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

12
votes
1answer
704 views

Relation between function discontinuities and Fourier transform at infinity

I have made the following assertion a few times in this space without ever having provided a proof: Let $m$ be the smallest number such that a function $f \in L^2(\mathbb{R})$ has a discontinuity in ...
1
vote
1answer
1k views

Fourier transform of a piecewise function

I am trying to find the Fourier transform of $$f(x)=Ae^{-\alpha|x|}$$ where $\alpha>0$. $f(x)$ becomes an even piecewise function defined over the intervals $-\infty$ to $0$ and $0$ to $\infty$. ...
1
vote
1answer
51 views

$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq \int_{-\infty}^{\infty}|f|^2 dx$?

Suppose complex function $f$ in the Schwartz Space, its definition see http://en.wikipedia.org/wiki/Schwartz_space how can we argue that $$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq ...
1
vote
1answer
254 views

Application of Parseval's theorem

Let $f(z)$ be holomorphic on the unit disc. Then by Parseval's theorem, $$\int_{\theta = 0}^{2\pi} |f'(r, \theta)|^2 \; d\theta = 2\pi \sum_{n=1}^\infty n^2 |a_n|^2 r^{2(n-1)}$$ where the $a_n$ are ...
2
votes
1answer
756 views

How can I prove that the Gibbs phenomenon overshoot for a Fourier Series is approximately 9%?

The question is pretty self explanatory, I'm studying Fourier Series with the book Mathematical Methods for Physicists written by Arfken and it does not explain that.
2
votes
0answers
62 views

Is there an intuitive understanding of what a walsh coefficient is?

I am working with Walsh coefficients. I know the intuitive understanding is almost that that they are the degree of connectivity, but it is there a better way of thinking about it? What is the ...
2
votes
1answer
108 views

Fourier transformation - connection between exponential and trigonomeric forms

On Wikipedia i have come across a Fourier transformation equation in exponential form and its inverse (Wiki): $$ \begin{split} \mathcal{F}(x) &= \int\limits^{\infty}_{-\infty}\mathcal{f}(k) \, ...
0
votes
2answers
111 views

Find the frequency response if i have the magnitude response?

if i have the transfer function of magnitude response is there a method that i could calculate the frequency response? For example the transfer function of the magnitude response is: $ 3db \pm ...
2
votes
1answer
123 views

Fourier Transform of this function

I am trying to find if the Fourier transform of the following function is positive of not. $$ f(x) = \frac{1}{2-e^{-x^2}} $$ How can I find the Fourier transform? I started with the classical ...
3
votes
1answer
62 views

Class of functions that the Fourier inversion holds

The following is from Stein and Shakarchi's Complex Analysis: For each $a>0$ we denote by ${\mathcal F}_a$ the class of all functions $f$ that satisfy the following two conditions: The function ...
2
votes
1answer
345 views

FFT Algorithm for an interpolating polynomial

I'm trying to use the Fast Fourier transform algorithm to determine the trigonometric interpolating polynomial of degree $16$ for $f(x) = x^2\cos(x)$ on $[-\pi,\pi]$ I see a computer result in my ...
1
vote
1answer
732 views

Find the Fourier series for $\cos^3(\theta)$

Find the Fourier series for $\cos^3(\theta)$. Ok, so i have calculated that $$A_0 = \frac{1}{2\pi} \int^{\pi}_{-\pi}\cos^3(\theta)d\theta= \frac{1}{2\pi} ...
1
vote
0answers
34 views

Compute frequency-wavesnumber plot from video

Hey StackExchange Math, I have a question regarding the computation of a wavenumber-frequency graph from a video. In the paper "Simulating Ocean Water" by Jerry Tessendorf (link), it is stated that a ...
1
vote
0answers
35 views

Relations between complex functions satisfying a specific condition

What is the relation between the following two complex functions: $$g(\theta)=\sum_n x[n]\ y[n]\ e^{in\theta}$$ and $$f(\theta)=\sum_n \left(x[n]\pm i\sqrt{1-x[n]^2}\right)\ y[n]\ e^{in\theta}$$ ...
12
votes
3answers
3k views

How was the Fourier Transform created?

