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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How to simulate the stroboscopic effect? [on hold]

Can anyone show me an example of estrobocópico effect in matlab? Or at least send me a link of any simulation. Thank you.
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18 views

Surface measure and the wave equation

I am having a annoying conceptual problem trying to solve problem 46 in Chapter 8 of Folland's "Real Analysis". I'll try to explain my problem as briefly as possible. Consider the wave equation ...
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Fourier Transform of mix partial derivative

I know FT{$\frac{\partial u}{\partial x}$} = (ik)FT{u}. Give a function $U(x,y)$. Is the following true? FT{ $\frac{\partial^2 U}{\partial y \partial x}$} = FT{$\frac{\partial U}{\partial y}$} ...
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Convergence of Fourier Series in $L^1(\mathbb{T})$

Suppose $f \in L^1(\mathbb{T})$ and the sequence of partial sums of its Fourier series converges (in $L^1(\mathbb{T})$) to $g$. How can I prove $f=g$?
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34 views

When orthogonal polynomials form an Hilbert basis?

Let $\mu$ be a probability measure on $\mathbb R$, and consider the sequence of orthonormal polynomials in $L^2(\mu)$. These polynomials are constructed by applying Gram-Schmidt to the sequence ...
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Frequency response and low pass filtering

I have one small doubt. I am studying filter banks, and hit upon by the frequency response of Haar wavelets. Which has the following form: $$\frac{1}{2} + \frac{1}{2}e^{j\omega}$$ This drops to to ...
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158 views

Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
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Calculate the fourier transformation of a function to determine the answer of another…?

I have an assignment which is: Show that $\int_{-\infty}^\infty (\frac{\sin x}{x})^2dx=\pi$ by calculate the Fourier Transformation of $ f(t) = \left\{ \begin{array} /1, |t| \leq 1 \\ 0, |t| > 1 ...
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Fourier inversion of $f(k) \Leftrightarrow \lim_{k \to -\infty } f(k)=0$

Let $k \to f(k)$ be a function and define its Fourier transform as $$ \hat{f} (u) = \int_{-\infty}^{\infty} e^{iux} f(x) dx $$ if $\hat{f} (u)$ is integrable we can get back $f$ by doing the inversion ...
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Fourier series with half range

Question What are $a_0,a_n,b_n$ equal to with range $-L\leq x \lt0$, rather than the standard $-L\leq x \leq L$? For example: $$f(x)=2x^2,\quad-1\leq x\leq0$$ Instead of $f(x)=2x^2,\quad-1\leq ...
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60 views

Fourier transform of Bessel Function Second Kind

How do I prove the following equation, $$\frac{1}{\sqrt{(x^2 + y^2)}}=\int_0^{\infty}\frac{2}{\pi}K_0(yt)(\cos(xt))\,dt $$ This is a Fourier transform of $K$, I proceeded as follows: \begin{align} ...
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30 views

Fourier transformation of $f(t) = \frac{1}{1+9t^2} $

The assignment is to determine the Fourier transform of the following function: $$f(t) = \frac{1}{1+9t^2} $$ I have some rules that I think I can use: $\frac{1}{1+t^2}$ has the transformation $\pi ...
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24 views

Evaluate Fouries transform using properties

$$x(t)=t\ \left[\frac{\sin(t)}{\pi t}\right]^2$$ How can I find the Fourier transform of the above signal without direct integral evaluation(using Fourier Transform properties) The answer will ...
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What is the Fourier Transform of $f'(x)/x$? [duplicate]

What is the Fourier Transform of $f'(x)/x$? Is it even possible to find? It's deceptively simple looking. What about $f(x)/x$?
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58 views

What is the Fourier Transform of $f'(x)/x$ [on hold]

What is the Fourier Transform of $f'(x)/x$? Is this even possible to find? It's deceptively simple looking.
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2answers
39 views

How to solve this coupled linear differential equations?

