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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Determining if something is a characteristic function

Suppose $X$ is a continuous random variable with pdf $f_X(x)$. We can compute its characteristic function as $\varphi_X(t)=\mathbb{E}[e^{itX}].$ Question: Given a function, say $\psi(t)$, how does ...
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107 views

How to express this convolution by the sum of integrals

If $$f\left(x\right)=\begin{cases} f_{1}\left(x\right), & x\in[0,1]\\ f_{2}\left(x\right), & x\in[1,\sqrt{5}]\\ 0, & \mbox{elsewhere} \end{cases}$$ what does the piecewise-defined function ...
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474 views

Fourier transform of compactly supported function on $\mathbb{R^n}$.

I am attempting to extend this question, which says that for a nonzero continuous $f$ with compact support on the real line, the Fourier transform cannot decay exponentially. Suppose $f \in ...
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2k views

Compactly supported function whose Fourier transform decays exponentially?

It's well known now that a function can not be compactly supported both on the space side and the frequency side (so-called uncertainty principle). On the other hand a function can have exponential ...
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1answer
101 views

Need numerical approximation for Fourier{max(0,f(x,y))} given Fourier{f(x,y)}

Given $\mathscr{F}\{f(x,y)\}$ is there a way to numerically approximate $\mathscr{F}\{max(0,f(x,y))\}$ ? I am not necessarily looking for a closed formula. Even some iterative method would be ...
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35 views

When to use other transforms?

maple code int(g*f, x=-infinity..infinity) when $g$ is $\large exp^{i*t*x}$, Fourier transform between density function and characteristic function If $g$ are $x^t$, $|x^{t}|$, $t^{x}$, what do they ...
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2answers
309 views

Finding the Fourier coefficients of this function

I need help finding the Fourier coefficients of: $f(x) =\begin{cases} \sum_{n=0}^\infty{\frac{e^{inx}}{1+n^2}} & \text{if } x\neq 2k\pi \\0& \text{if } x= 2k\pi ...
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3k views

Dirichlet Conditions and Fourier Analysis.

I read in my text book that the Dirichlet conditions are sufficient conditions for a real-valued, periodic function $f(x)$ to be equal to the sum of its Fourier series at each point where $f$ is ...
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1answer
435 views

Inverse Fourier transform of a hyperbolic cosine

This problem arises from trying to solve, by Fourier transform, the Cauchy problem $$\begin{cases} u_{tt}-u_{xxxx}=0 &x\in\mathbb{R},\, t\geq 0\\ \begin{cases} u(0,x)=f(x)\\ u_t(0,x)=0 ...
5
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1answer
549 views

Fourier transform of $ \frac{\sinh(kx)}{\sinh(x)}$ [duplicate]

Possible Duplicate: Calculating the Fourier transform of $\frac{\sinh(kx)}{\sinh(x)}$ Im trying to compute the integral of $$I = \int_{-\infty}^\infty ...
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2answers
170 views

Finding if the equation is even or odd

I am learning fourier transform and I came across this question in which author right away says the given equation is "even". How does this equation become "even"? $$x[n]=\begin{cases}A & -M\le ...
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1answer
141 views

Inverse Fourier Transform - left and right? [duplicate]

Possible Duplicate: Surjectivity of the Fourier Transform on Schwartz Space Consider the Fourier transform on Schwartz space, given by \begin{equation} \mathcal{F}(f)(\xi)= \hat{f}(\xi) = ...
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1answer
360 views

Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
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1answer
183 views

Fourier Transform of an Operator

I need to calculate the fourier transform of an Operator. meaning I need to calculate the transform of the Operator's corresponding convolution kernel. so the question is: 1.given a 2d fourier ...
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141 views

Is Fourier transform defined on $L^p(\mathbb{R})$ only for $p \in [1, 2]$?

Is Fourier transform defined on $L^p(\mathbb{R})$ only for $p \in [1, 2]$? From Lieb and Loss's Analysis, they extend the definition of Fourier transform from $L^1(\mathbb{R})$ to $L^p(\mathbb{R}), ...
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120 views

DFT shift theorem generalizations?

The DFT shift theorem implies that any circular shift in the input space is equivalent to a phase change in the frequency domain, while the absolute values are preserved. $$ ...
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3answers
1k views

Known proofs of Wirtinger's Inequality?

