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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Fourier Transform of a Polynomial

Lets say you are given \begin{equation} f(x)=1+x^3 \end{equation} and the definition of Fourier transform: \begin{equation} \hat{f}(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ikx}f(x)dx, ...
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55 views

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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33 views

Prove that the following function is $C^\infty$ [duplicate]

Prove that the following function is $C^\infty$ (and in the point $ξ=0$) : $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ for whichever $$λ>0$$ I am trying to find a ...
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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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97 views

Prove that the following function is $C^{\infty}$ [duplicate]

Prove that the following function: $$r:x \mapsto \begin{cases} e^{-{1\over (1-x^2)}}, & \text{if $|x|<1$} \\ 0, & \text{if $|x| \ge 1$} \end{cases}$$ is $C^{\infty}$ I found this problem ...
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55 views

On the Fourier transform of $f(x)=\ln(x^2+a^2)$

I would like to derive the Fourier transform of $f(x)=\ln(x^2+a^2)$, where $a\in \mathbb{R}^+$ by making use of the properties: \begin{equation} \mathcal{F}[f'(x)]=(ik)\hat{f}(k)\\ ...
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27 views

Property of complex borel measures with absolutely convergent Fourier series (Wiener algebra)

Let $\mu$ be a complex Borel measure on the circle $\mathbb{R}/\mathbb{Z}$ with $$ \sum_{n \in \mathbb{Z}} \lvert\hat{\mu}(n)\rvert < \infty. $$ How does it follow that $d\mu(x) = f(x) dx$ for $f$ ...
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13 views

Computing the laplacian Green function by Fourier transform and analytic continuation

I know that the Green function for the laplacian operator in $d$ space dimensions $$\Delta \equiv \sum_{i=1}^d \frac{\partial^2}{\partial x^2_i}$$ is given by $$ \Delta^{-1}(x-x')=\begin{cases} ...
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38 views

Find the fourier series of the function

Find the fourier series of the function $g(x) = \sum\limits_{n=1}^\infty \frac{sin(nx)}{6^n sin(x)}$ for $x \not= k\pi$, and $g(k\pi) = \lim_{x\to k\pi} g(x)$, $(k \in \mathbb{Z})$
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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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112 views

Fourier sine transform of $\frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert$

Show that $$ \int_0^{\infty} kF(k)\sin(ka)\,dk = \frac{\pi}{2}aG(a) $$ where $$ F(x) = \frac{1}{2}+\frac{1-x^2}{4x}\ln\vert\frac{1+x}{1-x}\vert $$ and $$ G(x) = \frac{\sin x-x\cos x}{x^4} $$ EDIT: ...
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39 views

Inverse fourier transform - Where did the Heaviside function come from?

I asked this question on another forum but no answers so I'm copy/pasting it here in hopes that someone can help out
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52 views

Fourier transform of $f(x)=1/x$

I would like to compute the Fourier transform for the function: \begin{equation} f(x)=\begin{cases} 1/x&, x\in [a,b] \\ 0,& x \notin [a,b]\end{cases} \end{equation} but I cannot do the ...
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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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65 views

Fourier transform with $\sin(t^2)$

This exercise gave me nightmares this night. I have $$ x(t)=\sin(t^2)e^{-2|t-2|} $$ to Fourier transform. First I though about solving the integral. (should I divide the signal in $2$, first for ...
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2answers
72 views

Replicating Kolmogorov's Counterexample for Fourier Series in Context of Fourier Transforms

It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More ...
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2answers
63 views

On the Fourier transform of $f(x)=e^{-x^2+2x}$

So, I have the $f(x)=e^{-x^2+2x}$ and to take the FT of it, I complete the square: \begin{equation} f(x)=e^{-x^2+2x \pm1}=e^{-(x-1)^2}e \end{equation} Then, by knowing that the FT of $g(x)=e^{-x^2}$ ...
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25 views

Express g's Fourier coefficients using f's ones, if $g(x)=f(x+c)$.

