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

2
votes
3answers
263 views

How to prove $(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$ for $x \in [0,1)$?

I tried to prove that $$(1-2x)^2=1/3+4/\pi^2\sum_1^\infty \cos(2n x \pi)/n^2$$ for $x \in [0,1)$ with Fourier analysis, but I just found a Fourier series which defines the function. I also found the ...
2
votes
1answer
364 views

Approximation of smooth periodic functions by trigonometric polynomials

I know that there is a following version of the Weierstrass approximation theorem for functions on $[a,b]$. For every $f:[a,b] \rightarrow \mathbb{R}$ there is a sequence $(P_n)$ of polynomials such ...
3
votes
1answer
265 views

Conditions for a finite Fourier series

Under what circumstances is the Fourier series of a function guaranteed to have a finite number of coefficients?
2
votes
1answer
81 views

question on fourier transform.

I ask myself what $$ {\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)'(\xi) )(s) $$ is. If it was just about $$ {\mathscr F}^{-1}( e^{it\xi} ({\mathscr F} \phi)(\xi) )(s) $$ it would ...
5
votes
1answer
186 views

Can the Fourier transform be defined as an integration over $\mathbb C$ instead of $\mathbb R$?

Can the Fourier transform of a whole function $f:\mathbb R\mapsto\mathbb C$ be defined as integration over $\mathbb C$ instead of $\mathbb R$ as well, such that $$\tilde f(k) = \frac{\mathcal ...
2
votes
1answer
89 views

A condition for $\hat f$ to be integrable

Let $f \in L^1 (\mathbb R^n)$. Suppose that $f$ is continuous at zero and that the fourier transform $\hat f$ of $f$ is non-negative. Does this imply that $\hat f \in L^1$ (and hence, by the inversion ...
2
votes
1answer
1k views

n-dimensional Fourier integral

let be $ \int_{R^{n}}dVf(r)e^{i(k.r)} $ the n-dimensional Fourier integral. $ dV=dxdydz.... $ the volume and $ (k.r)= \sum_{n} k_{n}.x_{n} $ is the scalar product of the position vector 'r' and the ...
1
vote
0answers
100 views

Dealing with integrals and Fourier transforms.

I have the following expression: $$\sum_{k}\left(\int_{-\infty}^{\infty}e^{-ikx}\, f(k')dk'-\int ...
3
votes
1answer
538 views

Fourier transform of Fourier coefficients, etc

I have some functions, which are periodic with period 1. Let one of them be $g$. Function $g$ has the following form $(K\rightarrow\infty)$: $$g(x)=\sum_{j=1}^K h\left(\frac{j}{K},x\right) $$ ...
4
votes
2answers
470 views

Pointwise, uniformly and absolute convergence of Fourier series

I'd really love your help with this one: I got this Fourier series for $f(x)=x$ in $[-\pi,\pi]$: $$\sum_{1}^{\infty}\frac{2(-1)^{n+1}\sin(nx)}{n}$$ and I need to check if it's (i) pointwise ...
2
votes
2answers
163 views

Why does the following Fourier series does not converge for $x \in R$, and does for $x \in [0,2\pi]$?

I would really love your help with the following facts that I can't understand. I can't understand why the following Fourier series does not converge: $$\sum_{0}^{\infty}\frac{e^{inx}}{n^2}.$$ 1.If ...
3
votes
1answer
352 views

Fourier transform (logarithm) question

Can we think, at least in the sense of distribution, about the Fourier transform of $\log(s+x^{2})$? Here '$s$' is a real and positive parameter However ...
2
votes
1answer
1k views

Intuition behind decay of Fourier coefficients

Many other posts have discussed the standard result that the smoothness of a function is related to the rate at which its Fourier coefficients decay. For example, there are proofs that show that if ...
1
vote
2answers
339 views

When is the convolution with a tempered distribution again a tempered distribution?

If $f$ is a Schwartz function on $\mathbb R^n$ and $g \in L^1(\mathbb R^n)$, then if $g$ is the Poisson kernel, is $f\ast g$ a Schwartz function? are there any known sufficient conditions on $g$ to ...
5
votes
2answers
7k views

FFT bins from exact frequencies

I'm trying to understand a few concepts about Fourier Transforms (mainly in the context of signal processing). Let's suppose a signal is sampled at 10kHz and that the FFT size is 1000. If 1000 ...
2
votes
0answers
79 views

Deduce the global differential equation from the pointwisely defined equation in Fourier space

Let $G\in \mathcal{F}(\mathbb{R}^{n+1})'$ be a distribution on the space of spatial Fourier transform'able function, ie an $L^1_{\mathrm{loc}}(\mathbb{R^{n+1}})$ function, $G = G(t,\xi)$. Assume ...
2
votes
0answers
401 views

How can I use the time-frequency uncertainty principle?

