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
2answers
44 views

Problem with the Fourier transform of a function

I'm having some troubles with this one: $$\mathcal F(e^{-|x|} +|x|e^{-|x|})$$ I know that $\displaystyle\mathcal F(e^{-|x|})={1\over \pi (1+w^2)}$ but the second part is where I get stuck.
0
votes
2answers
87 views

Convert phasors to sinusoidal waveform

I'm trying to convert this phasor into a sinusoidal waveform. $$ j6e^{-j\pi/4} $$ Here's what I have so far: $$ 6\sin(\omega t-\pi/4) = 6\cos(\omega t-\pi/4 - \pi/2) = 6\cos(\omega t-3\pi/4) $$ ...
1
vote
1answer
96 views

Inhomogeneous diffusion equation and initial conditions inversion

While working on a physical diffusion process, I encountered the following Fokker-Planck equation $$ \frac{\partial F}{\partial t} = D (x) \frac{\partial^2 F}{\partial x^2} \tag1$$ where $D(x) > ...
2
votes
1answer
209 views

Need to learn wavelet, suggest steps and resources

I am looking for a good introduction to wavelets and wavelet transforms. that covers the following: Basics Vector Spaces – Properties– Dot Product – Basis – Dimension, Orthogonality and ...
1
vote
0answers
111 views

How to interpret Fourier-Stieltjes transform on $\mathbb T$ (one dimesional torus)?

Let $\mu$ be a regular Borel measure on $\mathbb Z$ and we put, $$\|\mu\|:= |\mu| (\mathbb Z)= \text {total variation of} \ \mu . $$ and define $$M(\mathbb Z):= \{\mu: P(\mathbb Z)\to \mathbb C : ...
1
vote
1answer
38 views

Does there exists $f\in A(\mathbb T)$ such that $||f||=r$ and $||\mathrm{e}^{if}||= \mathrm{e}^{r}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t)\, \mathrm{e}^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb ...
0
votes
1answer
87 views

Solving a Fourier sine transform equation

Suppose we have the following Fourier sine transform equation $$\int_0^\infty f(x;p)\sin(\lambda x)dx \equiv 0,$$ where $f(x;p)$ has some parameters $p\in\mathbb{R}$ we can choose freely. Does this ...
0
votes
1answer
24 views

Finding Fourier coefficients of functions that are defined as integrals with known Fourier coefficients?

Given a continuous periodic function f, with a period of $2\pi$, and Fourier coefficients that are $\hat f(n) = \frac{1}{1+n^2}$ , what are the Fourier coefficients of $g(x) = \int_0^xf(t)dt $? So ...
1
vote
0answers
81 views

Is the definition of DTFT using $\omega$ wrong?

I'll briefly explain this problem I faced. Let's take this simple signal: $$x(n)=\cos(\pi n)$$ The signal is not absolutely summable, however we can define its DTFT in terms of distributions. That ...
6
votes
1answer
51 views

Derive Fourier transform from what it should do?

I was wondering about the following: Imagine you want to figure out whether there is a transform that exchanges differentiation with multiplication and convoution with pairwise transformation for ...
1
vote
0answers
102 views

Calculating Fourier magnitude spectrum for Local Binary Pattern histogram

I have the follwoing discrete Fourier transform function defined in my book (Computer Vision using Local Binary Patterns, Pietikainen et. al, 2011): $$H(n, u ) = \sum_{r=0}^{P-1}c_{nr} ...
1
vote
1answer
136 views

Decay of Fourier coefficients sequence

If $f:\Bbb R\to \Bbb R$ is a $2\pi-$ periodic, $C^1$ function, then $k^2a_{k}(f)\to 0$ where $$a_{k}(f)=\frac {1}{\pi}\int_{-\pi}^{\pi}f(x)\cos kx dx$$ are the Fourier coefficients. I ask if this is ...
1
vote
0answers
21 views

Fourier Operator and roots of Identity operator

I have seen that if Fourier operator is defined by $$ h(k) = \hat F(g(x)) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty} dx\:g(x)\:e^{ikx} $$ then $$ \hat F^2\{g(x)\}=g(-x) \implies \hat F^2 ...
3
votes
1answer
80 views

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x.