I am looking for proofs of the (Poincare-) Wirtinger inequality which states that if $f:[0,\pi]\to \mathbb{C}$ is $C^1$ and $f(0)=f(\pi)=0$ then \begin{equation} \int_0^\pi |f(t)|^2 dt \leq \int_0^\pi ...
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206 views

Uniqueness of Haar Measures

Haar measure on a LCA (locally compact abelian) group $G$ is said to be unique ... up to a scaling factor. This is not as elegant as it might be expected because this requires a choice of a unit ...
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467 views

Dirac delta forcing of a harmonic oscillator

Is it possible to solve this differential equation: $$\ddot{x}(t)+\omega^2x(t)=k\delta(t)$$ where $k$ is a constant and $\delta(t)$ the Dirac delta function? Is it possible alternatively, to know ...
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0answers
112 views

2-D DFT of a matrix PxP and 1-D DFT of a vector of size P^2?

What is the difference between the following two things: make a 2-D Discrete Fourier Transform of a certain matrix A[p,p], first reshape this matrix into a 1-D vector a[p^2,1], and compute the 1-D ...
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1answer
755 views

Normalizing scilab's FFT

I want to normalize the FFT used in scilab in a way so that the absolute values of the coefficients equal to the amplitudes of the time domain signal with that frequency. Example: I want an input ...
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182 views

convolution of L1 function with a harmonic oscillation

I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, ...
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1answer
249 views

Can fourier transform of a function with not empty support be zero on whole range?

We consider $f : [a, b] \subset \mathbb{R} \rightarrow D \subseteq \mathbb{C}$ and support may consist of only one point.
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340 views

Fourier Transform of f(t)=|t|

This is a question on taking the Fourier transform of a function. How does one find the fourier transform of $f$, where $f(t) = |t|$?
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578 views

Zeros of Fourier transform of a function in $C[-1,1]$

I am trying to prove the following: Let $g \in C[-1,1]$. Then the function $$G(z) = \int_{-1}^1 e^{itz}g(t)dt$$ has infinitely many zeros. I know that $G(z)$ is entire and $\lim_{x \to \pm ...
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163 views

Fourier transform terms explained

I know there are lots of tutorial on different webpages.. but I am not an engineering student... those look quite complex to me.. Could anybody explain what is "Fourier transform" in the very simple ...
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2answers
519 views

Combining bins of FFT ouput

I was trying to combine output of a $2n$ point Real FFT to generate custom FFT bins. For example the FFT generates components at equally spaced frequencies $f_0,f_1,f_2 ...f_{n-1}$ $f_0$ = ...
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1answer
316 views

Fourier transform of Kronecker deltas

I have a binary 2D image that consists of 95% black pixels with a few white pixels scattered about, and I want to convolve it with a 2D gaussian kernel. I'm hoping to exploit its sparsity to improve ...
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344 views

estimation for Dirichlet kernel.

I have the next questions that I am stuck in them. Let $f = 0 $ for $x\in [0,\pi]$ and $f=1$ for $x\in [\pi,2\pi]$. I need to find some constant $c$ such that for every natural N: $$f*D_N ...
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0answers
190 views

Fourier transform of $\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$? Not Gaussian like with Fermi-Dirac statistics?

This equation $\bar n_i=\frac{g_i}{e^{\frac{\epsilon_i-\mu}{kT}}-1}$ is Fermi-Dirac statistics where variables are defined here. The classical equation i.e. the Maxwell Boltzman equation is Gaussian ...
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103 views

Continuous, integrable fourier transform of an $L^{2}(\mathbb{R})$ function, integrable.

I've come across a number of sources claiming a smoothness-decay duality between a function and its Fourier transform. But most seem to give results about how the smoothness of a function leads to ...
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91 views

Suggestion for a project on Harmonic measure and Fourier analysis

I have a course project on harmonic measure and Fourier analysis. The goal is to give a presentation on a part of harmonic measure theory which relates to Fourier analysis. Harmonic measure is a vast ...
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1answer
194 views

Dirichlet Problem: Uniqueness of solution

Let $u$ be the solution to a Dirichlet Problem on a bounded open domain $D \subset \Bbb R^n$. Is the uniqueness of $u$ guaranteed by the maximum principle or by the smoothness of the boundary of $D$? ...
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723 views

Why is the absolute value needed with the scaling property of fourier tranforms?

