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

1
vote
1answer
31 views

How can we represent an image using basis images?

I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But i don't understand terms like "orthogonal " and "basis ...
0
votes
0answers
48 views

Fourier transform of a function

I'm struggling with FT, I just can't grasp the concept of it. Can somebody explain it on an example Ex 1: $f(t) = e^{-|t|}$ EX 2: $x(t) = \cos(\pi t/T)$ where it's different from $0$ just on ...
0
votes
0answers
7 views

Are the dominant frequencies preserved under fractional inversion

Let $f(t)$ be a signal that is a function of time. Let $F(f)=\mathcal{F}\{f(t)\}$ be the Fourier transform of $f(t)$. If $F(f)$ is dominated by a sparse set of frequencies $(f_1,f_2,\cdots,f_n)$ (only ...
1
vote
1answer
27 views

Does the integral in the formal 2D Fourier transform of the logarithm converge?

If $k$ is a nonzero vector in $\mathbb R^2$, how to interpret this integral: $$\int_{\mathbb R^2}e^{ik\cdot x}\ln{|x|}dx$$ Does it converge and in what sense? Thanks in advance.
3
votes
1answer
42 views

Fourier transform of $te^{-t^2}$?

How can I find the Fourier transform of: $$f(t) = te^{-t^2}$$
2
votes
0answers
39 views

Why are divergent Fourier series all so 'HARD'?

I'm not sure if this question is appropriate or even making sense, but I still feel curious: why are every example of divergent Fourier series SO COMPLICATED? It usually takes pages to construct and ...
1
vote
1answer
29 views

A function sequence converge to a Fourier series implies point-wise converge?

Assume $f(x)$ is a smooth $2\pi$ periodic function and can be decomposed as $$f(x)=\sum_{n=-\infty}^\infty a_ne^{inx}$$ A function sequence $f_m(x)$ satisfies $$\int_0^{2\pi}f_m(x)e^{-inx}dx\to ...
2
votes
0answers
42 views

Building up the Fourier inversion theorem on locally compact abelian groups

I'm reading through Folland's Abstract Harmonic Analysis and I've come to a bit of a road block with some machinery developed for the Fourier inversion theorem. We know that the Fourier transform of a ...
1
vote
1answer
39 views

Proof of Fourier series Theorem (k-continuous derivatives)

Here's the theorem: Theorem: If $f$ is periodic with Fourier coefficients $a_n,b_n$ and if the series $$\sum_{n=1}^\infty (|n^{k}a_n|+|n^{k}b_n|)$$ converges for some integer $k \geq 1$, then f ...
-2
votes
2answers
40 views

Fourier transform of t*(sent/pi*t)^2

Here's the function (I need it's fourier transform).
0
votes
1answer
19 views

Pseudodifferential operators and amplitudes

I am studying psudodifferential operators on $\mathbb{R}^n$. Let $U\subset \mathbb{R}^n$ an open subset. A function is $b\in C^\infty(U\times U\times U \times \mathbb{R}^n)$ is an amplitude of order ...
0
votes
1answer
32 views

Discrete Fourier Transform

I am studying DFT and am having trouble with the notation system. The frequency is from $0$ to $2B$ - in DFT the frequency domain does not have negative frequencies. But if this is the case, and we ...
0
votes
0answers
19 views

Fourier Transform by generalized distribution

I am having one little doubt on the subject. When we are defining Fourier Transform via generalized distribution on test functions, is this mandatory that the pairing has to be defined for all ...
0
votes
2answers
26 views

$\int\exp(-jnw_0t)\,dt$ integral calculus.

I seem to forgot these parts of integral calculus. I am trying to determine the Fourier coefficient in complex exponential form. Here, $t$ is the variable being integrated and $n$ is for all ...
0
votes
1answer
33 views

How to calculate the value of $\int_{-\infty}^{\infty} y(t)dt$?

For a function $g(t)$, $\int_{-\infty}^{\infty} g(t)e^{-j\omega t}dt=\omega e^{-2\omega ^2}$ for any real value $\omega$. If $y(t)=\int_{-\infty}^{t}g(\tau)d\tau$ then how to calculate the value of ...
0
votes
1answer
31 views

Fourier Transform and its Inverse

Could anyone show me how to prove the following results about Fourier Transform, please? It is stated in my book without proof. Thank you. Let $\mathcal F$ denote the Fourier linear operator and $f$ ...
0
votes
1answer
15 views

N-point FFT and 2-radix FFT

I am wondering what is the difference between a N-point FFT (output has same length as the input) and a 2-radix FFT (output is always of length $2^n$) For example a is a sequence: ...
0
votes
0answers
33 views

How are phase values able to capture motion from video?

