make a statement about arithmetic progressions in a given subset $A$ of an Abelian group $Z$ (the ...
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756 views

Is periodogram the same as DFT?

Is periodogram the same as DFT? What is the difference? http://en.wikipedia.org/wiki/Periodogram
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113 views

Fourier Series of $\sin^k(x/2)$

I'm stuck on a seemingly simple problem: What is the fourier Series for $\sin^k(x/2)$? I've tried Mathematica with no luck. Thanks for your help!
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185 views

fourier transform of $\operatorname{sinc}$ function

I have to do the fourier transform of this signal $\left(\frac{1}{10}\right)\operatorname{sinc}\left(\frac{t}{10}\right)$ where sinc function is defined as $\frac{\sin(\pi x)}{\pi x}$. the transform ...
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3answers
360 views

convolution of signals

I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals: $$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$ where $u(t)$ is ...
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1answer
577 views

Derivation of the fourier transform of $x^n f(x)$

Can anyone point me to a derivation of $x^n f(x)$? I know that the answer is $(i)^n$ times the $n$-th derivative of the transform of $f(x)$, but I've searched for a derivation and can't find it.
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707 views

Fourier series /spectrum of different cosine functions

I was given the following task. b) In this task you will concatenate the seven cosines from task a) into one 7 sec long vector. To concatenate vectors in MATLAB use: x=[x1 x2 x3 x4 x5 x6 x7]; ...
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2answers
134 views

Summation of a complex series

Is there a way to perform the finite sum $\sum_{m = 1}^n \exp(2 \pi i k (\sqrt5) ^m)$?, m even. I am trying to show a specific sequence is not equidistributed, and so I'd like to show that Weyl's ...
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2answers
234 views

An example of a “pathological” power-spectral density function?

Suppose that we are given a wide-sense stationary random process $X$ with autocorrelation function $R_X(t)$. Power spectral density $S_X(f)$ of $X$ is then given by the Fourier transform of $R_X(t)$, ...
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Is a Fourier transform a change of basis, or is it a linear transformation?

I've frequently heard that a Fourier transform is "just a change of basis". However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in ...
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Orthogonality relations of Characters

Could somebody please help me understand the jump from Proposition 10 to Proposition 11 in the following http://www.ms.uky.edu/~pkoester/research/charactersums.pdf Note: The orthogonality relations ...
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223 views

A function in BMO space

Let $\psi:[0;1]\to\mathbb R$ is a nonnegative measurable function. Let $b_d(x)=1_{B(0,1)}\cdot{\rm sgn}(\sin (\pi d|x|))$, where $d\in\mathbb N$. Here $1_{B(0,1)}$ is the charateristic function of the ...
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Fourier transform of $ (t^2-1)^n(\operatorname{sign}(t-1)-\operatorname{sign}(t+1)), $

I have trouble with finding the Fourier Transform of the following function: $$ (t^2-1)^n(\operatorname{sign}(t-1)-\operatorname{sign}(t+1)), $$ where $n\in N$. I know that the answer involve so ...
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1answer
85 views

Fourier transform of function decaying at $ae^{-bx^2+cy^2}$

I'm a bit stumped on a problem. The problem is as follows: Suppose $f(z)$ is entire and $|f(x+iy)|\leq ae^{-bx^2+cy^2}$ for $a,b,c>0$. If $\hat{f}(z)=\int_{-\infty}^\infty f(x)e^{-2\pi ixz}dx$, ...
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Clear explanation of heaviside function fourier transform

I know that fourier transform of Heaviside function is : $\hat{H}(x) = \pi \delta(\omega) + i (v.p. \frac{1}{\omega})$ How can i proof this result?
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154 views

Boundedness of supremum of an Integral operator

I am trying to find an $L_2$ - bound on a certain class of operators, and on my way I produced an estimate for which I need to show that \begin{equation} \sup_{x \in \mathbb{R}^n} \, ...
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Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
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Conceptual/Graphical understanding of the Fourier Series.

I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ...
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203 views

Showing $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2}=\frac{\pi^2}{\sin^2(\pi z)}$

I'm doing a homework problem, and so far I've proved $$\sum_{n=-\infty}^\infty \frac{1}{(z+n)^k}=\frac{(-2\pi i)^k}{(k-1)!}\sum_{m=1}^\infty m^{k-1}e^{2\pi imz}$$ for $k$ an integer $\geq 2$ and ...
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2answers
945 views

Computing a convolution using FFT

I have two sequences of the same length, $(x_i), i=1, 2, \ldots, N$ and $(y_i), i=1, 2, \ldots, N$ and a function $K(t) = -t \times \exp(-t^2 / 2)/ \sqrt{2 \pi}$. I need to compute the following ...
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169 views

the integral of the inverse of a Fourier series

Let $\{a_h\}$ be a double-sided complex sequence such that $\sum_{h=-\infty}^{\infty} |a_i| <\infty$ with $a_{0}\neq0$. Set $f(x) := \sum_{h=-\infty}^{\infty} a_h \exp(ixh)$ and assume that ...
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Fourier transform on $1/(x^2+a^2)$

I'm reading a book to review Fourier transforms, and I came across the following example, which is here: ...
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85 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
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232 views

An elegant non-technical account on the work of Joseph Fourier.