Express $(1+\cos(x-1))^3$ as a trigonometric polynomial in x. I keep doing this problem and somehow I keep messing up the constants, and it just jumbles up in my head. $$(1+\cos(x-1))^3$$ $$= ...
0
votes
1answer
46 views

$f(x) =\cos(x-y) -\cos(\delta)$ plotting

Ok, so this is a confusing one. I'm not sure what my teacher is looking for. The problem is: Plot any number $-\pi < y< \pi$ and pick a small number $\delta > 0$ so that the whole interval ...
3
votes
1answer
62 views

Basic question about Fourier transform

The problem is: Let $f \in L^{1}(R) \cap C(R)$ . Supose that $f$ is positive. Show that $|\hat{f}(\xi)| < |\hat{f}(0)|$ for all $\xi \neq 0$. My idea: By the definition of the Fourier transform we ...
0
votes
1answer
81 views

Integrating $\frac{\sin ^2(x)}{x^2}$

Fourier transforming the function: $$f(t) = \left\{ \begin{array}{ll} 1; & \mbox{ } |t| \leq 1 \\ 0; & \mbox{otherwise} \end{array} \right.$$ We get: $$F(y)=2 \frac{\sin y}{y}$$ And ...
0
votes
1answer
103 views

How to inter change of norm and limit in the Banach algebra?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
1
vote
1answer
40 views

Fourier analysis question

Let $f(t)=\frac 12 -t, t\in(0,1).$ Calculate the Fourier coefficients of the function $f$ and the sum $\sum_{n=1}^{\infty} \frac {1}{n^2}$. Note that $L^2 (\Bbb{T}) \to l^2(\Bbb{Z})$ and ...
1
vote
0answers
69 views

construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
1
vote
1answer
175 views

Convergence of the Fourier serie of $f(x)=e^{2\pi i \alpha x}$

I have some difficulties with the last part of an old exam exercise. For the 1-periodic function $f$ defined on $[0, 1[$ by $f(x)=e^{2 \pi i \alpha x}$ with $0<\alpha <1$. I have found that its ...
0
votes
0answers
73 views

calculate the integral using Fourier transform

I am asked to calculate the integral $$\int_{-\infty}^{\infty}\frac{\sin^2(\omega)}{\omega^2}e^{i\omega}d\omega.$$ I read all the posts on this site about the integral ...
2
votes
1answer
91 views

Relationships between growth rates of a distribution and smoothness of its Fourier transform

Let $f\in \mathcal{S}^\prime(\mathbb{R})$ be a tempered distribution, and $\hat{f}$ be its Fourier transform. It is known that when both $f$ and $\hat{f}$ are $L^2$ functions, there are relationships ...
0
votes
1answer
57 views

Fourier series using other bases?

The theory of Fourier series, representing a reasonable function by an infinite sum of exponential functions, is very well-developed. In addition to basic functional-analytic results there are things ...
1
vote
0answers
108 views

Fourier Transform of a Gaussian Signal?

As far as I know this is the formula for FT : On this question on part b) I fint on the answer the part with e^-jwt is changed with cos(wt) I have no idea how cos(wt) came in ... would you please ...
1
vote
0answers
57 views

Number of roots of sine-and-cosine expression

Is there an easy proof of the following fact? Let $a_0, \ldots, a_n, b_1, \ldots, b_n$ be real numbers, not all zero. Then, the function $$a_0 + a_1 \cos x + b_1 \sin x + a_2\cos 2x+b_2\sin ...
1
vote
1answer
59 views

Fourier Transform Identity

I'm trying to verify the following: $$ \int_{\mathbb{R}} e^{-z\xi^2} \hat{f}(\xi) \; d\xi = \sqrt{\frac{\pi}{z}} \int_{\mathbb{R}} e^{-\pi^2x^2/z} f(x) dx, $$ for $z = \alpha i$ purely imaginary and ...
1
vote
1answer
49 views

About a function and its Fourier transform being zero at the same time.

If we have $f\in L^{2}(\mathbb{R})$. How can I prove that (if $f$ is not zero), $f$ and its Fourier transform $\cal{F}(f)$ can't be zero out of a bounded interval? I think it involves the inversion ...
3
votes
1answer
618 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
3
votes
2answers
2k views

Multidimensional Fourier transform of the laplacian

In my course on electromagnetic field theory we use the Fourier transform to simplify Maxwell's equations, for example: $$\frac{\partial ^2\vec E(\vec r,t)}{\partial t^2} \rightleftharpoons ...
0
votes
1answer
28 views

Fourier coefficient problem

Calculate fourier coefficient $\hat{s}(-1)$, where 1-periodic signal $s$ :$\Bbb{R}/\Bbb{Z}\to\Bbb{C}$ is defined with equation $s(t)=(2cos(\pi t))^{16}$
1
vote
1answer
62 views

DFT matlab problem

We have signal $\sin(2\pi v_1 t)+\sin(2\pi v_2 t)$ and we know $ν_1\in{700,780,860,940}$ and $ν_2\in{1200,1340,1480}$. Also we have vector here: $$h(k)=\sin(2π ν_1 k Δt)+\sin(2π ν_2 k Δt)$$ where ...
1
vote
0answers
20 views

Fast Fourier Transform for non-trigoniometric bases

The fast fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other basis, e.g. orthogonal polynomial bases ...
3
votes
1answer
96 views

Ques from exam: sequence of functions and improper integrals

$P_n(x):R\rightarrow R$ is a sequence of functions defined by: $$P_n(x)= \frac{n}{1+n^2x^2}$$ f:R->C is continuous and 2pi periodic. We define: $$f_n(x)=\frac{1}{\pi}\int ...
3
votes
1answer
180 views

Artifacts and low frequencies FFT.