I understand how to prove the scaling property of Fourier Transforms, except the use of the absolute value: If I transform $f(at)$ then I get $F\{f(at)\}(w) = \int f(at) e^{-jwt} dt$ where I can ...
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1answer
141 views

General questions on Cayley graphs

In Graph Theory mainly in Cayley graphs there are four general questions " according to Audery Terras" : 'Suppose A is the adjacency operator of a connected regular (undirected) graph $X$ of degree ...
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1answer
36 views

Quantization Uniform

Let the output of image sensor take values between 0 to 10. If the samples are quantized uniformly to 256 levels, show that transition and reconstruction levels are $$t_k=\dfrac{10(k-1)}{256},\, ...
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1answer
82 views

DFT of basis functions

Suppose functions $u_{1}(x)$,..$u_{K}(x)$ are a basis of $H[0,1]$( some space of real-valued functions). Define Discrete Fourier transform $$ U_{l}(x)=\sum_{j=1}^{K}u_{j}(x)\exp(2\pi i lj/K) $$ and ...
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1answer
202 views

Sobolev space exercise

I need to show $|f|_{L^\infty}\leq c|f|_{H^2} = c(\int_{\mathbb R^n} (1+|\xi|^2)^2|\hat f(\xi)|^2 d\xi )^{1/2}$, assume $f\in H^2(\mathbb R^2)$ I think I can trasnfer $f\ = \int \hat f(\xi)e^{2\pi i ...
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189 views

Autocorrelation derivation using fourier transform

I am stuck with basic understanding of the Auto-correlation derivation of a simple signal and I would be pleased if you could help me out with that. Lets have a signal $x(t)=\cos(2\pi{f_{0}}{t})$. ...
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1answer
146 views

What is this Hilbert space?

The space is $H^s(\mathbb R^d)$. If $f$ is in this space, it means $\int_\mathbb {R^n} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi < \infty$ where $\hat f$ is the fourier transform of $f$: $\hat ...
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697 views

Notes for Beginner Fourier Analysis?

Are there any good lecture notes or books on basic fourier analysis that authors publish freely online? It is very difficult to find rigorous mathematical theory of fourier analysis because google is ...
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105 views

Sampling Theorem Poisson Formula

Theorem If the Fourier transform $\hat{f}(w)$ of a signal function $f(x)$ is zero for all frequencies ouside the interval $-w_c\leq w \leq w_c$, then $f(x)$ can be uniquely determined from its sampled ...
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234 views

Orthogonal complete set of functions

Every square-integrable function on an interval can be written as a linear combination of e^inx (Fourier series). Are there any other orthogonal and complete set of functions for square integrable ...
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49 views

Is $C(B^c)$ an open set?

Assume that $B$ is an open set, if $C:=\{x=\sum_{i=1}^{n}\lambda_{i}x_{i},\lambda_{i} \geq 0,\sum_{i=1}^{n}\lambda_{i}=1,x_i \in B\}$ is a convex that contains $B$, $C$ is an open set? What's ...
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Calculate $\int_{R^{3}}e^{-2\pi ix \cdot \xi}\left(\frac{1}{4\pi}\int_{S^2}f(x-\gamma t)d\sigma(\gamma)\right)dx$

I should derive $$\int_{R^{3}}e^{-2\pi ix \cdot \xi}\left(\frac{1}{4\pi}\int_{S^2}f(x-\gamma t)d\sigma(\gamma)\right)dx=\hat{f}(\xi)\frac{\sin(2\pi |\xi| t)}{2\pi |\xi|t}$$ I already calculate ...
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63 views

Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
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1answer
151 views

Finding the Fourier transform of tf''(t)

I'm trying to find the fourier transform of $t\cdot f''$. The rules I've found that relates to this seems to be that for a function $f(t)$ and it's Fourier transform $F(\omega)$ the following holds: ...
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3answers
494 views

Heat equation, separation of variables and Fourier transform

I have a question about the heat equation $\frac{\partial \varphi}{\partial t} = \frac{\partial^2 \varphi}{\partial x^2}$ with the conditions that $\varphi(x,t=0) = f_0(x)$ and $\lim_{x \rightarrow\pm ...
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407 views

A Fourier transform using contour integral

I try to evaluate $$\int_{-\infty}^\infty \frac{\sin^2 x}{x^2}e^{itx}\,dx$$ ($t$ real) using contour integrals, but encounter some difficulty. Perhaps someone can provide a hint. (I do not want to use ...
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2answers
195 views

fourier series and heat equation

Let $v$ a solution of he heat equation, given by $\frac{\partial v}{\partial t}(t,x)=\frac{\partial^2v}{\partial x^2}(t,x)$ for $t>0,x\in\mathbb R$ with the following properties $v(0,x)=u_0(0) ...