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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$L^2$ implies tempered distribution

I am reading a book and I don't understand some of the statements in the proof. It says $1_{B(0,1)}f\in L^2(\mathbb{R}^d;(1+|\xi|)^sd\xi))$ so it also belongs to $S'(\mathbb{R^d})$ thus it is a ...
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Fourier transform of raised cosine

I want to find the Fourier transform of the raised cosine with $\alpha=1$, i.e. $g(t) = \text{sinc}(t/T)\frac{\cos(\pi t/T)}{1-4t^2/T^2}$ We can with substitution $u=t/T$, a trig identity and ...
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12 views

twisting rings for FFT

I am implementing an FFT described by Daniel Bernstein in http://cr.yp.to/papers.html#multapps on page 332 (8 in pdf) he states the following: One can multiply in $R[x]/(x^{2n} +1)$ with $(34/3)n ...
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16 views

Fourier transformation in division

I am trying to figure out the channel impulse of a doppler shift channel. This will lead to a frequency shift. Supposing the input signal is $x(t)$ and its Fourier transform is $X(f)$. Then the ...
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21 views

How to model time changing random variables

Lets say I have a random variable $X(t)$ which describes some unit of motion of a living organism and $X(t)$ is itself a timeseries since this unit of motion changes in time. I would like to be able ...
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25 views

Fourier transform is unitary proof and other unitary integral operators

There is this old unanswered question: Proof the Fourier Transform is Unitary/Not Unitary What is the easiest way to see that the Fourier transform is unitary and why it is important to have constant ...
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17 views

Inverse Fourier Transform involving inverse square root

I'm currently working on this paper: http://web.calstatela.edu/faculty/rcooper2/article.pdf and I want to proof Lemma 3.0 in the case of $n=2$, on page 441. It seems that the Ph.D. thesis the author ...
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52 views

Series of $\csc(x)$ or $(\sin(x))^{-1}$

In some cases I found that $$\csc(x)= \lim\limits_{k\rightarrow \infty}\sum_{n=-k}^{k}(-1)^{n}\frac{1}{x-n\pi}$$ Is anything to prove or disprove that?
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49 views

Getting 0 solving Schrodinger equation with Dirac delta by Fourier transform

I am attempting to solve the Schrödinger equation with the potential $V = - \delta (x)$. This leads to a differential equation $$ \alpha \psi''(x) + (E + \delta(x)) \psi(x) = 0 $$ where $$ \alpha ...
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Solving Poisson's equation for a point charge in 1-D

I Apologize that this is a continuation of a question that I just asked. Anyway here is where I am: Ok so I was trying to solve the Poisson's equation for a point charge with a Fourier transform to ...
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21 views

Solving Poissons equation in 1D with Fourier Transforms

So ultimately I am trying to solve this in 3 dimensions but I am embarrassingly struggling with the 1-D solution right now. $\frac{\partial^{2}}{\partial x^{2}} f(x) = \rho(x) $ I express f and ρ ...
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51 views

Why is $\overline{\operatorname{span}\{e_n\mid n\in\mathbb{Z}\}}=L^2(\mathbb{T})$?

I want to know, why $\{e_n\mid n\in\mathbb{Z}\}$ is an orthonormal basis of $L^2(\mathbb{T})$, where $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_{\mathbb{T}} ...
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42 views

Orthonormal basis of $L^2(T)$

Why is $\{e_n\mid n\in\mathbb{Z}\}$ an orthonormal basis of $L^2(T)$, where $T=\{z\in\mathbb{C}\mid |z|=1\}$, $e_n(z)=z^n$, and $\int_T f(z)\,dz:=\int_0^1f(e^{2\pi i t})\,dt$? My try: If $n=m$, ...
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54 views

Wavelet admissibility

This question is about Daubechies' book "Ten Lectures on Wavelets", section 2.4. This author says, if $\psi \in L^2(\mathbf{R})$, $\int_{\mathbf{R}}dx\ \psi(x) = 0$, and \begin{align*} \exists ...
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57 views

Help with notations from 2D to 3D FFT representations as 1D FFT

I need some help and clarifications for my notations in 3D centered fft. Consider the 1D centered fft of a 2D image of size $(N+1)\times (N+1)$, where $N$ is even, along X axis it can be computed ...
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24 views

Fourier kernels relations to windowing in signal processing.