I know that the phase spectrum contains most of the structural information about the image. But I want to know more about importance of phase spectrum related to video signals. I have read that ...
1
vote
1answer
49 views

Is $\langle f,g\rangle$ defined for distributions $f,g$?

Consider a standard setting for the development of the theory of distributions. Let $D(\Omega)$ be the space of test functions and $D'(\Omega)$ be the space of distributions ("generalized functions"). ...
3
votes
0answers
55 views

How to do this Sum? Poisson Resummation?

In the paper hep-th/0812.2909 page 34-35, there's a sum that I've been trying to do explicitly but I can't find a way. The sum is $$ \frac{2l}{\pi l! (l-1)!} \sum_{k\in\mathbb{Z}} \sum_{n=0}^{\infty} ...
0
votes
0answers
12 views

Calculating h-ellipticity

How do we calculate h-ellipticity $E_{h}$ of standard five point discrete Laplacian of two dimensional partial differential equation?
3
votes
2answers
91 views

What does the Fourier transform of $1/x^2$ mean?

If I ask Mathematica to compute the Fourier transform of $\frac{1}{x^2}$ using the FourierTransform function, it gives me a result of ...
0
votes
1answer
38 views

intuition behind an identity related to fourier transforms

I saw the proof of this identity in a question about Fourier transforms : $F(f(−t))w=F(f(t))(−w)$ Can someone give the intuition behind it ? What I understand of Fourier transform of a function ...
0
votes
0answers
24 views

Proving something is a convolution operator…

If we define the operator $K(a)=F^{−1}aF$ where $ F:L^2({\mathbb R})\to L^2({\mathbb R})$, is the fourier transform given by $$\left(Ff\right)\left(x\right)=\int_{{\mathbb ...
1
vote
1answer
50 views

Fourier transform of exp(cos)

How do I calculate the Fourier transform ($t \rightarrow \omega$) of the following: $\exp(A\cos(\omega_0 t))$ $A$ is a real constant, and $\omega_0$ is a real and positive constant. I know that this ...
2
votes
1answer
33 views

a question about Fourier transforms

I know it s simple but how to show that $\mathcal F(f(-t))w=\mathcal F(f(t))(-w)$ ? $\mathcal F(f(-t))w=\int^\infty_{-\infty}f(-t)e^{-iwt}dt$ $if -t=x\to -dt=dx$ ...
1
vote
1answer
14 views

2D Fourier Transform proof of Similarity Theorem

I have to solve an exercise, but if i could use the following theorem, it would be piece of cake Similarity Theorem if $ \mathscr{F}\{g(x,y)\}= G( f_x,f_y)$ then $ \mathscr{F}\{g(ax,by)\}= \frac ...
-1
votes
1answer
21 views

Aperiodic signals fourier transform short question?

What is the fourier transform of the aperiodic signals with infinite sequence? How about the transform of aperiodic fourier signals with finite sequence?
2
votes
1answer
59 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
3
votes
2answers
24 views

Is there any commonality between the use of Parseval's Identity in two different contexts?

In Fourier analysis, Parseval's Identity relates to "the summability of the Fourier series as a function." In inner product space analysis, the "identity" works as a "Pythagorean theorem" relating ...
0
votes
0answers
13 views

Fourier transform Piecewise function

i have this piecewise function and i need to get the fourier transform $$f(n)=\begin{cases} 0, & t<-2 \\ -1, & -2\le t<-1\\ 1, & -1\le t \le 1\\ -1, & 1<t\le2\\ 0, & ...
1
vote
1answer
11 views

infinite discrete abelian group

I am not too familiar with Fourier analysis, but I needed to use a certain result. I would appreciate any assistance. I was reading a literature in Foruier analysis and it said something like "Every ...
0
votes
1answer
31 views

Is there an equivalent of Plancherel's theorem with wavelets transform?

For the Fourier transform, we know that: $$||f||_2=c||\hat{f}||_2$$ where $c$ depends on the normalization. Is there an equivalent with wavelet transform? Thanks.
1
vote
1answer
46 views

Proving this Corollary regarding Fourier Series

Okay so here's the the problem: Let $k \in \mathbb{N}$. If $f$ is periodic, with Fourier coefficients $a_n,b_n$ and the series $\sum_{n=1}^\infty{(|a_n| + |b_n|)n^k}$ converges for some $k$, then ...
0
votes
2answers
21 views

How to show something is a convolution operator?

