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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61 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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2answers
28 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 ...
0
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1answer
31 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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0answers
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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48 views

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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2answers
45 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
15 views

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 ...
0
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1answer
18 views

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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0answers
8 views

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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0answers
10 views

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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1answer
235 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
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6 views

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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1answer
13 views

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)] = ...
2
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1answer
58 views

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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1answer
42 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
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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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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0answers
27 views

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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757 views

Sobolev space is an algebra

How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have ...
0
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1answer
27 views

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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2answers
42 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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1answer
35 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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0answers
27 views

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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1answer
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!
4
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1answer
82 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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1answer
50 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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What is the Fourier transform of the product of two functions?

Given $x(t) = f(t) \cdot g(t)$, what is the Fourier transform of $x(t)$? If possible, please explain your answer. The motivation behind the question is homework, but this is a basic principle in ...
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1answer
16 views

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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1answer
187 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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0answers
39 views

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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1answer
39 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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5answers
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Difference between Fourier transform and Wavelets

While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia. The main difference is that wavelets are localized in both time and frequency whereas ...
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333 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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0answers
29 views

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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0answers
63 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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2answers
113 views

What is difference between Fourier Transform and Fast Fourier Transform?

If you think about Fourier Transform, in the classical cases, say on the real line, what it is? Just a waded sum. Right? You take a function $f$, and you take it's Fourier Transform at particular ...
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0answers
20 views

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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1answer
33 views

Degrees of freedom in each domain in Discrete, Continuous and Mixed Fourier Transforms

I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms. In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which ...
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1answer
214 views

Fourier transform of $f(x)=\frac{1}{e^x+e^{-x}+2}$

Let $$f(x)=\large \frac{1}{e^x+e^{-x}+2}$$ Compute the Fourier transform of $f$. We can factor the denominator to get $$f(x)=\frac1{(\exp(x/2)+\exp(-x/2))^2}=\frac1{(2\cosh(x/2))^2}$$ I'm thinking ...
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1answer
52 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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0answers
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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1answer
228 views

Is there a combinatoric identity for the multiplicities of the following set?

Are you ready for some psychedelic pictures? Define the multiset$$S_n=\left\{\sum_{j=1}^n(-1)^{\left\lfloor(k-1)/2^{j-1}\right\rfloor}u_n^j\mbox{ for }1\leq k\leq2^n\right\}$$ where ...
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0answers
26 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., ...
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0answers
17 views

Need some help computing Fourier Transforms of a couple of functions

I have the following functions and would like to find the Fourier Transform: a) $x(t) = (t + 2)^2 e ^ {-2t} $ b) $x(t) = e^{-5it} [σ (t+7) - σ (t+1)]$ I don't really know what $σ$ stands for so I ...
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1answer
19 views

Frequency response of unit impulse function

Could someone throw some light on how to get the frequency response of unit impulse function. I am not from EE, but I need it for my wavelet study.
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2answers
319 views

Multidimensional Fourier transform of the laplacian

In my course on electromagnetic field theory we use the Fourier transform to simplify Maxwell's equations, for example: $$\frac{\partial ^2\vec E(\vec r,t)}{\partial t^2} \rightleftharpoons ...
4
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0answers
28 views

Differentiation in Besov-Zigmund spaces

This is my second question in a short time on Besov spaces. I apologize. I am having a rough time with them and I really need to understand this spaces quickly. The besov spaces ...
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1answer
34 views

Power series and Fourier identity approximated in two or three iterations

I understand that Fourier has proven that the sum of sines and cosines can be used to describe (almost) any curve. The power series describe that the sum of polynoms can be used to describe (almost) ...
0
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1answer
19 views

Heat Equation Two Conditions

I'm currently working on solving the Heat Equation in a one dimensional rod of length $L$. However, instead of the 'usual' singular condition $u(x,0)=f(x)$ for all $0\leq x\leq L$, I am given ...