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).

learn more… | top users | synonyms

0
votes
1answer
22 views

Fourier coefficients of a triangle function

I'm trying to find the Fourier coefficients ($c_n$), of the following function : for $x$ in $[-\pi/2;\pi/2[$ $f(x)=x$ for $x$ in $[\pi/2;3\pi/2[$ $f(x)=\pi-x$ I think its not that hard, but I keep ...
2
votes
2answers
40 views

Value Proposition of Fourier Analysis?

I am a software engineer trying to wrap his head around Fast Fourier Transform (FFT). Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of ...
1
vote
1answer
34 views

If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$

I'm learning about Fourier analysis and need help with the following problem: Suppose $f \in L^1[-\pi, \pi]$ and $\alpha_n, \beta_n$ are the Fourier coefficients of $f$. Show that if $f$ is odd ...
1
vote
1answer
14 views

Laplace equation with boundary conditions in polar coordinates

Show that the problem with this boundary conditions $u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}=0$, $\quad 0 < r < 1, \quad 0 < \theta < \pi$ $u(r,0)=0$ $u(r,\pi) =T_0$ $u(1,\theta) =T_0 $ ...
2
votes
1answer
51 views

Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots = \frac{1}{2}\cot\frac{x}{2}$ for $x \neq 2k\pi, k \in \mathbb{Z}$

I'm learning about Fourier series (specifically Cesàro summation) and need help with the following problem: Show that the Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots$ is equal to ...
1
vote
2answers
51 views

Why is $A(\mathbb{T})\subset C(\mathbb{T})$?

where $A(\mathbb{T})$ is the space of Absolutely Converging Fourier Series and $C(\mathbb{T})$ is the space of Continuous Functions, both over $\mathbb{T} = [0,1)$. If $ f\in A(\mathbb{T})$ is $f\in ...
0
votes
0answers
8 views

squared bicoherence matlab parameters

I would like to use a modified version of the squared bicoherence formula in Matlab. HOSA does not contain the squared bicoherence formula. The bispectrum is given by the following equation $$\ ...
1
vote
1answer
47 views

Given $n$ points in the plane, prove that less than $2n^{\frac{3}{2}}$ pairs of points are a distance 1 apart.

Given $n$ points in the plane, prove that less than $2n^{\frac{3}{2}}$ pairs of points are a distance 1 apart. It seems like Piegeon Hole Principle but I don't know how to proceed.
6
votes
2answers
101 views

Does $ \int_a^b |f(x) - f_1(x)| = 0$ imply $ \int_a^b |f(x) - f_1(x)|^2 = 0$?

Context:I'm trying to solve this problem: Suppose $f, f_1, g, g_1$ all Riemann integrable complex valued functions on $[a, b]$ such that $f \sim f_1$ and $g \sim g_1$. Prove $\langle f, g \rangle ...
0
votes
0answers
20 views

Laplace equation with Boundary value conditions by parts

I don´t know how to procced in this problem by parts $u_{xx}+u_{yy}=0$, $\quad 0 < x < \pi, \quad 0 < y < \pi$ $u(x,0)=0$ $u(0,y) = \begin{cases} y, & \text{for } 0 < y < ...
1
vote
0answers
22 views

A Hilbert transform that takes several functions

While playing with some PDE I came across a singular integral that looks something like ...
0
votes
1answer
38 views

Mean-Square Fourier Convergence

Let $ \left \{X_n\right \} ^{\infty}_{n=1}$ be any orthogonal (in the $L^2$ sense) set of functions. Let $$S_N(f) = \sum^{N}_{n=1} \frac{(f, X_n)}{ \left \|X_n\right \|^2} X_n$$ be the “Fourier ...
4
votes
2answers
60 views

Taking inverse Fourier transform of $\frac{\sin^2(\pi s)}{(\pi s)^2}$ [duplicate]

How do I show that $$\int_{-\infty}^\infty \frac{\sin^2(\pi s)}{(\pi s)^2} e^{2\pi isx} \, ds = \begin{cases} 1+x & \text{if }-1 \le x \le 0 \\ 1-x & \text{if }0 \le x \le 1 \\ 0 & ...
1
vote
0answers
30 views

Fourier transformed multiplication operator leaves $L^2([-C,C])$ invariant?

Let $C > 0$ be some constant and $L^2([-C,C])$ the square integrable functions on $[-C,C]$. Let $\delta > 0$ and let $M_{|\cdot |^\delta}$ denote the multiplication operator on $L^2(\mathbb R)$ ...
0
votes
1answer
28 views

How to solve this Laplace boundary value problem by Fourier series

can someone help me?, I don't know how to proceed in the last boundary condition $u_{y}(x,1)=x(1-x)\ $ $u_{xx}+u_{yy}=0\ $, $\ 0<x<1,\ 0<y<1$ $u(0,y)=0$ $u(1,y)=0$ $u_{y}(x,0)=0$ ...
0
votes
0answers
16 views

Computing eigenfunctions, difference between beta and beta prime.

