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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CT fourier Transform Proof

I am confused trying to understand the Proof of Fourier Transform from Oppenheim book Signals and Systems. I am pasting the equations directly from the book: ...
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Criterium forSubspace of tempered distributions

I have a question concerning the subspace $\mathcal{S}'_h$ of tempered distributions defined by $u\in\mathcal{S}'_h\Leftrightarrow\lim_{\lambda\rightarrow\infty}\Vert\theta(\lambda ...
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Convergence of the sum of fourier coefficients

I have a function $f$ defined on $[-\pi, \pi]$, $f(\pi) = f(-\pi)$, and has continuous first derivative. How can I prove that the sum of the absolute value of the Fourier coefficients converges? The ...
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Finding a Fourier pair satisying some conditions

I want to find a Fourier pair $(r,\hat{r})$ satisfying some conditions listed below and making $\hat{r}(0)$ as small as possible. The requirements for $(r,\hat{r})$: $r,\hat{r}\in L^1(\mathbb{R})$, ...
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Eigenfunctions of a 4th Order PDE on 2D domain

What technique would you use to find the eigenfunctions of this problem? $$\nabla^2(\nabla^2 u) = 10$$ $0 < y < G$ $0 < x < L$ With Pure Homogeneous Dirichlet boundary conditions and ...
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Piecewise linear approximation of a (low order) trigonometric polynomial, with quantization

The Problem Given a trigonometric polynomial of order $K$: $$y(t)=\sum_{k=-K}^K c_k \ e ^ {j k \bar \omega t} \ , \qquad c_{-k}=\bar c_k$$ we want to find the best approximation to it using a ...
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wave front set - directions of singularities

I am learning about the wave front set of a distribution but am having difficulty understanding some details, which to me seem counter intuitive. We know the fourier transform of a smooth function ...
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Fourier transforms of distributions

I am reading a proof claiming that every partial differential operator $P(D)$ has a fundamental solution $E$. It says that "if we have a distribution $u$ on $R^n$ with $u(P(D)\phi)=\phi(0)$ for ...
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A continuous function whose Fourier Series diverges

I have read several times that there exist many continuous functions whose Fourier series diverge at some points (sometimes even on a dense subset of the domain!). I tried to find an explicit example ...
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I have derived a short proof that answers 'DTFT assumes the sequence to be periodic and hence the expression for the same'. Does my proof make sense?

Define: Function '$f$' defined at all (-oo, oo). $U(to)$ is a step function that equals 1 when $t \geq to$ $\sum_{n \epsilon Z}\delta(t - nT)$ is an impulse train that is non-zero for all $t = nT|{n ...
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Question about Fourier Transform?

Could someone please explain how they got from the first step to the next? I have no idea how the second step follows...
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61 views

The derivative of the trigonometric polynomial

For $n\in\mathbb{N}$, let $\varphi:\mathbb{R}\to\mathbb{R}$ be a $4n-$ periodic function s.t. $$\varphi(t)=n-|n-t|,\quad -n\leq t\leq 3n$$ For each $j\in\mathbb{Z}$ define $$c_j=\int_0^1\varphi(4n ...
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Derivation of First order derivation of Fourier transform

Let $R$ be $(-\infty,+\infty)$. Starting with: $$x(t)=\int_{R}X(f)e^{2i\pi ft}df$$We have its Fourier transform: $$F\{x(t)\}(f)=X(f)=\int_{R}x(t)e^{-2i\pi ft}dt$$ Its derivation is derived as ...
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Fourier Transform of a CFT two point function

In the study of Conformal Field Theory in physics, one encounters the following function $$ \left( \frac{1}{\sinh(x+i\epsilon)} \right)^{2\Delta}. $$ It appears as the correlation function of certain ...
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Find the Fourier Sine series of $x-\frac{1}{2}$ on the interval $0<x<1$

Two Parts to this question are: (a) Find the Fourier Sine series of $x-\frac{1}{2}$ on the interval $0<x<1$ (b) Let $a \notin Z$ and find the Fourier Cosine Series of $\cos ax$ on the ...
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Fourier transform on Laplace equations

We can solve some Laplace equations on the lectangular area by using Fourier transform. However many textbooks only introduce the cases when the area is given as a half plane or a strip. ; $y>0, ...
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Generalization of a FFT with powers of 3

I'm not satisfied with the current answers asked about this on MathSE. So I'm going to ask: Describe the generalization of the FFT algorithm to the case in which n is a power of 3. What's the ...
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problem 9.26 from Folland's real analysis Fourier Transform of $ G(x,t) = (4t\pi)^{(-n/2)} e^{{-|x|^2}/{4t}} \chi_{(0,\infty)}(t)$

I was just given this question from Folland's real analysis second edition dealing with tempered distributions and their Fourier transforms Exercise 26 on page 300 : On $ R^n \times R $ let $ ...
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Extension of the Fourier transform, proof-read

I'm writing a Bachelor-thesis in mathematics which is to be submitted in a couple of days, and would be thankfull if the following arguments concerning the extension of the Fourier transform from the ...
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64 views

Techniques in analytic number theory

I'm fairly new to the subject and trying to figure things out. Would be nice to hear some ideas and trickery for what follows. Suppose we wish to show there exists an integer $x$ in some finite set ...
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Autocorrelation of a signal known only on an interval of finite size

Let's consider we have a continuous random signal ${ t \in ] - \infty \,;\, + \infty [ \mapsto b (t)}$. We assume this signal to be stationary, so that when ensemble-averaged, one may introduce the ...
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Zero padding property of FFT