The Fourier Transform is a very useful and ingenious thing. But how was it initiated? How did Joseph Fourier composed the Fourier Transform formula and the idea of a transformation between periodic ...
0
votes
2answers
63 views

Trigonometric proof query

I am having trouble proving the following identity (where $m,n \in \mathbb{R}$ are arbitrary): $$\sin(mx)\sin(nx) = \frac{1}{2}[\cos(m -n )x - \cos(m + n)x] \quad (1)$$ By expanding the RHS, I can ...
6
votes
2answers
436 views

log sin and log cos integral, maybe relate to fourier series

I try to use the method of differentiation under integral sign for the first one And integrate it back, but I failed to find the constant $c$ .... Anyone hav other method? $$\begin{align} & ...
1
vote
1answer
158 views

Stuck with Fourier transform

I'm trying to solve a simple mass-dashpot-spring system $$m\ddot{u}(t) + c\dot{u}(t) + ku(t) = F(t)$$ by making use of the Fourier transform defined as $$\tilde{f}(s) = \int_{-\infty}^{\infty} f(t) \, ...
0
votes
1answer
882 views

Finding the complex exponential form of the fourier series of a function

Find the complex exponential form (i.e. $\sum_{n=-\infty}^{\infty}c_n e^{\frac{2\pi}{T}nt}$) of the Fourier series of $$2+\frac{1}{2}\cos(t+45^\circ)+2\cos(3t)-2\sin(4t+30^\circ)$$ EDIT: Some info on ...
2
votes
1answer
127 views

Fourier transform of locally integrable function

I have a question and I don't know if it is true. Is any locally integrable function a sum of an $L_{1}$ function and another nice function (perhaps, an $L_{2}$ function). This is related to the ...
2
votes
1answer
82 views

Show its a continuous linear functional (distribution)

I want to show that for $\phi\in C_0^{\infty}(\mathbb{R})$, the following map $$\Lambda(\phi) = \int_{-a}^a\frac{\phi(x)-\phi(0)}{x}dx $$ for $a>0$, is a continuous linear functional. (i.e. a ...
0
votes
1answer
305 views

Power spectrum for discrete signals.

If $x(t)$ is a real (aperiodic) power signal, i.e. \begin{equation} 0<\lim_{T\rightarrow\infty} \frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2 dt<\infty \end{equation} $x_T (t)$ is a truncated version of ...
5
votes
2answers
422 views

How to visualise Fourier Transform of a function?

I solved many problems on Fourier series,transforms and inverse fourier transforms as part of my academics. And i am aware that FT converts a time domain signal to frequency domain and IFT is vice ...
3
votes
2answers
53 views

Fourier sums in cosine and sine and Borel resummation

is there a method to evaluate the fourier sums ?? $$ \sum_{n=0}^\infty t^n \sin(nx)= F(x,t) $$ $$ \sum_{n=0}^\infty t^n \cos(nx)= G(x,t) $$ my idea is that i need to use these sums to apply Borel ...
2
votes
2answers
971 views

How can I apply a low pass filter to a Fourier Series?

I have a square wave signal and its Fourier Series. This signal pass through an ideal low pass filter,H(f), which has a cutoff frequency of 4KHz. What is the resulting baseband bandwidth of the ...
2
votes
2answers
887 views

3D Fourier transform

I don't know how to evaluate an integral of the form $$\int d^3 r \exp(-i \vec r\cdot\vec q)\exp(-a^2 r^2)$$ where $a\in \mathbb R$. Could anyone please teach me how to do this integral? Many ...
3
votes
1answer
301 views

Solving the heat equation using Fourier series

I'm interested in using the Fourier transform to solve the heat equation. I've been poring over this wikipedia article: ...
1
vote
1answer
56 views

Why does $\sin{\alpha}\cdot i\sin{\alpha x}$ disappear from this integral?

In a section on fourier transforms, my textbook contains these steps for an example: $$f(x) = \int_{-\infty}^\infty \frac{\sin{\alpha}}{\pi \alpha}e^{i\alpha x}d\alpha$$ $$= ...
4
votes
1answer
391 views

proof of Poisson formula by T. Tao

I do not understand one thing in an article on the blog of Terence Tao: For instance, restricting a function $f: G \rightarrow \mathbb{C}$ to a subgroup $H$ causes the Fourier transform $\hat f$ ...
3
votes
3answers
235 views

Two improper log integrals

Evaluate $$\int_0^{\frac{\pi}{2}}\ln ^2(\tan x)\text{d}x$$ $$\int_0^{\frac{\pi}{2}}\ln ^2(\sin x)\text{d}x$$
10
votes
1answer
317 views

Regularizing effect of the heat equation

Consider the heat equation on $\mathbb{R}_+\times\mathbb{R}^d$ \begin{align*} \partial_t u -\Delta_x u &= f, \\ u(0,x)&=u_0(x). \end{align*} In the case where $u_0\in L^2(\mathbb{R}^d)$ ...
4
votes
2answers
1k views

Obtaining the $\frac{1}{2\pi}$ factor in the Fourier transform

This MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ ...
2
votes
2answers
175 views

A integral with polygamma

I was doing a integral, the last part is $$\int_0^{\frac{\pi}{2}}x^3\csc x\text{d}x$$ I ran this on Maple, it turns into polygammas...How we evaluate this? I think there should be a way to evaluate ...
1
vote
1answer
70 views

Do we have a general form for this integral?