I am trying to implement the method described in the following paper. I am just kind of confused as to the difference between beta and beta prime. I am much more a computer programmer than ...
0
votes
0answers
17 views

How to find the Fourier series of the following function?

I know how to calculate a Fourier series for odd and even functions but I wish to evaluate the $2\pi$ periodic function given by $$f(x)=\begin{Bmatrix} 0~~~-\pi \leq x \leq 0 \\ 1~~~0 < x \leq ...
0
votes
0answers
22 views

$f(t)=\sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}$ for $f\in L^{2}(\mathbb T)$?

Let $f\in L^{2}(\mathbb T).$ Define $g(t):= \sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}, (t\in \mathbb T).$ Since $\hat{f} \in \ell^{2}(\mathbb Z),$ we note that $g\in L^{2}(\mathbb T).$ My ...
2
votes
1answer
31 views

When is the Fourier Transform of a function periodic?

Using the duality property, I guess it happens whenever the original signal is composed of a sum of dirac delta functions spaced at equal intervals of time. I conclude this as the Fourier transform ...
3
votes
0answers
28 views

Relations between Laplacian and Fourier transform

I have been thinking this question for a couple of days. To be more specific, I want to derive an explicit expression of Fourier transform under some coordinate chart of a manifold. What I do have ...
-1
votes
1answer
38 views

Fourier Analysis and its applications [duplicate]

My question has two parts: $1)$ Could anyone explain in simple terms what a Fourier Transform is? $2)$ What are some of the applications of Fourier Analysis in the field of high school mathematics? ...
0
votes
0answers
14 views

Decay of fourier series implies existence of (non contiuous) derivative

Let $a_n=\mathcal{O}(\lvert n\rvert^{-\alpha})$ where $\alpha>\frac{3}{2}$ then \begin{equation} f(x):=\sum_{n\in\mathbb{Z}}a_n e^{inx}, \end{equation} is differentiable and its derivative is in ...
2
votes
1answer
32 views

How to compute the fourier transform of $\operatorname{sgn}$ directly?

I've been trying to compute the fourier transform of $\operatorname{sgn}(x)$, but I'm having trouble with the complex exponential at infinity. The issue is the following: by definition we have ...
0
votes
0answers
4 views

Can a Fourier series have a recurring “sub-sequence” in a single period?

Let $f(x) = \frac{a_0}{2} + \sum_{n=1}^N a_n \sin(\frac{2\pi}{P}nx + \phi_n)$ be a Fourier series with period $P$. Obviously $f(x)$ is repeating every period $P$, but I'm wondering if within one full ...
2
votes
1answer
29 views

Poisson equation in semi-infinite domain

Im trying to solve the following Poisson equation: $$\nabla^2\phi = F(x,y)$$ $$for\ x\in(0,\infty)\ and\ y\in(0,L)$$ $$\frac{\partial\phi(x,0)}{\partial y}=0\ , \ \frac{\partial\phi(x,L)}{\partial ...
1
vote
0answers
70 views

Why is this theorem equivalent to the informal explanation given by Tao?

I will copy-paste the statement and the theorem from this paper by Tao about an uncertainty principle for groups of prime order. http://arxiv.org/pdf/math/0308286.pdf Theorem 1.1: Let $p$ be a prime ...
1
vote
1answer
30 views

Convolution identity in Schwartz space

I met a convolution identity in section 7.4, page 185, Functional Analysis by Rudin, which I cannot justify . $$(P(D)f)*e^{itx}=f*P(D)e^{itx}$$ where $D$ is differential operator and ...
0
votes
1answer
25 views

Differential operator- Equality

Suppose that $L$ is a linear differential operator such that $Lu(x)=f(x)$. Why does the following equality hold? $$L \frac{1}{\sqrt{2 \pi}} \int_{\mathbb{R}^n} \hat{y}(\omega) e^{i \omega t} d ...
2
votes
1answer
24 views

Fourier Transform of a Series is the Series of Fourier Transforms?

If $\phi_k$ ($k\in\mathbb{Z})$ are in $L^2(\mathbb{R})$ and $\displaystyle\sum_{k\in\mathbb{Z}}\phi_k$ converges in $L^2(\mathbb{R})$, then is it true that $$ ...
7
votes
1answer
99 views

Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains $$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$ The standard ...
0
votes
0answers
32 views

Expansion of an eigenfunction $\psi$ into a Fourier series

I was going through a paper by Jean Bourgain and it states that an eigenfunction $\psi$ of the Laplacian $\Delta$ with eigenvalue $-\lambda$ on the flat $n-$torus ...
2
votes
2answers
41 views

Find Fourier Transform without the use of integration?

