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
2answers
51 views

What is the realtionship between a cosine function and a Dirac delta? [closed]

I want to compute the Fourier transformation of a cosine function $\cos(2\pi f_0 t)$, but I don't know if there are any relationships between trigonometric functions and Dirac delta's.
2
votes
1answer
37 views

Fourier transform of a divergent function

During a calculation of Feynman diagram, I encountered an integral which is diverging: $$\int d^{2}p\frac{p^{2}p_{n}}{\left(p^{2}+\alpha k^{2}\right)^{2}}e^{ip\cdot y}$$ where $p$, $k$ and $y$ are ...
2
votes
1answer
32 views

Expanding in a Fourier series $y = |\cos x|$

How to expanding in a Fourier series function $y = |\cos x|$? Especially interested in how to find $$a_n= \frac{2}{\pi}\int\limits_{0}^{\pi}|\cos x|\cos(nx)dx$$
0
votes
1answer
48 views

Jump discontinuity of a function and its analytic phase

Let $f:\mathbb{R}\to \mathbb{R}$ and $f \in BV(0,1)$ with a jump discontinuity at $x = a\in(0,1)$. Let $f_h$ be its Hilbert transform and let $$f_A(x) = f(x) + i f_h(x)$$ Is it true that the function ...
1
vote
0answers
20 views

Is this a viable generalization of Newton series?

I wonder if the following formula a viable generalization of Newton's series. $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} (- i \omega)^x \int_{-\infty}^{+\infty}e^{i\omega t}\sum_{m=0}^\infty ...
1
vote
1answer
28 views

Help understanding the output from Apache Commons Math's Discrete Fourier Transform

I'm using a discrete Fourier transform to translate a finite set of samples to the frequency domain. I'm trying to start with a very simple set, but am still getting confused. I'm starting with this ...
2
votes
2answers
47 views

Evaluate fourier coefficient of $f(t)=t$.

Evaluate the Fourier coefficient of $f(t)=t$. $$\hat{f}(n) = \frac{1}{2\pi}\int_0^{2\pi} te^{-int}dt$$ I'd be glad for help with this calculation. My integration skills need an improvement. My ...
2
votes
1answer
77 views

Fourier transform of $f[n]$ and $f[-n]$

Hi I am just wondering, If I have a signal $f[n]\in \mathbb{C}^L$, i.e. $f$ is $L$-periodic, i can also define $h[n]=f[-n]$. Is it true that the Fourier transform of $f$, say $\hat{F}$, and the ...
1
vote
2answers
165 views

How one can implement the equation with “$i$” in it?

I have an equation: $$f(t)=c(e^{i2\pi\frac{n}{T}t}+e^{-i2\pi\frac{n}{T}t})$$ ...for $t\in(-\pi,\pi)$, and with $T=2\pi$. I have to draw a plot of the function $f(t)$ for $n\in\left \{0,1,2,5 \right ...
2
votes
2answers
133 views

Finding the complex fourier series of the function $x^2\sin(x)$ in the interval $[{-\pi}, \pi]$?

This forms part of a project I am doing and I wish to see how well complex fourier series approximates a smooth curve such as this one. After tedious integration by parts, I have attained an answer ...
0
votes
1answer
45 views

How to prove that cosine squared is a positive-definite function?

I need some help with proving that function $f: \mathbb{R} \to \mathbb C$, $f(t)=(\cos(t))^2$ is a positive-definite function. I know that if $\sum_{k,l\le n}(f(t_k-t_l)z_k\overline z_l)\ge0$ then ...
9
votes
1answer
142 views

Closed-form of sums from Fourier series of $\sqrt{1-k^2 \sin^2 x}$

Consider the even $\pi$-periodic function $f(x,k)=\sqrt{1-k^2 \sin^2 x}$ with Fourier cosine series $$f(x,k)=\frac{1}{2}a_0+\sum_{n=1}^\infty a_n \cos2nx,\quad a_n=\frac{2}{\pi}\int_0^{\pi} ...
2
votes
0answers
37 views

Is the Fourier transform a tame linear operator?