The Fourier coefficients are defined (in our course) as: $$\hat{f(n)}={1\over 2\pi}\int_{0}^{2\pi}{f(t)e^{-int}dt}$$ I am asked to express g's coefficients as a combination of f's ones, given ...
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61 views

how to solve this inverse fourier $ f(x) =\int^{\infty}_{-\infty} 1/\sqrt{2\pi}\ e^{-2\pi^2/s^2} e^{ i \ s\ x}ds$

I have two functions f(x) and f(s). f(s) is the fourier transform of f(x) and tends to $$e^{-2\pi^2/s^2}$$ I need to take inverse transform of this f(s) to get to f(x). (i need to prove f(x) tends to ...
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$L^{2}$ Approximation Error of Fourier Series of Union of Disjoint Arcs

Given $N$ disjoint arcs $\{I_{\alpha}\}_{\alpha=1}^{N}\subset\mathbb{T} $,set $f=\displaystyle\sum_{\alpha=1}^{N}\chi_{I_{\alpha}}$ show that $$\sum_{|v|>k}|\hat{f}(v)|^2\le\dfrac{CN}{k}$$ This ...
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7 views

Increasing order of fourier coefficients on the boolean cube

Given a function $f:\{0,1\}^n\rightarrow \{0,1\}$, is it true that for any $S,T\subseteq[n]$, such that $S\cap T =\phi$, then $\hat{f}(S\cup T)\leq \hat{f}(S)$? It seems so to me cause, if if you just ...
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14 views

How to introduce a new equivalent using two separate boxes in variables?

I am reading a paper in mathematics about Fourier Transform. It consider two boxes A and B with centers of $ x_0(A) $ and $p_0(B)$ respectively. It introduce a new function $R^{AB}(x,p)$ as follows: ...
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1answer
48 views

Representation of Heaviside function's Fourier transform

I've seen here that the Fourier transform of Heaviside function $\Theta(t)$ is $$ \Theta(\omega) = \frac{1}{i\omega} + \pi \delta(\omega) \tag{1}$$ But in some physics texts and here I've seen the ...
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50 views

Solve one dimensional wave equation using fourier transform

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
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1answer
56 views

Solving wave equation by fourier method

I'm trying solve this wave equation using fourier method, but I am stuck... $${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \ 0<x\le L, t >0 $$ $${ u }( 0,t) = { u }( L,t) = 0$$ $${ ...
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98 views

Sets of Divergence for Fourier Partial Integals

It is a consequence of Carleson's theorem together with a transference argument that (see Section 4.3.5 in L Grafakos, Classical Fourier Analysis for proof) that the Fourier partial integrals of a ...
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52 views

Fourier transform of cosine with square root

In relativistic mechanics, i came across the Fourier transform of the following function : $\cos \left(t \sqrt{x^2+m^2} \right)$ or $e^{it \sqrt{x^2+m^2}}$ ($t$ and $m$ are constants). Is there a way ...
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When is it appropriate to neglect all terms after the first non-zero term of a Taylor expansion series?

Suppose I am interested in the Taylor expansion series of a Cosine function at the neighbourhood of a=0. In computing the series from n=0 to n = infinity, when would it be appropriate to neglect all ...
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4answers
99 views

Dirac Delta function inverse Fourier transform

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
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Dirac Delta function inverse Fourier transform [duplicate]

We know that the Fourier transform of the Dirac Delta function is defined as $$\int_{-\infty}^{\infty} \delta(t) e^{-j\omega t} dt = 1,$$ and if I were to reconstruct the function back in time ...
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21 views

Cropping off the Taylor Series

We know that the Taylor series is for expansion of any function, but for digitization we need to crop off some parts? How can we determine upto which derivative should we consider.. I am mainly ...
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1answer
38 views

An equality about Fourier transform

I have read an equality about Fourier transforms which I can not proof. It is as following: Let $u\in C_0(\mathbb{R}^n)$ and \begin{equation} g(x_1,x_2,...,x_{n-1}):=u(x_1,x_2,...,x_{n-1},0). ...
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1answer
51 views

What if the Fourier series of a periodic function also has periodic coefficients $a_k$

If given that $x(t)$ is a periodic continuous time signal, with periodic $T$. It can be expressed by the Fourier series, i.e. $x(t)=\sum\limits_{k=-\infty}^{+\infty}\,a_k\cdot e^{j k \frac{2 ...
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27 views

In what sense is the Schwarz class of functions a “class”?

