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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Laplace Transform: Basis

I tend to think of the Fourier Transform (FT) as projecting a function onto a basis of cosines and sines. The Laplace Transform (LT) has a similar form to the FT, except it has been generalised. ...
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45 views

Is this function square-integrable? Able to be Fourier expanded?

I want to do a 3-dimensional Fourier series expansion on this function$$\frac{\cos (x) \cos (y) \cos (z)-\sin (x) \sin (y) \sin (z)}{\left[(a+\sin (y)+\cos (z))^2+(b+\cos (x)+\sin (z))^2+(c+\sin ...
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116 views

Easiest way to prove integral of $e^{ikx}$ is $\delta(k)$

What is the easiest way to to derive the following equation: $$\int_{-\infty}^{\infty}e^{ikx}dx = 2\pi\delta(k)$$ I understand the equation can be derived by assuming the Fourier integral theorem, ...
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61 views

The Fourier sine transform of $f(x)/\sin x$

Is the following result $$\lim_{\lambda \to \infty} \frac{2}{\pi} \int_0^\infty \frac{f(x)}{\sin x} \sin(\lambda x) \, dx = f(0) + 2\sum_{k = 1}^\infty f(k\pi),$$ where $\lambda$ is an odd integer, ...
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124 views

Square-summable sequence and Fourier series

Every square-summable sequence $(a_{n})_{n}$ is represented by $a_{n}=\widehat{f}(4^n)$, where $\widehat{f}(i)$ is Fourier coefficient of continuous function $f$. Where can I find proof of this ...
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143 views

Autocorrelation Function and Power spectrum from ACF

In my assignment I am required to write or use a C code to find the autocorrelation function of a given function and then find the power spectrum from it. The function is as follows: $$f(t) = \cos(10 ...
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25 views

Vector*Matrix multiplication through Fast Transforms

I have recently read a paper in which the authors indicated they used a Fast Cosine Transform to implement a Vector*Matrix multiplication. The idea is to decrease complexity when implementing such ...
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148 views

Limit of an integration formula

Let $f$ be a smooth real (or complex) valued function defined on $S^2$. Then a direct calculation shows that $$\int_{S^2}f(x)e^{ixy}\, ...
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51 views

Half range expansion

This is a exercise of sine half range expansion I do the bn expansion and is not the final result
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51 views

Fourier Series and sum help

I have to find the Fourier series expansion of the function $f(x)$=$x^2$ for $-\pi <x< \pi$ and using it I have to show that, i) $1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}...$ = ...
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343 views

Using the Modulation property of the Fourier Transform

I'm working on a problem: Let $X(w)$ be the Fourier transform of $x(t)$. Find the transform of $y(t)=x(5t+3)\sin(2t)$ in terms of X(w). I am table to take the Fourier transform of $x(5t+3)$ and ...
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21 views

Let $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show uniform convergence against $f$.

Let $\sum^{\infty}_{-\infty} |d_n| < \infty$ and define $f(x):= \sum^{\infty}_{-\infty} d_n e^{inx}$ for $x \in \mathbb R$. Find the Fourier series for $f$ and show it converge uniformly on ...
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89 views

Using Fourier Transforms to Solve the Heat Equation PDE In Infinite Three Dimensions

Problem: Using Fourier transforms, solve for $u(x,y,z,t)$, where $$u_t=D\nabla^2 u$$ $$ -\infty<x,y,z<\infty,t>0$$ $$D>0, u(x,y,z,0)=f(x)f(y)f(z)$$ and $u\rightarrow 0$ as ...
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17 views

How to solving for t in the first order derivative of a Fourier Synthesis.

How can I solve for $t$ given, $$ 0 = \frac{2\Pi}{T} \sum_{k=1}^N k(a_k cos(2\Pi\frac{k}{T}t) - b_k sin(2\Pi\frac{k}{T}t)) $$ What I am trying to do is, given a complex wave in the form of a Fourier ...
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324 views

Fourier Transform on compact groups

I'm trying to get my head around the concept of Fourier Transform on a compact group. The standard definition is $$\widehat{f}(\pi)=\int_Gdg\,f(g)\pi(g)$$ where $\pi\in\widehat G$, the Pontryagin ...
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24 views

$f\mapsto \sum_{n\in \mathbb Z} |\widehat{F(f)}(n)|$ lower semi continuous?