In engineering a common practice is to "window" a signal (by multiplying a function which decays smoothly at each end) before applying a Fourier transform. Windowing is done to avoid false frequency ...
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Fourier transform of $1 - \cos(xe^{-x^2})$

Is there a closed form expression or maybe an infinite series? If not is there a "good" approximation to it? Even a "good" approximation of the fourier transform close to zero frequency would do. Can ...
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Why can I change the integration unit in Fourier transform>

I was hoping for an explanation of the following steps in taking the Fourier transform of the Klein-Gordon equation. \begin{equation} \phi (\boldsymbol x,t)=\int \frac{d^3p}{(2\pi)^3}e^{ip\cdot ...
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1answer
31 views

Deriving the inverse Fourier transform without knowledge of the form it will take

I've run by several proofs of the Fourier inversion theorem. However, every proof I have come across starts by assuming the form that the inverse transform will take. For example, Ron Gordon's ...
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27 views

A Fourier cosine series of type $f(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k \cos(2^k x)$.

For a function $f(x)$ defined on $x \in [0,\pi]$, one can write $f(x)$ as $$f(x)=\frac{a_0}{2}+\sum_{k=1}^\infty a_k \cos(kx)$$ for some coefficients $a_k$. Fourier claimed that any function $f(x)$ ...
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Study materials for Differential equations and Fourier analysis

In two days, on Monday, a new course called Introduction to differential equations starts, and when that ends in one month another called Fourier analysis and its application starts (Both are actually ...
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Are the Spherical harmonics the S^2 equivalent of the exp(i \pi n) function series?

As I understand it, the Spherical harmonics and the "Fourier functions" $\exp(i\pi n)$ with $n\in\mathbb{N}$ have much in common: Both are eigenfunctions of the angle part of the Laplace operator. ...
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Fourier transform is an isomorphism over the Schwartz space

Last term we defined the (one-dimensional) Schwartz space as the following: $$\mathscr{S}(\mathbb{R}) := \left\{ f \in C^{\infty}(\mathbb{R}) \mid \forall p,q \in \mathbb{N}_0 \,\exists c_{p,q} \in ...
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Double integral definition of periodic Sobolev spaces

Preliminaries: For $s \geq 0$ define the periodic Sobolev spaces $H^s(\mathbb T)$ with norm $$ \| f \|_{H^s(\mathbb T)}^2 = \sum_{k \in \mathbb Z} \big(1 + |k|^2 \big)^s \, \big|\widehat{f}_k \big|^2, ...
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29 views

Given a spectrum, what can we know about its function?

Say we are given a well-behaved function $f(t)$ (either discrete or continuous) and are able to compute its spectrum $F(k)$ using the (discrete) Fourier transform. Then say we lose $f(t)$ and know ...
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22 views

Fourier expansion and transform - what about the phase of the waves that i am adding?

Say we have a wave on the surface of the water and we want to describe it as a sum of other waves. So we use Fourier expansion to add waves of different wavelengths. For simplicity, say we have to ...
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61 views

Fourier series of half of $\sin(\pi x)$

So my question is: Find the Fourier series (using integrals) for the half wave rectified sine function: $$f(x)= \begin{cases}0&-1<x<0\\ \sin(\pi x)& 0<x<1\end{cases}$$ ...
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27 views

Identical Summation and Integration of specific functions

A strange coincidence which I discovered recently, is that $$\int_{-\infty}^{\infty}{\tan^{-1}{\frac{1}{(x-\alpha)^2+\frac{3}{4}}}}dx = ...
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21 views

Growth of Fourier coefficients of piecewise linear function

Suppose $f$ is a periodic continuous piecewise linear function. What can be said about the growth (or decay, rather) of the Fourier coefficients $\hat f(n)$ as $n\to\infty$, other than the ...
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35 views

'Converse' of Riemann-Lebesgue Lemma

So for a periodic function $f$ (of period $1$, say), I know the Riemann-Lebesgue Lemma which states that if $f$ is $L^1$ then the Fourier coefficients $F(n)$ go to zero as $n$ goes to infinity. And as ...
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29 views

Why the difference between definitions of the discrete/continuous Fourier transforms?

I should preface this question with the fact that I'm not familiar with the meaning/utility of the Fourier transform. Perhaps more accurately: I may have learned them, but have since forgotten; in any ...
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FFT multiplication

I'm currently implementing a specific polynomial multiplication algorithm for a project. The current goal is to implement chapter 2 of Daniel Bernstein's paper ...
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45 views

Where does the $2 \pi$ come from in the Fourier Transform Equation?