I am working on analyzing a time signal and want to preform a FFT. However I run in to some artifacts at low frequencies. I have managed to reproduce the behavior in a test signal. Given by $S(t) = ...
2
votes
1answer
76 views

Integrating the Fourier Transform

I am trying to show that $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{- \infty}^w \hat{f}(w') \, d w'.$$ Shouldn't it be $$\mathcal{F}\left\{ \frac{f(t)}{t} \right\} = - i \int_{w}^{+ ...
1
vote
0answers
43 views

Why the space of Fourier-Stieltjes transforms on a compact groups is same as the space of FT on compact groups?

Let $G$ be a locally compact group and we put $L^{1}(G)= \ \text {The space of all complex functions which are integrable with respect to the Haar measure of } \ G.$ Let $\Gamma$ is the dual group ...
1
vote
0answers
80 views

Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

I asked yesterday on math.stackexchange a question and received no answer. Since I'm very interested in an answer, I'm reposting it here: "Let $k$ be a fixed integer, and $\mathcal{F}$ the set of ...
1
vote
1answer
70 views

Fourier transform of $\frac{d}{dt}\ln\frac{1}{it}$

I'd like to proove the identity $$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$ with $H=\mathbb{I}_{\mathbb{R}^+}$ ie the Heaviside step function, $\mathcal{F}$ denote the Fourier ...
2
votes
0answers
38 views

Pointwise convergence of a sequence of trigonometric polynomials with bounded number of nonzero terms

Let $K$ be a fixed integer, and $\mathcal{F}$ the set of trigonometric polynomials with at most $K$ nonzero terms. Let $(f_n)$ be a sequence in $\mathcal{F}$ converging pointwise (on $\mathbb{R}$) to ...
0
votes
1answer
30 views

question involving integration of fourier transform

I was reading a paper and I came across one equation, in which I had a problem deriving this equation. ...
3
votes
1answer
69 views

Using the Discrete and Fast Fourier Transform for Polynomial Multiplication

I need to multiply $ f(x) = x^2-3$ by $ g(x) = -2x$ using both Fourier transformations. I think I have found the roots of some equation, and it gives f(x) $$= 1,\frac{-1+i\sqrt3}{2} and ...
2
votes
1answer
72 views

proof by induction to fourier problem

So if $h_n (t) = e^{\pi t^2}\frac{d^n}{dt^n}(e^{-2\pi t^2})$. Show proof by induction that $$\widehat{h_n}=(-i)^n h_n$$ Any ideas how to go about with this one? When $n=0 \to \widehat{h_0}=h_0$.
0
votes
1answer
60 views

Fourier conjugate problem?

Do I need to use conjugation rules to show that $\overline{h(t)}=\hat{g}(t)$ and when $g(t)=\overline{\hat{h}(t)}$? Trying to prove parsevals identity with this one. Edit: Something like this: ...
0
votes
2answers
619 views

how to find fourier transform of $\exp(-x^2/2)$

How can we find the fourier transform of $e^\frac{-x^2}{2}$ where -$\infty $ < x < $\infty $. I tried applying the standard formulae but ended up in un defined form..
1
vote
1answer
32 views

Fourier coefficients of $e^{xe^{it}}$

I am given an arbitrary $x \in \mathbb R$ and the $2 \pi$-periodic function $$f(t) = e^{xe^{it}}.$$ The Fourier coefficients are for any $n$ given by \begin{equation*} 2 \pi c_n ( f) = ...
2
votes
0answers
80 views

Proof of alternate characterization of Schwartz functions

In a book that I am reading called "Integral Geometry and Radon Transforms" by Sigurdur Helgason, Schwartz functions are defined by $f \in \mathcal{S}(\mathbb{R}^n)$ if and only if for every ...
1
vote
1answer
67 views

simple question about a result of Fourier series

I am studying the proof of this result but i am with a problem in a part of the proof: Result: Let $f \in L^{p}(T) = \{ h : R \rightarrow C , \text{of period 1 such that } \int_{0}^{1}|f|^p < ...
0
votes
1answer
87 views

Frequency result of FFT for data that does not start at t=0

I know there are already a lot of questions about frequency bins in FFT. However I have one that doesn't really fit to the ones I read. I have time dependent data where the time does not start at t=0 ...
0
votes
1answer
38 views

Maximize a sum of sinusoids with comensurable periods

I'm writing a program that requires finding $$\text{argmax}_\theta\sum_{k=1}^na_k\cos(k\theta+b_k),$$ where $a_k$ and $b_k$ can be any real numbers. How can I do this efficiently?