If I have $\mathcal{F}(f(t))=F(\omega)$ with $$f(t)= \begin{cases} 1, & 0 \leq t < \pi\\ -1,&- \pi \leq t <0 \\ 0 & \text{otherwise}\end{cases}$$ I have found $F(\omega)= ...
0
votes
0answers
28 views

Fourier transform of $\sin(2\pi fm t) \sin(2\pi fc t) $

How do I find the Fourier transform of $\sin(2\pi f_m t)\sin(2\pi f_c t) $? My main confusion comes from $fm$ and $fc$. If I had the same frequency I could of used a trig identity and then Fourier ...
1
vote
1answer
36 views

About Hardy-Littlewood maximal function from Grafakos' “Classical Fourier Analysis”

The Hardy-Littlewood maximal operator $M$ is defined by $$ M f(x)=\sup_{r>0}|B(x,r)|^{-1}\int_{B(x,r)} |f(y)|dy. $$ There is the following example in Grafakos's "Classical Fourier Analysis" book ...
3
votes
2answers
111 views

Intuition behind Fourier and Hilbert transform

In these days, I am studying a little bit of Fourier analysis and in particular Fourier series and Fourier/Hilbert transforms. Now, I am confident with the mathematical definitions and all the ...
2
votes
1answer
38 views

Error in the statement of Wirtinger's inequality?

Theorem. Suppose that $f(x)$ has a continuous derivative on the interval $[0, 1]$, and that $\int_0^1 f(x)\, dx=0$. Then $$\int_0^1 |f'(x)|^2\, dx\ge 4\pi^2\int_0^1 |f(x)|^2\, dx.$$ Proof. ...
0
votes
1answer
35 views

Inverse Fourier transforms with Heaviside step function

I want to find the inverse Fourier transforms of: $$u(\nu + 1) \ \exp(-\nu)$$ Attempt: So the inverse Fourier transform is given by: $$\int^\infty_{-\infty} u(\nu + 1) \ e^{-\nu} e^{j2 \pi t} \ ...
1
vote
2answers
28 views

Using Paley-Wiener theorem and Fourier inversion formula to get this result

I want to solve the problem #8 of Stein's book Complex analysis in Chapter 4, for the first part I've got the following: We know that the coefficients of a series are: ...
1
vote
0answers
35 views

Proof that fixed points of a null field are zero

In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2 $ Suppose we have a ...
0
votes
1answer
29 views

Inverse Fourier Transforms

Find the inverse Fourier transform of the following: $$\sin(2 \pi \nu T) \cos (10 \pi \nu T) / (\nu T)$$ Attempt: I was told it was easier if we rewrite this in terms of a $sinc$ function. I think ...
1
vote
1answer
48 views

How can I write this in a more convenient way?

This question is part of a question deriving Fourier coefficients, so one of them is $$b_n = \frac{1-\cos(\frac{n\pi}{2})}{n\pi}$$ I think this is ugly, so $$ b_n = \begin{cases} ...
1
vote
1answer
28 views

Using Fourier Series to find the sum of a numerical series

I have to use a Fourier series to compute the sum of the series $$\frac{1}{2} + \sum_{n=1}^\infty (-1)^n \frac{1}{n^2 + a^2}$$ My guesses are the Fourier series $$e^{ax} = \frac{e^{a\pi} - ...
1
vote
1answer
17 views

Prove the following about the integral of Fourier coefficients

I'm having a difficult time going from $$\sum_{n=1}^\infty (\cos nx \int_{-\pi}^\pi f(t) \cos nt dt + sin nx \int_{-\pi}^\pi f(t) \sin nt dt)$$ to $$\sum_{n=1}^{\infty}(\int_{-\pi}^\pi f(t) \cos ...
0
votes
0answers
17 views

Application of Abel's Method to Summation of Fourier Series Question

The series $$f(x, r) = \frac{a_0}{2} + \sum_{n=1}^\infty r^n (a_n \cos nx + b_n \sin nx)$$ where $0 \le r \lt 1$ clearly converges, as the terms monotonically decrease. My question: My textbook ...
0
votes
1answer
27 views

Periodic version is constant implies f is constant

Let $f:\mathbb R\to \mathbb R$ be a uniformly continuous function with finite measure support. Let $g$ be the periodic version of $f$ defined on [0,1), that is: $$ g(x)=\sum_{k\in \mathbb Z}f(x+k) $$ ...
1
vote
2answers
39 views

Is $|x|^{-r}$ tempered distribution?

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |(1+|x|)^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ ...
0
votes
1answer
28 views

Is there is notion of Fourier transform of distribution?

We note that every tempered distribution is a distribution. Can we find a example of distribution which is NOT a tempered distribution? Can we talk of Fourier transform of that ...
0
votes
0answers
13 views

Determining wave algorithm based on sine wave

I have some data that I've noticed conforms to a sine wave and I want to approximate it as closely as I can. In the graph, the blue line is the data I want to model as closely as possible. From ...
0
votes
0answers
32 views

Derive the Fourier Transform

I have been asked to derive the Fourier Transform for $$f(x)=\frac{1}{x^2+a^2}$$ where $a>0$. I know the Fourier Transform is equal to ...
1
vote
0answers
18 views

Time-Shifted Trigonometric Fourier Series Coefficients

I'm trying to find the Trigonometric Fourier series coefficients for a particular periodic function. Given $$f(t) = 2 - \frac9\pi \sum_{n=1}^\infty\frac1{2n-1}\sin\left(\frac{2n-1}2 \pi t\right)$$ ...