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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Given its spectrum, what can we know about a 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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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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1answer
35 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
14 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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1answer
24 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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12 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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19 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 ...
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43 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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18 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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1answer
18 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
18 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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30 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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25 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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29 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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56 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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Complex Fourier integral of cosht [on hold]

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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15 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 ?
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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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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 $T = 3$? ...
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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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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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1answer
33 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 ...
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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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20 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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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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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}) = ...
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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 ...
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1answer
29 views

in Maxima how to get Fourier transform identity ($\mathcal{F}^{-1}(\mathcal{F}(f)) = f$)?

I'm currently trying to figure out something related to signal processing and wanted to draft Maxima for the task. As a starting point I tried to make Maxima evaluate the identity ...
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31 views

Cosine Fourier series solution of semi-major axis nonlinear integral equaton

Consider an integral equation $$ \frac{1}{z(t)}=f(t)+\alpha\int_0 ^\infty \cos(ts)z(s)\,ds $$ I am required to solve for $z(t)$. I approached this problem by considering the integral on right hand ...
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32 views

Fourier transform of an inverse function.

If for a given function $f(x)$, the Fourier transform is $\hat{f}(p)$; Is there a way to find the Fourier transform of $f(x)^{-1}$ in terms of $\hat{f}(p)$?
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23 views

Convolution and space-time Fourier transform

I have a general function $u(x,y,z,t)$. Then, (1) what would be the space-time Fourier transform of $$G \star \frac{\partial^n u}{ \partial t^n }$$ and (2) would the relation $$G \star ...
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when Wiener amalgam space is a subset of Lebesgue space?

Let $X=\mathcal{F}L^{p}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{p}(\mathbb R)\},$ and $\|f\|_{X}= \|\hat{f}\|_{L^{p}}.$ In the definition of Wiener amalgam spaces $W(X, L^p)$, I am taking ...
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31 views

Is there a way to find an in-between 1D FFT with one less point?

Given a discrete 1D signal $f(n)$ over the support $-N/2 \leq n \leq N/2$, where $N$ is even, and given an arbitrary scalar value $\alpha$, the definition of 1-D fractional Fourier transform (FrFT) ...
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18 views

Proof about fundamental frequency for periodic functions

I know that in the Fourier series expansion for $f$, we have $$f(t) = \sum_{n=-\infty}^{\infty} c_n \exp\left(\frac{2\pi int}{T}\right)$$ where the lowest frequency term (ignoring the constant) has ...
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1answer
54 views

Condition that Fourier inversion formula holds

Recently I am reading Stein's Complex Analysis but I haven't read the book Fourier Analysis before. So I do not have any knowledge about Fourier Transformation. In Chapter 4.3 (p.121), it tells that ...
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2answers
25 views

Fourier series on general interval $[a,b]$

Currently I'm studying Fourier series and the first thing I've read is the definition of the series for a function $f : [-\pi,\pi]\to \mathbb{R}$. In that case the Fourier series is ...
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32 views

Is there a difference between cosine and sine transform?

Surely both should work with the same set of functions. Why is only cosine transform used in JPEG? Why not sine? It seems that using fourier transform rather than cosine transform would result in ...
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30 views

Integral (Fourier transform) of Heaviside radial function in 3D

I am trying to calculate the following integral: $ \int \frac{d k_x d k_y d k_z}{(2 \pi)^3} \left[ \exp( - \frac{(k_x^2 + k_y^2 + k_z^2) \sigma^2}{2}) + \frac{1}{2} H(\sqrt{k_x^2 + k_y^2 + k_z^2} - ...
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46 views

Approximation by sinc functions in L2

I wish to find the best approximation in $ L_{2} (\Re )$ of $f(x)=\frac{sin(ax)}{ax}$ for $0<a<\pi$ and for $a>\pi$ , Using the system of sinc functions: $$g_{n}(x)=sinc(\pi x-\pi n) = ...
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1answer
39 views

Why this formula doesn't work for $n=1$?

I've been studying Fourier series and in trying to compute the Fourier series for the function $f: (-\pi,\pi)\to \mathbb{R}$ given by $f(x)=|\sin x|$ I've found something quite strange that I'm not ...
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45 views

For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$.

As the title states: For $f \in C_c^\infty(\mathbb{R})$, does $\hat{f}(k)\sum_{j=0}^n \frac{(-k^2)^j}{j!}$ converge to $\hat{f}(k)e^{-k^2}$ in $L^2(\mathbb{R})$ where $C_c^\infty(\mathbb{R})$ is the ...
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39 views

how to disprove uniform convergence

I've been asked to check the uniform convergence of the following function sequence on the real line: $$ f_{N}(t)=\sum_{n=-N}^{n=N}\sin(n) \,\frac{\sin(\pi t-\pi n)}{\pi t-\pi n} $$ It is asked in a ...
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34 views

Find the Fourier transform of the given memory function in the limit volume $V\rightarrow\infty$

The memory function is given by, \begin{equation} \mu (t)=(8\pi e^{2}/3V)\sum_{\vec{k}}|f_{\vec{k}}|^{2}\cos (ckt) \end{equation} where $V$ is the volume, $f_{\vec{k}}$ is the form factor. In this ...
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Fourier transform of a constant - Is it a function of distance only?

I have the following power spectrum $P(k)$, function of the modulus $k$ of the vector $\vec{k}$ in Fourier space: $$P(k) = \begin{cases} P_0 \exp (-\frac{k^2 \sigma^2}{2}) & \text{for} \; k \leq ...