Let $T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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62 views

Method of PDE solution by Fourier transform

In Rudin's Functional Analysis (Chapter 7, exercise 17), Rudin claims that for $n=1$ or $2$, if $u$ is a distribution on $R^n$ with compact support $K$, whose Fourier transform $\hat{u}$ is a bounded ...
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122 views

What is the Fourier series of $e^{\mu\cos\theta}$?

Motivation: I want to solve this convolution problem on the circle: find $f$, given $g$ and $$ g(\theta) = \int_{S^1} e^{\mu\cos(\theta-\phi)}g(\phi)\ d\phi. $$ To do this, I want to find the Fourier ...
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59 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetic functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
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56 views

When do Fourier series and Fourier transform coincide

The other day I proved that if $f \in \ell^1 (\mathbb Z)$ then its Gelfand transform $\widehat{f}$ is a map $S^1 \to S^1$ such that $$ \widehat{f}(z) = \sum_{k \in \mathbb Z}f(k) z^k$$ and that ...
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52 views

Calculate the Fourier Transform of the function

I have to calculate the Fourier transform of the function $f(x)=sgn(x) e^{-a |x|}, a \geq 0$. After that I have to take the limit $ a \rightarrow 0$ to calculate the Fourier transform of the sign ...
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42 views

Fourier transform for pde

I solved the following PDE: $u_{t}-(u_{t})_{x}+(b+1)uu_{x}=bu_{x}u_{xx}+uu_{xxx}$ numerically, using Fourier Transform method. For this i wrote it in the following way: ...
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21 views

fourier-transform: where is my mistake?

I am trying to do a fourier-transform of the function $$\psi(x,t=0) = \frac{1}{\sqrt{\sigma}(2\pi)^{1/4}}e^{-\frac{x^2}{4\sigma^2}}e^{ik_0x}$$ My calculation is $$\int_{-\infty}^\infty ...
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51 views

Converting this sum to integral (possible?). The goal is to get error function

The solution to the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}$$ with boundary conditions and ini condition as followed: $$u(0\ or\ 1, t)=0\qquad u(x,0)=1$$ ...
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102 views

Reconstruction formula for a function of moderate decrease

I want to show that, for a function $f$ of moderate decrease and $\hat{f}(\xi)$ supported in $I=[-\frac{1}{2},\frac{1}{2}]$, $$f(x) = \sum^{\infty}_{n=-\infty} f(n)K(x-n)$$ where $K(y) ...
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140 views

Finding the natural frequency of a system via a Fourier Transform

This is a mechanical engineering question with its roots in Mathematics and so I felt this was the best stack exchange site to post my question. In short, I have used a device to measure the ...
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24 views

A Question on Fourier analysis

The following is an exercise in Knapp's book, representation theory of Semisimple Groups. For $f\in C_c^\infty(\mathbb R^2-\{0\})$ with norm $$\|f\|^2=\int_{-\infty}^\infty\int_{-\infty}^\infty ...
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95 views

Is $\int_0^\infty \sin(Kx)f_K(x)\mathrm dx$ of larger order than $\int_0^\infty \cos(Kx)f_K(x)\mathrm dx$?

Suppose we have a function $f$, such that $f$ is of some smoothness degree $m$, and $f,f^{(k)} \in L_1[0,\infty)$ $k=1,...,m$. Now if $f^{(k)}(0) = \lim_{x\rightarrow\infty}f^{(k)}(x) = 0$ for ...
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68 views

On $\int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t}$

How to count this? $$ \int_{-\infty}^{+\infty} {\frac{\tan(t-t_0)}{\cosh^2(t-t_0)} \cos(\omega t) \,\mathrm{d}t} $$ Can we use residue formula?
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Decay of Fourier coefficients of $\frac{1}{f}$

Let $\alpha > 0$ and define \begin{equation*} \mathbb{H}^{\alpha}\left[-\pi,\pi\right] = \left\{f:\left[-\pi,\pi\right]\mapsto\mathbb{R} \;s.t.\; \sum\limits_{n\in\mathbb{Z}} \left\lvert ...
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40 views

Uniform Boundedness in N of $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$

Show that $\int_0^\infty \frac{\sin(x)}{x}\,\mathrm{d}x = \frac{\pi}{2}$, and using that show that $\sum\nolimits_{0<\mid n \mid \leq N} \frac{\mathrm{e}^{inx}}{n}$ is uniformly bounded in N and ...
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104 views