So I was working through the Fourier transform equations that arise. I was wondering where the radical outside the integral originated from? $\hat{f}(k) = \sqrt{\frac{|b|}{(2 \pi)^{1-a}}} ...
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22 views

Von Neumann stability analysis of non-linear systems

The von-neumann stability analysis is based on the time and space discretisation schemes, what if the schemes are non-linear and too complicated to analyse. Is there a way to look at the matrices of ...
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18 views

Positive-definiteness of a specific function

Is the following function positive-definite $$\varphi(t)=\max\left\{{1-\frac{n-\left|2|t|-n\right|}{2(m+1)},0}\right\}$$ where $0<m<n$ and $0\le |t|<n$. I'm aware of the Bochner's theorem ...
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Ambiguity in the Fourier transform of $f(x)=\cos(ax)$

I am slightly confused about two contradictory answers I am getting with regard to the Fourier transform of the function $f(x)=\cos(ax)$. The first method I used was \begin{align} ...
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19 views

Fourier transform of finite aperiodic signals

I know the fourier transform of most of the signals,but how about the fourier transform of aperiodic finite signals?
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33 views

Bochner-style theorem for SO(3)

Bochner's Theorem essentially provides necessary/sufficient conditions for when something is the Fourier transform of a nonnegative measure on a compact abelian group. I'm looking for a similar ...
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A discussion on fourier and laplace transforms and differential equations …?

i have read many of the answers and explanations about the similarities and differences between laplace and fourier transform. Laplace can be used to analyze unstable systems. Fourier is a subset of ...
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44 views

Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
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58 views

How to manipulate this functions to an identity involving the Riemann zeta function

The identity I want to prove is the following (from Stein's book, an introduction to Fourier analysis): $$\pi^{-s/2} \Gamma(s/2) \zeta (s)=\frac{1}{2} \int_{0}^{\infty}t^{\frac{s}{2}-1}(v(t)-1)dt$$ ...
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Converting Walsh coefficients to values of a function

I assume I know the Walsh coefficients of a function f: $\mathbb{F}_{2^n}$ to $\mathbb{F}_{2}$. Is there any efficient possibility to get the values of the function f ?
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Why wavelets based transmitter/receiver digital signal processing aren't common? [migrated]

I have seen this thread: Difference between Fourier transform and Wavelets AFAIK there is no common usage of wavelets in the real-time DSP world (excluding image and video processing). I am curious ...
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Difficulties of convergence of the partial sums of the Fourier inversion formula

Define the Fourier inversion formula over $\mathbb{R}^n$ by $$ f(x)=\int_{\mathbb{R}^n}e^{2\pi ix\cdot\xi}\hat{f}d\xi $$ where $\hat{f}$ is the Fourier transform over $\mathbb{R}^n$ $$ ...
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Finding $a_0$ for the function $s(t)=1-e^{-2t}$.

I am working on multiple Fourier series questions about the function $s(t)=1-e^{-2t}$. How do I find a naught as in $a_0=\dfrac 1T\displaystyle \int\limits_{t_0}^{t_0+T}s(t)\,dt$, when $T = 3$? ...
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38 views

How to derive the complex Fourier series of $s(t) = 1-e^{-2t}$? [closed]

I have the periodic function $s(t)=1-e^{-2t}$. I am required to derive the complex Fourier series of $s(t)$. I have some knowledge of Fourier series but not enough to know if I am doing it correctly. ...
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32 views

Fourier transform of the Cosine function with Phase Shift?

How can i calculate the Fourier transform of a delayed cosine? I haven't found anywhere how to do that. This is my attempt in hoping for a way to find it without using the definition: $$ x(t) = ...
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34 views

Fourier transform of 2D function in terms of 1D function

$$f\left(x,y\right)=g\left(\frac{c_2 \cdot \left(x-y\right)}{c_1}+y\right)$$ $$\mathcal{F}\left(X,Y\right)=\frac{c_1}{c_2} \cdot \delta\left(Y-\left(\frac{c_1}{c_2}-1\right) \cdot X\right) \cdot ...
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22 views

Connection between autocovariances and Fourier series of a continous function.

Let $f(w)$ be a continuous function of period $2 \pi$ then it's Fourier series is $$f(w) = \sum_{k = 0}^j \left(a_k \cos(kw) + b_k \sin(kw)\right)$$ I wrote that the autocovariances $\gamma(k)$ (of ...
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What are the statistics of the discrete Fourier transform of a Bernoilli process?

The problem I would like to understand the statistics of the discrete Fourier transform of a sequence of uncorrelated events $\{x_n\}$ each of which takes the value $\pm1$ with probability $1/2$. In ...