It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
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DTFT of a triangle function in closed form

I am sampling a continuous signal $x_c(t)$ that follows a triangle function in the time domain, meaning: $$x_c(t)=\left\{\begin{array}{rl}1-|t/a|,&|t|<|a|\\ ...
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2answers
679 views

Which function's Fourier transform is the function itself?

We know that the Fourier transform of a Gaussian function is Gaussian function itself. Can anyone give one or more functions which have themselves as Fourier transform?
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363 views

Fourier matrix - multiplicity of eigenvalues?

This question is Miscellaneous Exercise M.10 in Chapter 8 (Bilinear Forms) of Artin's Algebra. (The sentences in italics are due to me.) The row and column indices in the $n \times n$ Fourier ...
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117 views

How can one prove that integrating $\cos{(ix)}\cos{(jx)}$ cancels in Fourier Analysis?

This portion of the Wikipedia entry on Fourier Analysis details a formula, and later says that the terms for $j \ne k$ vanish. Could someone please provide a proof of this? I actually would like to ...
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432 views

Dirac delta forcing of a harmonic oscillator

Is it possible to solve this differential equation: $$\ddot{x}(t)+\omega^2x(t)=k\delta(t)$$ where $k$ is a constant and $\delta(t)$ the Dirac delta function? Is it possible alternatively, to know ...
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1answer
997 views

Computing Coefficients of Complex Form Fourier Series

I am having some trouble knowing how to correctly start a problem of finding the Fourier Coefficients using complex exponential form. The problem is given below: $$g_1(t)=\begin{cases} 1,~~\qquad ...
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2k views

Fourier transform of the characteristic function

My qustion is about the Fourier transform of the characteristic function $\chi_{[0,1]}$. How can I find what it is? The problem is I got something really messy, so I think I didn't get it right.
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Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
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How to do a statistical analysis?

I am sorry for my profound knowledge of statistics and for this candid question. Your help is valued. I have the following data. Data1 (stress-I):: 24 35 53 15 40 37 58 11 ...
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216 views

Do Fourier transforms of $\min$ and $\max$ exist (in closed form)?

I am wondering if there are Fourier transforms of $\min(x,a)$ and $\max(x,a)$ functions. Please forgive me if this is a dumb question, I don't normally use Fourier transforms. I attempted to simply ...
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920 views

time-frequency domain

im confused on how these folks seems to like convert a frequency into a time function, and a time function into a frequency function. i know that time function uses amplitude that varies over time, ...
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How to find inverse Fourier transform

I have the function $$ \delta(f-2) $$ How can we inverse Fourier transform it? It's easy if $f$ is replaced with $w$. But based on my knowledge, $w = 2\pi f$. The correct answer is $$ e^{4\pi i ...
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750 views

Fourier transforms of cos and sin

I have the function of time $f(t)=\cos(t10\pi) + \sin(t10\pi)$ and i wish to transform it. By using the tables, i have $\pi [\delta(w-10\pi) + \delta(w+10\pi)] + (\frac{\pi}{j})[ \delta (w-10\pi) = ...
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93 views

Introductory book on Distribution theory [duplicate]

Possible Duplicate: Distribution theory book Is there a good alternative to Friedlander Introduction to the Theory of Distributions ? Many thanks !
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294 views

How to solve a linear PDEs with trigonometric function as coefficients

Is a general method for solving a system of linear partial differential equation with trigonometric function as coefficients exist ? For example something like that: $q$ is the unknown function, $2 ...
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1answer
222 views

Solution of the Dirichlet problem

I'm reading Jones' book Lebesgue integration on Euclidean space. Let $u(x, y)$ be a harmonic function on the half space $\mathbb{R}^n \times (0, \infty)$, with boundary condition $f(x) = u(x, 0)$. On ...
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168 views

What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
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671 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
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231 views

Functions whose Fourier transform vanishes outside of a small interval

Suppose $f(t)$ is a function whose Fourier transform $\hat{f}(\omega) = \int_{-\infty}^{+\infty} f(t) e^{- \omega t} dt$ is supported on the interval $[-\epsilon,+\epsilon]$. Is there a theorem to the ...