I have a signal composed of the summation of a set of sine waves of different frequencies. The amplitude of these sub-signals can change so many times a second. I have been told that, if I want to ...
1
vote
1answer
91 views

Asymptotic boundary on Fourier coefficients of absolutely continuous function

Let $f$ be absolutely continuous. Prove that $\hat{f}(n)=o\left(\frac{1}{n}\right)$. Any hint will be appreciated, thanks.
1
vote
1answer
947 views

Convergence of Fourier series for $|\sin{x}|$

I was solving this question I saw in a textbook. The question is : Calculate the Fourier series for $ f(x) = |\sin x| $ for $-\pi \leq x \leq \pi$. Which I had $ f(x) = \frac{a_{0}}{2} + \sum ...
1
vote
1answer
86 views

Tempered Distribution Calculation

I hope you don't mind that rather than typing this question up I took a screenshot and uploaded it: http://www.math.ualberta.ca/~schlitt/stackexchangeproblems/tempered-distributions-calc.png The ...
1
vote
2answers
354 views

Problem concerning continuous probability distribution

How do you prove that the real part of the characteristic function of the continuous probability distribution $f(x)$ is a characteristic function, but the imaginary part is not? The second part is ...
6
votes
1answer
246 views

Convergence of Fourier Series

Is there an $f\in L^1(\mathbb{T})$ whose Fourier series converges a.e. on $\mathbb{T}$ but not a.e. to $f$?
1
vote
0answers
113 views

Finding Transfer Function with an intermediate variable

How do I find the transfer function (using the bilateral z-transform) of the problem below. A stable LTI system with input x[n] and output y[n] is modeled by the difference equations c[n] + ...
2
votes
1answer
536 views

Fourier transform in Mathematica

When I calculate the Fourier transform of the function $$f(t) = \mathrm e^{-|t|/\tau} \text{ with } \tau >0$$ in Mathematica once via the function FourierTransform and once by hand, I get different ...
1
vote
0answers
59 views

Slowly varying vectors and coefficients of a sine transform

Let $u_k$ be the vector in $\mathbb{R}^n$ whose $i$'th entry is $\sin(\pi ki/n)$. The vectors $u_1,\ldots, u_n$ are orthogonal and correspondingly every vector in $\mathbb{R}^n$ can be decomposed as a ...
5
votes
1answer
564 views

Comparing/Contrasting Cosine and Fourier Transforms

What are the differences between a (discrete) cosine transform and a (discrete) Fourier transform? I know the former is used in JPEG encoding, while the latter plays a big part in signal and image ...
0
votes
0answers
237 views

Sample Sinusoidal Signal, Determine its Frequency

I am tasked with simulating an analog signal, and sampling it isochronously to determine its frequency. By simulate I mean we are using math.c's sin function, so no need (I think) to dive into the ...
4
votes
3answers
1k views

Does the phrase “instantaneous frequency” make sense?

I had always thought of time and frequency as being two different (yet complete) descriptions of the same system, so to me, the phrase "instantaneous frequency" didn't make sense -- frequency is a ...
2
votes
1answer
339 views

Is there an expression for the inverse Fourier transform of a log-normal function?

i.e. is there a simple solution for the following integral? $$\int_{-\infty}^{\infty} \exp(-\log^2(|\omega|/\omega_0)) \; \exp(i \omega t) \; d\omega$$ where $\omega_0 > 0$ Failing that, is ...
3
votes
1answer
169 views

If the Fourier transform of a signed measure is identically zero, is the same true of the measure?

I am trying to prove the following seemingly obvious fact: Let $\mu$ be a finite signed measure on $\mathbb R$. Suppose that $\hat\mu(u) = \int_\mathbb R e^{iux} d\mu(x) = 0$ for all $u$. Then ...
4
votes
1answer
411 views

Fourier Transforms

I'm having a terrible time trying to understand Fourier transforms. I'm very visual so leaving the $X,Y,Z,t$ domain is not working form me :) I'm trying to figure out the basics at the moment. ...
0
votes
1answer
4k views

Fourier transform of $f(x) = e^{-x^2}$ [duplicate]

Possible Duplicate: Fourier Transform of complicated product: $(1+x)^2 e^{-x^2/2}$ I calculate the Fourier Transform of $f(x)$ by $$\mathbb{F}(t) =\int_{-\infty}^{\infty}e^{-x^2} \cdot ...
1
vote
1answer
154 views

On a duality Fefferman-Stein's inequality

Let $M$ be the Hardy-Littlewood maximal operator. In the book "Weighted norm inequalities and Related Topics" by Rubio de Francia and J. Cuerva, page 150, theorem 2.1.2 states as the following: *For ...
3
votes
1answer
356 views

Fourier-like expansion of a closed curve in 2D

Fourier expansion can be used to represent any periodic function in one variable. Closed surfaces in 3D can be built out of spherical harmonics. Is there a similar expansion to represent a curve of ...
3
votes
1answer
507 views

Application of Fubini's Theorem

I am trying to show that for $f,g\in L_1(\mathbb{R}^d)$, $f*g\in L_1(\mathbb{R}^d)$. Somewhere along the way I need to switch the order of integration in the following integral (I know this for sure ...
3
votes
1answer
273 views

Applications of Young's convolution inequality

Recall that the convolution of two functions is given by $$f*g(y)=\int f(x)g(y-x)dx.$$ The well known inequality known as Young's inequality, say that $$\|f*g\|_r\leq\|f\|_p\cdot\|g\|_q $$ provided ...
3
votes
2answers
139 views

Solving this Fourier transform?

