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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Fourier transform probability distribution

Suppose I have 1,000 independent random values with a uniform distribution $[+1, -1]$. Now suppose I take the discrete Fourier transform of this data. What the heck is the probability distribution of ...
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How to generalise the Fourier transform

The Fourier transform approximates a signal using a bunch of sine and cosine waves. The inverse Fourier transform then reconstructs the original signal from this information. I am told that it's ...
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A Paley-Wiener like theorem in real-analysis

I try to identify conditions for the Fourier-transformation $\mathcal{F}(f)$ of some function $f \in L^1(\mathbb{R}^n)$ to be real-analytic. Namely I want to show that one of the following two ...
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Properties of the Toeplitz matrices formed by square-summable sequence (as opposed to absolutely summable)

I've been reading a wonderful monograph by Robert Gray on the Toeplitz and circulant matrices and am curious about the assumption (4.3) of absolute summability of the sequences $\{t_k\}$ that form the ...
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white noise has flat power spectral density.

I am trying to prove that the white noise has constant power spectral density using matlab but the amplitude of the spectrum looks like random amplitude. can anyone tell me why? here is my code. ...
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Fourier Transform of $xf(x)$

I am not able to get the Fourier Transform of $xf(x)$ if $<f(x)>$ is the Fourier transform of $f(x)$ . BTW i tried using convolution theorem but didn't work out .
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Primitive $2^k$-th roots of unity in $GF(p)$

While debugging an NTT implementation I've noticed something. Looks like if I have a primitive $(n = 2^k)$-th root of unity $\omega$ in a $GF(p)$, then $\omega ^0 = -\omega ^{n/2} + p,$ $\omega ^1 = ...
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Help with an inverse Fourier transform calculation

I'm trying to do an exercise in Folland (8.45), and I'm having to calculate $W_t=\left[\frac{\sin (2\pi t|\xi|)}{2\pi|\xi|}\right]^\vee,$ where $\vee$ is the inverse Fourier transform. We can make ...
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A basic estimate for Sobolev spaces

Here is a statement that I came upon whilst studying Sobolev spaces, which I cannot quite fill in the gaps: If $s>t>u$ then we can estimate: \begin{equation} (1 + |\xi|)^{2t} \leq \varepsilon ...
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The digit base and the NTT convolution

Suppose I'm using a number theoretic transform (NTT) in an integer field $GF(p)$. I assume that $2n$-th root of unity exists for such a $p$, and I want to compute a convolution of two $n$-length ...
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Finding an ON basis of $L_2$

The set $\{f_n : n \in \mathbb{Z}\}$ with $f_n(x) = e^{2πinx}$ forms an orthonormal basis of the complex space $L_2([0,1])$. I understand why its ON but not why its a basis?
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DFT and DWT difference?

what is the basic difference between the Discrete Fourier Transform and the Wavelet Transform ? and why does JPEG2000 preferred DWT over DCT or DFT ?
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Question from Stein's Singular Integrals and Differentiability Properties of Functions.

My question is in regards of Stein's proof that Hilbert transform is of weak $(1,1)$ property, on page 30 of the textbook I mentioned in my title. On page 32 he writes that because $|\nabla K| \leq B ...
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873 views

Why is Parseval's Equality and Bessel's Inequality Different?

Bessel's Inequality: $\sum_n |\langle x, e_n \rangle |^2 \leq \|x\|^2$ Parseval: $\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ $\sum_n |\langle x, e_n \rangle |^2 = \|x\|^2$
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Discrete Fourier transform - period and other questions

I am trying to get understand few things about DFT, from wikipedia: $X_k=\sum_{n=0}^{N-1}x_ne^{-i2\pi\frac{k}{N}n}$, we have N numbers $x_0$ to $x_{N-1}$ Why there is $\frac{k}{N}*n$ in the ...
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Need help with Discrete Fourier Transform

Yes, this is a homework. I've been told to perform Fourier transform on the following sequence of values: a=[0 2 -1 3] I think I'm supposed to use Discrete ...
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Physical meaning of Fourier transform of complex signal?

I understand what is meaning of Fourier transform over function that returns only real values — it can be thought of function taking time and returning real ...
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limiting behaviour of the heat kernel on the real line

I am lost with the following exercise that is posed in Steve Rosenberg's book "The Laplacian on a Riemannian Manifold": Show that for a continuous function f, \begin{equation} \lim_{t \to 0} \quad ...
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480 views

Long integer multiplication using FFT in integer rings

I would like to perform long integer (~= polynomial) multiplication using the FFT or its direct analogue, but never leave integer rings. Please excuse in advance all my mistakes in formulation and ...
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Altering L^{2} functions so that the FTC holds

I'm studying for a real analysis exam by doing a previous year's test. I have come across one question which I have banged my head against for a few days but can't seem to make any progress. I have ...
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Instructive proofs in functional analysis

I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
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462 views

Discrete Fourier Transform - Definition?

On the internet I have found the definition of the DFT to be : $$ F(k) = \frac{1}{\sqrt{N}} \sum\limits_{n=0}^{N-1} f(n) e^{-\frac{2\pi}{N}jkn} $$ But in this article I have found an implementation ...
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The graph of Fourier Transform

I am trying to grasp Fourier transform, I read few websites about it, and I think I don't understand it very good. I know how I can transform simple functions but there is few things that are puzzling ...
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completeness of orthonormal set

I am currently working through some lecture notes on the Geometry of Hilbert spaces, and I am stuck with the following comment: If we are given the inner product space $C([0,1])$ of continuous ...
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Singular measures - approximate characteristic function

One can decompose a $\sigma$-finite measure $\mu$ on $\mathbb{R}$ in three parts: $\mu_{ac}$: absolutely continuous $\mu_{sc}$: singular continuous $\mu_{pp}$: pure point A common example for a ...
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Inverse Fourier Transform

How can you find the inverse Fourier transform or $f(w) = \frac{\exp(-iw)}{2+iw}$ ? I started out by using $f(t)= \frac{1}{2\pi} \int f(w)\exp(jwt)\delta$ but I'm not sure how to go about it..
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heat equation solution

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
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165 views

About $\sum_{k=1}^\infty \frac{b_k}{k}$, where $b_k$ are Fourier coefficients

This is my first post here. I have some troubles with this property of the Fourier coefficients. Indeed, let $f(x)$ be a continuous real function, with compact support $[a,b] \subset (0,2\pi)$, and ...
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Why the Fourier and Laplace transforms of the Heaviside (unit) step function do not match?

The Fourier transform of the Heaviside step function $u(t)$ is $\dfrac{1}{iω} + π δ(ω)$. The Laplace transform of the same function is $\dfrac{1}{s}$. I remember the proof came from derivatives and ...
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756 views

FFT Characteristics?

I'm looking at this regarding a simple spectrogram 1) my question is about this line % Take the square of the magnitude of fft of x. mx = mx.^2; why do ...
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426 views

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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314 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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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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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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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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363 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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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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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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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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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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656 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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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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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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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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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 ...