$\partial_t f(x,t)= \alpha \partial_x^2f+\beta f + \gamma g \\ \partial_t g(x,t)= \alpha \partial_x^2g -\beta f - \gamma g$ With everything real. I tried to take the first equation and express ...
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1answer
19 views

Minimum phase non-rational transfer function: Hilbert transform between log magnitude and phase

In Signal Processing literature, it is well known that a minimum phase sequence with rational transfer function ('zeros' and 'poles' in unit circle) has Hilbert transform relation between log ...
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23 views

Finding function given its fourier coefficients

Let $f:[-\pi, \pi] \to \mathbb{R}$ be the step function $f(x) = -1$ if $x<0$, $f(x) = 1$ if $x>0$. The Fourier coefficient of $f(x)$ is given as $\widehat{f}(n) = -\frac{2i}{\pi n}$ if $n$ is ...
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What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
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25 views

Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} ...
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Inverse Fourier transform of complex hyperbolic functions

I'm trying to solve a boundary condition problem and I got the solution in frequency regime: $$f(w)=\frac{\sinh(a|w|)}{b|w|\cosh(c|w|)-iw\sinh(c|w|)}$$ I'm wondering if there's any analytical form ...
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Finding a discrete Kalman-type process that produces a given Frequency spectrum

Given a power spectral density from f = -1/2 .. 1/2, is it possible to find a 1st order process that produces this series? In other words, x_i+1 = G x_i + W r_i ...
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Prove $\mathscr{F}[{x^nf(x)}] = (j)^n\times \mathscr{F}^n[f(x)]$

How to prove this property of Fourier Transform: $$\mathscr{F}[{x^nf(x)}] = (j)^n\times \mathscr{F}^n[f(x)]$$ Fourier Transform's definition is: $$\mathscr{F}[f(x)] = ...
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Why does the point spread function not violate the linearity of the Fourier transform?

In radio astronomy the point spread function is the Fourier inverse of the $uv$-sampling function of a telescope. The $uv$-sampling function is a sum of sampling functions (one for each baseline). So ...
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Are FT and LT both isomorphic?

As the following diagram:(from a textbook) Note: 1. L2: L2 space, H2: H2 space 2. The upper one is in t-domain; the lower one, f-domain 3. : the Laplas transform operator : the fourier tansform ...
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The non-existence of one distribution

The problem is to prove that does not exists a distribution $u$ on $\mathbb{R}$ such that $$ \langle u, \varphi \rangle = \int e^{1/x^2} \varphi(x) \, dx, \hspace{0.9cm} \varphi \in ...
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39 views

Pulsating waves of zeta function

Below is an animation of the partial sums of $\operatorname{li}x-2\Re\sum_{k=1}^{N}\operatorname{Ei}(\rho_k \log x)-\log2+\int_{x}^{\infty}\dfrac{\text{d}t}{t(t^2-1)\log t},$ for $1\leq N\leq100$ ...
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1answer
39 views

Evaluate the integral $\int_{-\infty}^\infty e^{-(t²+2t)/2}e^{-i\omega t}dt$

I have this integral: $$\int_{-\infty}^\infty e^{-(t²+2t)/2}e^{-i\omega t}dt$$ I don't know how to solve it, but I have tried, like this: $$ \int_{-\infty}^\infty e^{(-(t²+2t)/2)+(-i\omega t)}dt $$ ...
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18 views

The continous of a function in the Sobolev class

Let $f\in S$ with $S= \left\{ {f:\mathbb{R} \to \left[ {0, + \infty } \right):\int_{ - \infty }^{+\infty} {{{\left| {\hat f\left( t \right)} \right|}^2}{{\left( {1 + {{\left| t \right|}^2}} ...
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Exercise 31 from chapter 4 (“Hilbert Spaces: An Introduction”) of Stein & Shakarchi's “Real Analysis”

The following problem shows that $\{e^{inx}\}_{n \in \mathbb{Z}}$ is an orthonormal basis of $L^2([-\pi, \pi])$. It is taken from [1] (exercise 31 of chapter 4: "Hilbert Spaces: An Introduction", pp. ...
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Why does the discrete cosine transform compact the information at the “low frequencies”?

I've been investigating about the discrete cosine transform. I think I understand the practical applications it has and how it is used in image/audio compression. I also know it is related with the ...
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41 views

A simple Fourier Transformation

I am a bit stuck with this small basic signal. I have this $$y(t)=\frac{\sin(200\pi\,t)}{\pi\,t}$$ and I want to take its Fourier Transformation. Obviously it looks like the sinc function. But that ...
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Why is the Fourier transform better than the Black Scholes model for options pricing?