Is there a general formula or recursion for this integral? $$\int_0^1\left(\frac{\arcsin x}{x}\right)^n\text{d}x,\ \ n\in\mathbb{N}$$
1
vote
0answers
67 views

Questions about Fourier transform.

I am reading the notes Lecture notes on representation theory. I have some difficulty in proving a) in Exercise 1.2. We need to prove that $$ \mathcal{F}^2(f)(x) = qf^{-}. $$ I compute as follows: ...
3
votes
2answers
168 views

Another integral with Catalan

Show that: $$\int_0^1\frac{\arcsin^3 x}{x^2}\text{d}x=6\pi G-\frac{\pi^3}{8}-\frac{21}{2}\zeta(3)$$ I evaluated this by some Fourier series. Is there any other method? Start with substitution of ...
2
votes
0answers
76 views

The variance of a square integrable function

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable, symmetric, has infinite support ($\text{supp}(f)= \mathbb{R}\backslash U$, where $U$ is a set of points), and decays at infinity. ...
1
vote
1answer
252 views

Fourier transform of a function is square integrable

Is there a result stating that if a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable and decays at infinity, then its Fourier transform is also square integrable?
3
votes
1answer
232 views

Solution of the Dirichlet problem

I'm reading Jones' book Lebesgue integration on Euclidean space. Let $u(x, y)$ be a harmonic function on the half space $\mathbb{R}^n \times (0, \infty)$, with boundary condition $f(x) = u(x, 0)$. On ...
3
votes
2answers
189 views

Solve $y'' + 4y = e^{-x^2}$ using Fourier transforms

I need to solve the equation $y'' + 4y = e^{-x^2}$ using Fourier transforms. I was able to take the Fourier transform of both sides and solve for $\hat y$. I have $\hat y = ...
2
votes
1answer
709 views

Matlab FFT-algorithm example, one simple question

This is a question about an example for the software Matlab, but I still chose to ask it here, since I suspect that the question is more about the math involved than the software itself. I want to ...
6
votes
0answers
137 views

Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
0
votes
1answer
29 views

sup-Inequality for function

Hoi, I want to show that for $\phi\in C_0^{\infty}(\mathbb{R}^n)$ where supp $\phi = \overline{B}(0,r):=B$ we have $$\sup_{x\in B}|\phi(x)|\leq 2R \sup|\partial_{x_1}\phi(x)| $$ I dont quite ...
2
votes
1answer
66 views

What is the degree of the fourier expansion

Let $ f:\{-1,1\}^3 \rightarrow \{-1,1\} $ , $f(x)= \operatorname{sgn}(x_1+x_2+x_3)$; (Majority function), then Fourier expansion of $f$ is $f(x)= \frac{1}{2} ...
2
votes
1answer
100 views

Norm of this linear operator

Let $X$ be the Banach space of all continuous functions $f(x)$ on $[0,2\pi]$, provided by the uniform norm $$ \|f\|=\max_{x\in [0,2\pi]}|f(x)|. $$ Let $$ ...
1
vote
0answers
25 views

$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...
2
votes
1answer
117 views

More on the generalized integral

Refer to my previous topic: A generalized integral need help I think we get this : $$\frac{\sin \theta}{1-2\cos \theta x+x^2}=\sum_{k=1}^{\infty}\sin (k\theta )x^{k-1}$$ Then $$\int_0^1\frac{\ln ...
1
vote
0answers
226 views

Fourier Coefficients of Complex Measure

For my homework I am trying to prove the following: Suppose $\mu$ is a complex Borel measure on $[0,2\pi)$, and define the Fourier coefficients of $\mu$ by $\hat{\mu}(n)=\displaystyle\int ...
2
votes
2answers
365 views

the fourier transform of a “double convolution”

Suppose I have a function $$ m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x) $$ I want to find the Fourier transform of m(x) in terms of the Fourier transforms of $f,h,g$ but for the ...
5
votes
0answers
347 views

How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?

I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g. $P(t) = IFT_w \{ ...