As an engineer who has not learned very much modern algebra, I recently learned about "class" in the algebra sense. Then I remembered our professor calling the set of Fourier transformable functions ...
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40 views

Finding the eigenvalues and eigenfunction (tricky)

I'm given $$X"- vX' +X \lambda=0$$ (v is a constant) I have worked x' to be: X'(x) = $$\frac{1}{2} B v e^{\frac{v x}{2}} \sin \left(\frac{1}{2} x \sqrt{v^2-4 \beta ^2}\right)+\frac{1}{2} B ...
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73 views

Fourier transform is real if $f$

I want to prove that the Fourier transform $F(\xi)$ of a function $f$ will be a real function when, and only when, $f(x)$ is an even function. I'm using the following definition of Fourier ...
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36 views

Fourier analysis notation - Sh and Ch

I reading something dealing with Fourier analysis and don't know what "Sh" and "Ch" indicate. Thanks!
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22 views

Bessel equation of half-order (asymptotic)

Not really optimistic about getting a reply for a question tagged under "Bessel function" but here goes, I have $$J_{\frac{1}{2}} = (a_1 \cos(z) + a_2 \sin(z))Z^{-\frac{1}{2}} $$ and ...
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67 views

What is $\lim_{n \to \infty} n^3 a_n$? [duplicate]

$a_n$ is the Fourier coefficient of $$f(x) = \left(1 - \frac{|x|}{\pi}\right)^4$$ The answer is infinity, but can someone give an answer that doesn't require explicit computation of the $a_n$? I'm ...
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22 views

1D FFT on rotated image column by column

I am facing a problem: performing 1D FFT on a rotated column by column on a rotated image, described as following: Original Image: Rotated Image: What I have: original image convolution ...
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49 views

A Hölder continuous function whose Fourier coefficients do not decay very fast

At Stein's book of Fourier analysis (Chapter 3, page 91, exercise 15) I was trying to solve the following problem I have to prove that the result ...
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13 views

When would a Fourier Product (made up term) exist for a finite sequence of the form $C_{\text{Max}}\prod _{i=1}^k A_i \cos \left( B_i n\right)$

Let us say that we are given a finite list of points of the form C = {i,$x_i$} where i goes from 0 to the card(C) that when plotted in the Euclidean plane has some vertical axis that splits the graph ...
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The bond between Fourier Transform and Epicycle theory

Can someone help me understanding the bond between the Fourier Transform and the epicycle theory? I have searched in many places such as: http://math.stackexchange.com/a/72479/185138 ...
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39 views

Fourier transform division theorem in $\mathbb R^n$

It is known that if $f \in L^1(\mathbb R)$, $\widehat f(\xi) \neq 0$ for any $\xi \in \mathbb R$, then for any $h \in L^1(\mathbb R)$ such that $\widehat h$ is compactly supported there exists $g \in ...
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Rate of convergence of a Weyl-Heisenberg (Gabor) frame expansion

If $\{g_{m,n}\}$ is a Gabor frame for $L^2(R)$, with window function $g$, and $f \in L^2(R)$, is there a bound on the approximation error of $f$ using a finite subset of the frame? That is, is ...
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28 views

Central Limit theorem: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected [duplicate]

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
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6 views

A theorem regarding epicycles

Can somebody help me understanding the theorem on the last page of that article about Fourier Series and Epicyles? ...
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1answer
19 views

Functions of polynomial growth and the Schwartz space

A smooth function $m \in \mathcal C^\infty(\mathbb R^n)$ is said to be slowly increasing if for all $\alpha \in \mathbb N^n_0$ there exists $C_\alpha, k_\alpha$ such that $|\partial_\alpha f(x)| \leq ...
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1answer
67 views

Central Limit Theorem proof: Taylor series diverges for harmonics with higher number and those harmonics can't be neglected

I've read several proofs of Central Limit Theorem and they all seemed inaccurate to me, because they drop last members of Taylor series, whereas those members are not infinitesimal. The classical ...
3
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2answers
60 views

Fourier transform of the 1-d Coulomb potential

Though it may sound like a physical problem, but the thing I will introduce is rather mathematical. For the Fourier transform of Coulomb potential $$ V(\vec{x})=\frac{1}{\vert x\vert} $$ I can ...