I have the operator $W(a)$ defined by $$W(a)=F^{-1}aF$$ where $F$ denotes the fourier transform and $a$ is a function on $L^{\infty}$. I need to prove that this is convolution operator, but I don't ...
1
vote
1answer
30 views

When is a function a Fourier transform of an integrable function?

Specifically, in the case $f(\xi)=\frac{1}{(1+\xi ^2)^\epsilon}$ where $0<\epsilon<1$. I wish to prove this is a Fourier transform of a $L_1$ function. Any insight into the manner would be ...
0
votes
0answers
15 views

In signal processing, every where you see infinity. Why?

Everywhere, in signal processing you see infinity. For example, in Fouriers, correlations. But no body would live to see infinity. Why do we aritificially talk about infinite time signals and then ...
0
votes
1answer
42 views

Proving that $\sum_{n=1}^\infty{[a_ncos(nx)]^{(m)}+[b_nsin(nx)]^{(m)}} \leq \sum_{n=1}^\infty{(|a_nn^m| + |b_nn^m|)}$

Okay so here's the the problem: Let $k \in \mathbb{Z}.$ Then, $\forall$ $m \in [1,k]$, $$f^{(m)}(x) = \sum_{n=1}^\infty{[a_ncos(nx)]^{(m)}+[b_nsin(nx)]^{(m)}} \leq \sum_{n=1}^\infty{(|a_nn^m| + ...
1
vote
1answer
55 views

Fourier series problems

I've got an "interesting" problem. I've gotten a way through it, but I'd like someone to look if what I've done so far is correct, and what to do next. We've got a function that is $0$ on the ...
2
votes
1answer
42 views

Sobolev spaces and Holder continuity (or, fractional derivatives and singularities)

I have two specific questions. The first is the result I actually need, and the second would let me prove it. EDIT: The second statement was wrong. I am keeping it for posterity. I am adding a third ...
1
vote
2answers
17 views

How to get the real frequency from an FFT output graph?

I am working in Origin 7.0 and trying to understand how the FFT Analysis works. In an attempt to understand, I decided to run a test of the program using known frequencies. The function I used was: ...
0
votes
1answer
20 views

Fourier transform of function similar to a Riesz kernel

I am trying to prove that the Fourier transform of $$\frac{x_1 x_2} {|x|^4}$$ in $\mathcal{R}^2$ (in the sense of distributions) is a bounded multiplier given by $\frac{\xi_1 \xi_2}{|\xi|^2}$ but am ...
6
votes
3answers
102 views

Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
2
votes
2answers
31 views

Uniform bound on Fourier series

This is from Fourier Analysis by Stein and Shakarchi, section 3, exercise 19. I am trying to prove that $\sum_{0<|n|\le N} e^{inx}/n$ is uniformly bounded in $N$ and $x\in [-\pi,\pi]$. Following ...
1
vote
0answers
6 views

convolution of three functions of two variables

Give three functions of two variables $a(x,y),b(x,y),c(x,y)$ one can construct the following convolution like integral: $y(x,y) = \int dx' dy' a(x',y')b(x-x',y') c(x-x',y-y')$ which I have a hard ...
0
votes
1answer
35 views

Fourier Transform, and identities

I need to calculate the fourier transform of this $ t \cdot sin(t-3)+2 \cdot t \cdot cos(3t) \cdot rect(6t) $ is the following valid, based on the first fourier identity i've read in a book, and ...
1
vote
0answers
31 views

Three Dimensional Fourier transform of a raidal function

Scaling analysis shows that the three dimensional Fourier transform of the function $f(\mathbf{r})=1/r$ is proportional to $1/k^2$. On the other hand, when working with spherical coordinates ...
0
votes
0answers
21 views

Fourier transform theorem

I have an expression: $c(s) = \int_{-\infty}^{\infty} d\omega e^{-i*\omega*s}*F(\omega)$ $G(W)=\int_{0}^{\infty} ds e^{i*W*s}*c(s) $ Is $G(W) = F(W)$? What is the relation of G(W) and F(W)? ...
1
vote
0answers
22 views

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. ...
0
votes
1answer
47 views

Berlekamp Massey and DFT

I was looking into the Berlekamp Massey algortihm, for LFSR, over GF(2) wondering if there was any DFT(alternately FFT), for the above scheme. Also, is there any generalization to Fn, ie, start ...