Is there any way to compute in closed form (in terms of known functions) the Fourier integral $$ \int_{-\infty}^{\infty} \frac{\cos(ux)}{(x^{2}+a^{2})^{s}} dx$$ where $u$ and $a$ are real positive ...
3
votes
1answer
150 views

Fourier transform inequalities on a probability distribution

I am reading a paper and the following came up: Given a probability density function, $\rho(x)$, such that for $\epsilon > 0$ $$ \int_{-\infty}^{\infty} |\rho(x)|^{1+\epsilon}dx < \infty ...
1
vote
2answers
1k views

Duality and the Fourier transform

Regarding Fourier transform, I read that the translation property and frequency-shift property are a duality. What does that mean and why is it true? Is there a physical implications? Thanks.
6
votes
1answer
638 views

Pointwise but not uniform convergence of a Fourier series

What is an example of a continuous, or even better, differentiable, $2\pi$ (or 1) periodic function whose Fourier series converges pointwise but not uniformly? (Such function cannot be of Hölder ...
8
votes
1answer
787 views

How do I compute the eigenfunctions of the Fourier Transform?

In Andy's answer to the question "What are fixed points of the Fourier Transform" on Math Overflow, he shows that the Fourier Transform has eigenvalues $\{+1, +i, -1, -i \}$ and that the projections ...
4
votes
4answers
558 views

Fourier series analog of a formula in Fourier transform

Every Fourier transform formula that I know of has a corresponding Fourier series analog, except the multiplication formula $$\int_{-\infty}^\infty f(x)\hat{g}(x) dx=\int_{-\infty}^\infty ...
4
votes
2answers
109 views

inequality with roots of unity

Do you know proofs or references for the following inequality: There exists a positive constant $C>0$ such that for any complex numbers $a_1,\ldots,a_n$ $$ |a_1|+\cdots+|a_n| \leq ...
0
votes
1answer
310 views

Fourier transform of exp(exp(x))

I am interested in the Fourier transform of a function of the form $f(x) = \exp(g(\exp(x)))$, where g has a "simple" form, for example $g(y) = \frac{(y-1)^2}{y^2 - 1}$. Has anyone a starting point ...
3
votes
1answer
141 views

If a Fourier Transform is continuous in frequency, then what are the “harmonics”?

The basic idea of a Fourier series is that you use integer multiples of some fundamental frequency to represent any time domain signal. Ok, so if the Fourier Transform (Non periodic, continuous in ...
1
vote
1answer
2k views

convolution a continuous function?

define $$h(x)=\int_0^{2\pi}f(x-y)g(y)dy=f*g(x)$$ if $f,g \in L^2$ are $2\pi$ periodic, show that h is continuous on $[0,2\pi)$ so let $x_n \to x$, then $$|h(x)-h(x_n)|=|\int f(x-y)g(y)-\int ...
7
votes
3answers
312 views

For what sequences of real numbers $\left\{ k_{n}\right\}$ is the set of functions $\left\{ e^{ik_{n}x}\right\}$ a basis?

It is well known that the set of functions $\left\{ e^{^{inx}}\right\}$, for integer $n$, is an othonormal basis for the space of square integrable real functions in the interval $[-\pi,\pi]$. Now ...
0
votes
1answer
45 views

Bessels and initial conditions

I'd like to know if I have got the following ideas right: 1) $f(r,\theta,t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty a_{nk}J_n(j_{nk}r)\exp[in\theta-j^2_{nk}t]$ subjected to initial ...
1
vote
0answers
81 views

Fourier analysis confusion

I think I may have misinterpreted this question, anyhow I am very confused. Here it is in its full glory: Let $f(r,\theta, t)=\sum\limits_{n=-\infty}^\infty \sum\limits_{k=1}^\infty ...
4
votes
1answer
212 views

Conditions for the Convolution $f \ast g$ to be Continuous at a Point

Let $f$ and $g$ be functions on $\mathbb{R}^n$. Let $x_0$ be a given point in the unit ball $B(0,1)$. I am looking for sufficient conditions for the convolution $$ (f \ast g)(x) = \int_{B(0,1)} ...