Fourier Transform Help Needed

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one. The only way I could think of to solve this was by considering it ...
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53 views

Rate of convergence of Fourier series

I am having a bit of a confusion regarding convergence results. Suppose $f$ is Lipschitz, or $f \in C^\infty$ and let $S_{N}f$ be its truncated Fourier series. In the wikipedia page ...
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32 views

Fourier Transform of $sin(5t - \frac{\pi}{4})U(t+8)$

I have this function $$ sin(5t - \frac{\pi}{4})U(t+8) $$ I know the Fourier Transform of $sin(5t - \frac{\pi}{4})$, which is $$ \frac{e^{-\frac{\pi^2}{2}fj}}{2j}\left [\delta (f-\frac{5}{2 \pi}) ...
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38 views

Rearrangement of Fourier Series Sum to attain convergence

Let $f$ be a continuous function with diverging partial Fourier sums $S_N(f)(0)$ : $$ f(\theta) = \sum \limits_{k=1}^\infty \alpha_k P_{N_k}(\theta)$$ Let $f(x) \sim \sum \limits_{n=-\infty}^\infty ...
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How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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91 views

Spatio-temporal triple correlation

I would like to simplify if possible the spatio-temporal triple correlation of the following function: $$f(\vec{x},t)=\delta(\vec{x}-\vec{x}_0(t)) \otimes f_p(\vec{x})$$ where $\delta$ is the Dirac ...
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32 views

Why is this an inverse fourier cosine transform?

I would like to understand the principle of Fourier transform spectroscopy. This is explained in Wikipedia. I did all the modeling of the system and I got the same formula: $$ I(p,\tilde{\nu}) = ...
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71 views

Inverse Fourier transform of two variable function $F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}$

I am trying to find the inverse Fourier transform of: $$ F(k_x,k_y)=e^{ikz} e^{-ik_\rho ^2 z /2k}, $$ where $k^2 = k_x^2 +k_y^2 +k_z^2 = k_\rho ^2 +k_z^2 $ is a constant. I am getting confused as to ...
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73 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
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42 views

When it is possible to integrate an oscillatory integral?

Let $\phi(x,t;\theta) = \theta f(x,t)$, $\theta \in \mathbb R$, $x \in \Omega \subset \mathbb R^k$ ($\Omega$ is a domain), $t \in (0,+\infty)$, be a phase function and define an oscillatory integral ...
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46 views

Fourier transform evaluation

Let $x\in \mathbb{R}^n$ and $h(x)=\left\{\begin{array}{cc} 1 & ,|x|<1\\ 0 & ,|x|\geq 1 \end{array}\right.$ I want to find the Fourier transform , $\hat{h}(\xi).$ Here is how I proceed ...
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52 views

A quantity depending on two independent variables must be a constant, why?

I'm studying about Fourier analysis and there is one part in my book about partial differential equations I don't understand. It states that a quantity, which depends on two independent variables $x$ ...
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76 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
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71 views

Is the definition of DTFT using $\omega$ wrong?

I'll briefly explain this problem I faced. Let's take this simple signal: $$x(n)=\cos(\pi n)$$ The signal is not absolutely summable, however we can define its DTFT in terms of distributions. That ...
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98 views

Calculating Fourier magnitude spectrum for Local Binary Pattern histogram

I have the follwoing discrete Fourier transform function defined in my book (Computer Vision using Local Binary Patterns, Pietikainen et. al, 2011): $$H(n, u ) = \sum_{r=0}^{P-1}c_{nr} ...
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18 views

Fourier Operator and roots of Identity operator

I have seen that if Fourier operator is defined by $$ h(k) = \hat F(g(x)) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty} dx\:g(x)\:e^{ikx} $$ then $$ \hat F^2\{g(x)\}=g(-x) \implies \hat F^2 ...
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69 views

construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
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93 views

Fourier Transform of a Gaussian Signal?

As far as I know this is the formula for FT : On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please ...
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55 views

Number of roots of sine-and-cosine expression

Is there an easy proof of the following fact? Let $a_0, \ldots, a_n, b_1, \ldots, b_n$ be real numbers, not all zero. Then, the function $$a_0 + a_1 \cos x + b_1 \sin x + a_2\cos 2x+b_2\sin ...