I am doing a research paper about using the Fourier transform to price options and I need a good place to start. I understand the properties of the Black Scholes model which make it inaccurate. So ...
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54 views

Fourier transform $\frac1{t^2+2t+2}$

I need help to calculate the Fourier transform of this function: $$\frac1{t^2+2t+2}$$ Thanks!
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32 views

Decay of Fourier Coefficients implies Holder Continuity?

This is an exercise problem. I got stuck here and would like to get a hint. The problem is Suppose $f$ is continuous and $2\pi$-periodic, and $|\hat{f}(n)|\leq |n|^{-3/2}$ for all non-zero ...
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Solving the wave equation bounded by one free end and one fixed end

Given that $\{\sin\left[\frac{(2n-1)\pi}{2L}x\right] : n\in\mathbb N\}$ is the complete set of eigenfunctions of a regular Sturm-Liouville with boundary points $0$ and $L$ and weight function $1$, and ...
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47 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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Fourier transform and Z transform question?

Lets suppose we have an exercise where I have to find the Z transform and its region of convergence.I find the Z transform and the region.How do I determine if the Fourier transform exists from this ? ...
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81 views

How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
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Mellin transform with compact support

Mellin transform for $f(x)$ is usually defined as: $$F(s)=\int_0^\infty f(x)x^{s-1}dx$$ Is there a Mellin transform with compact support? For example like $$F(s,a,b)=\int_a^{b} ...
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36 views

Convergence of a sum of sines

If $ s_N(x) := \sum_{n = 1}^N c_n \sin(n x) $ converges uniformly on $[0, \pi]$ as $N \to \infty$ then $c_n = o(n^{-1})$. a) Is $c_n = o(n^{-1})$ sufficient for uniform convergence? b) Is $\sum_n n ...
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1answer
96 views

How to get fourier series of 8-bit character to be transmitted?

I have been reading this in a book, but can't understand how he used the 8-bit in fourier series equation to get the result below. The transmission of the ASCII character ‘‘b’’ encoded in an 8-bit ...
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331 views

Solving a tough integral

I am studying telecommunications theory and I was doing an exercise where it's required to find the (infinite) taps of a zero forcing equalizer. Here's the point where I am stuck at: $$ ...
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Fourier transform of $e^{if(x)}$

I'm trying to find an explicit result for the following Fourier transform: $$\mathcal{F}\left[e^{if(x)}\right](k)=\int_{\mathbb{R}^n} e^{if(x)}e^{-ik\cdot x} dx$$ So far I could come up only with a ...
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61 views

An entropy inequality

Let $f:[0,2\pi]\to \mathbb{R}$ be a smooth, positive function such that $f(0)=f(2\pi)$, and $\int_0^{2\pi}fd\theta=2\pi.$ Is it true that $$2\int_0^{2\pi}f\ln fd\theta- 2\int_0^{2\pi}\ln ...
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Need a closed form for fourier coefficients (if it exists)

i have a set of 53 fourier coefficients. the dc term is 0. the 26 positive frequency amplitudes (coefficients) are given below. the 26 negative frequency amplitudes are the same. {0.014451, ...
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3answers
36 views

Fourier Series Trig Functions

I need assistance finding the fourier series for the following function: $$ f(x)=3\cos^2(5x) $$ I know that $$ a_0={1\over 2\pi}\int_{-\pi}^\pi 3\cos^2(5x)\,dx={3 \over 2} $$ and $$ ...
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29 views

a question about integration by parts

Suppose that $t f(t) \to 0$ when $t \to \infty$ and $t f(t)\to 0$ when $t \to 0$. For the following integral, $$I(z)=\int_0^{\infty} f(t) \cos (z t) \mathrm{d}t,\qquad z>0 \tag{1}$$ We can apply ...
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51 views

Fourier series of rescaled cosine function [closed]

How would I find the Fourier series of $\cos\left(\, 5x/2\,\right) $ on $\left[-\pi,\pi\right]$? Progress $$A_0={1\over 2\pi}\int_{-\pi}^\pi \cos(5x/2)dx={2\over 5\pi}$$ $$A_n = {1\over \pi} ...
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25 views

Solution of nonhomogeneous problem using semigroup of linear operators

Let $X$ be a complex Hilbert space; and let $A:D(A)\subset X \to X$ be a $\mathbb C-$linear operator. Assume that $A$ is self-adjoint(so that $D(A)$ is a dense subset of $X$) and that $A\leq 0$ (i.e., ...