I wonder if the following basic property I thought up is a real property of FFT (or more specifically the discrete version of Fourier series), and if so what is it officially called? The property ...
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54 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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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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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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132 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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102 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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999 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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49 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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176 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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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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Extending an identity for the Dirac delta function

The identity $$x^p \; \delta^{(n)}(x) = (-1)^p \frac{n!}{(n-p)!} \; \delta^{(n-p)}(x)$$ can easily be derived from the generalized Leibnitz formula for $n$ and $p$ positive integers: $$\int \; x^p ...
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Cross-correlation of Gaussian and Jacobian sums

I recently came upon the following kind of sum and I'm wondering if anyone has seen it before, or could point out something interesting about them. Let $F$ be a finite field with $q > 2$ elements ...
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259 views

Good Kernel's Properties

I am recently studying properties about a good kernel, and came across a problem. Definition: A kernel $K_\delta$ is 'good' if they are Lebesgue integrable and satisfy the following conditions for ...
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Gauss sum of a multiplication of two multiplicative characters of a finite field

Let $F$ be a finite field with $q$ elements and characteristic $p$. Let $E$ be a proper extension over $F$ of degree $n$. Let $\psi$ be the canonical additive character of $E$ defined by $\psi(x) = ...
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How to show that $\int_{-\infty}^{\infty} \mathrm{d}^3 \textbf{k} \frac {e^{i \textbf{k x}}} {(2 \pi)^3} = \delta^3(x)$ in spherical coordinates?

Recently I had to deal with Fourier transformations and delta functions, and I was wondering how about that. I know, that its trivial to show in cartesian coordinates, but i couldn't do it in ...
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Is $X(T) = A \sin(\omega_0 t + \Phi)$ mean ergodic?

This is an example of a tutorial but I think has not been solved properly. Please help me! $X(T) = A \sin(\omega_0 t + \Phi)$ $A$ and $\phi$ are independent $A$ is uniformly distributed over ...
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Possible Connections between Harmonic Analysis, Potential Theory and Analytic Capacity for a Fourier Analyst

So, Folks, here's the deal: After looking at this question, posted a little earlier on this site, and getting quite inspired by the beauty of this kind of result, I have got quite interested on this ...
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53 views

Convolution of a gabor function and gaussian noise?

I am convolving the same image with a 2D Gabor over different gaussian noise masks that are generated in every trial. The convolution naturally takes time, is there any way to speed up the process by ...
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54 views

Is there a Fourier invariant basis?

There are some functions which are invariant under Fourier transformation up to scaling factors, eg. sech(pi*x), Gaussian function etc.. Is there a set of basis functions, which form an invariant ...
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Fourier Transform of Positive and Negative Parts of Functions

Suppose I have a function of the form: $G(t,x) = \alpha\left(P(t,x) - \Theta(t,x) \right)^+ + \beta \left( P(t,x) - \Theta(t,x) \right)^-$. Here, $P(t,x)$ and $\Theta(t,x)$ have compact support and ...
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Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
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Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$?

Why is $\sum_{n=0}^{N}{\cos nx}={1\over 2}+{1\over 2}\sum_{-N}^{N}e^{inx}$? I have gone through all the identities relating Fourier series and I can't seem to understand why. In this question, the ...
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How do you find the Fourier series of $\max(0, \sqrt{1 - \cos{\theta}})$?

I was trying to express the following periodic function: $$ f(x) = \max \left( 0, \sqrt{1 - \cos{x}} - \frac{\sqrt{2}}{2} \right)$$ as a summation of cosines and sine waves $f(x) \approx a_0 + ...
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Fractional derivative of $e^{-x^2/2}$ using Fourier transform and Taylor series

I am not familiar with fractional calculus, so I want to know what I am doing wrong. The convention I use $$\int^\infty_{-\infty}e^{-\frac{x^2}{2}}e^{-i k x}dx=\sqrt{2 \pi}e^{-\frac{k^2}{2}}$$ I am ...
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Let $A,B:V\to V$ positive definite operators in complex linear space with inner product $V$, $dimV<\infty$

Let $$A,B:V\to V$$ positive definite operators in complex linear space with inner product $$V$$, $$dimV<\infty$$ Show that $$log det(A\cdot B^{-1})=-\int_{0}^\infty tr(e^{-t\cdot A}-e^{-t\cdot ...
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On the convolution of $f(x)=\sin x/x$ and $g(x)=1-|x|$

I am having trouble with computing the convolution of $f(x)=\sin x/x$ and: \begin{equation} g(x)=\begin{cases} 1-|x|,& -1 \leq x \leq 1 \\ 0, & x \notin [-1,1] \end{cases} \end{equation} I ...
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Recovering Time Shift Using DFT of Translated Square Pulse?

As an exercise, I attempted to manually translate a pulse $n_0$ steps to the right and recover the translation using the time-shift property. The problem I'm encountering is that the phase unwrapping ...
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104 views

Fourier transform of the realization of a stationary process in the space of tempered distributions?

A path of a stationary sequence of random variables $y_t$ does not have a discrete-time Fourier transform in the classical sense because it is not summable. This leads to considering the spectral ...
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How can I show the approximate version of the fourier inversion formula?

Let f be $L^1(R) \cap C_0(R)$ and satisfies $|\hat{f}(\alpha)|\leq A\frac{1}{|\alpha|}$, for all non zero real $\alpha$, for some positive A. Then, show that for any $x \in R$, $f(x) = ...