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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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 ...
4
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1answer
27 views

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 ...
2
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1answer
20 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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1answer
44 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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1answer
33 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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1answer
43 views

Wavelet admissibility

This question is about Daubechies' book "Ten Lectures on Wavelets", section 2.4. This author says, if $\psi \in L^1(\mathbf{R})$, $\int_{\mathbf{R}}dx\ \psi(x) = 0$, and \begin{align*} \exists ...
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37 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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22 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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13 views

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 ...
2
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1answer
21 views

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
28 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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23 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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15 views

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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27 views

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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1answer
18 views

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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1answer
32 views

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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28 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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1answer
21 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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0answers
60 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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1answer
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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1answer
19 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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2answers
34 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 ...
2
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1answer
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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20 views

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 ...
2
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1answer
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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0answers
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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0answers
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 ...
0
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1answer
19 views

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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1answer
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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31 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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31 views

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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1answer
39 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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1answer
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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34 views

Complex Fourier integral of cosht [closed]

Hi please help me with finding complex exponential Fourier integral for $f(t) = \begin{cases}cosht & |t|<p \\ 0 & |t| > p \end{cases}$
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0answers
18 views

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 ?
2
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22 views

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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1answer
40 views

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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17 views

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 $T = 3$? ...
0
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1answer
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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0answers
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) = ...
2
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1answer
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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1answer
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 ...
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17 views

Prooving that multiply by exponent in time domain yields a frequency shift in frequency domain using convolution.

im trying to proove that $F[x(t)e^{-jat}] = X(w-a)$ using convolution. using the convolution property i know i should get a convolution of $F(x(t))$ and $F(e^{-jat})$ So: $$ F[x(t)e^{-jat}]= 1/2\pi ...
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27 views

Diagnalization of block matrix with circulat blocks

I have the following Matrix $A = \begin{pmatrix} X \\ Y \end{pmatrix}$ Where X, and Y are circulant Matrices. I want to diaganlize $AA^T$. I tried the following: $AA^T = \begin{pmatrix} ...
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0answers
55 views

Reducing or avoiding the Gibbs phenomenon.

What is your favourite method which would help reduce the Gibbs phenomenon in Fourier Series and Fourier Transforms. This could mean pre-processing or post-processing or altering the transform. With ...
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22 views

Fourier methods and a conductor bar

I was doing this question bellow: I tried: Could you help me in the 3 (second Picture) and how to solve the problem?
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1answer
34 views

Methods for solving definite trig. integrals?

I am studying Fourier series and there is a lot of integration going on, specifically with trigonometric functions involved. When solving for the Fourier coefficients, often times, the definite ...
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2answers
31 views

How to do Fourier transform for these 2 questions?

I don't get certain of parts of these two questions 1) I'm trying to do the Fourier transform of: $$f(x) = \, xe^{-x^2} $$ In the problem it said to use: $$F \, (e^{-tx^2}) = ...
0
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0answers
20 views

Compute phase-shifted variant of a real-valued function

I'm trying to compute a phase-shifted (by angle $\phi_0$) version of a general real-valued function $f(x)$. I realize that the phase shift is convenient to perform in frequency domain, so first I ...