$\mathcal{F}:C^{\infty}_{0}(B^d)\to L_{1}^{\infty}(\mathbb{R}^{d},\mu,w)$ $\mathcal{F}(f)=\hat{f}$ I'd like show that $\left\|\mathcal{F}(f)\right\|_{n}\leq\left\| f ...
3
votes
3answers
4k views

Understanding Fourier transform example in Matlab

I'm studying about Fourier series and transform and I get confused with the following Matlab example of Fourier transformation: ...
5
votes
1answer
81 views

Why it is true for rapid decreasing function $g$ that: $\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x-y)|\leq A_{l,k}(1+|y|)^{l}$

If $g$ is of rapid decrease, that is $\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq 0)}(x)|<\infty$, then we have: $$\displaystyle\sup_{x\in\mathbb{R}}|x|^{l\geq 0}|g^{(k\geq ...
0
votes
0answers
19 views

Fast Fourier transfrom

What are the prerequisites for understanding the fast fourier transform for fast multiplication? What topics should I be familiar with first?
0
votes
0answers
33 views

DFT of vector $(0, 1, 2, 3)$

The problem is that my answer is different from answer i get in MATLAB. My answer is $(6, -2-2i, -2, -2+2i)$ while MATLAB answer is $(6, -2+2i, -2, -2-2i).$ In MATLAB i use command ...
1
vote
1answer
37 views

Why is a wave with high FM aperiodic?

I was playing with sound synthesis in a program I wrote and I had a wave of the form $\sin(2\cdot\pi\cdot(f_c+\sin(2\cdot\pi\cdot f_m \cdot t)) \cdot t) $ So, just simple frequency modulation. When ...
1
vote
0answers
61 views

Characterization of $H^{-1}(\mathbb{R}^N)$ by Fourier transform

We denote by $H^{-1}(\mathbb{R}^N)$ the dual space to $H_0^1(\mathbb{R^N})$. Let $$\langle f,v \rangle = \int_{\mathbb{R}^N} fv \hspace{1cm} (f,v \in L^2(\mathbb{R}^N)).$$ It is known that if $f \in ...
2
votes
3answers
54 views

Is the Fourier series a “linear transform”?

Fourier series fundamentally involve the sine and cosine functions: $$a_0+\sum_{k=1}^\infty \left(a_k \cos kx+b_k \sin kx\right)$$ These functions are about as non-linear as you can get. But... is ...
0
votes
1answer
36 views

Fourier transform, quadratic function

I'm trying to compute this convolution: $\frac{2 \alpha}{\alpha ^2 + 4 \pi ^2 x^2} * \frac{2 \beta}{\beta ^2 + 4 \pi ^2 x^2}$ I know that the Fourier transform of a convolution of two functions is ...
1
vote
1answer
121 views

Phase shift of two sine curves

How do I determine phase shift of two sine curves (discrete time sampled sine curves) in Matlab. Currently, I have the FFT of these two sine curves, the phase shift is just the delay in time, which ...
0
votes
0answers
20 views

Fourier transform of $be^{i k y^b}/y^{1-b}$

I'm trying to compute the Fourier transform of $$ \frac{ be^{i k y^b}}{y^{1-b}}$$, i.e. $$ F(z) = \int_{-\infty}^\infty \frac{ be^{i k y^b}}{y^{1-b}} e^{i z y}dy$$ I tried using Mathematica for ...
3
votes
3answers
348 views

Inverse fourier transform of $ 1/(1+s^2)$

Hoi, I want to have the inverse fourier transform $\mathcal{F}^{-1}(\frac{1}{1+s^2})$. So I thought about using some properties of fourier-transform. But knowing the answer I must make some sort of ...
1
vote
0answers
77 views

Calculating an integral with sine, cosine

I've recently calculated the Fourier transform of $\dfrac{\sin \pi ax}{\pi x}$. Now I'm trying to calculate $$\int _{\mathbb{R}} \frac{\sin ^2 \pi ax}{\pi ^2 x^3} \cos \pi bx\;\mathrm dx$$ The ...
1
vote
1answer
30 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
0
votes
1answer
15 views

2D DFT of Hexagonally Sampled Grid

Is there a good way to perform a 2D FFT of discrete data on a hexagonally sampled grid? The best method I've got so far involves oversampling the hex grid to a rectangular grid, and performing the 2D ...
0
votes
1answer
38 views

Convergence of a sequence involving integral

Consider $f:[-\pi, \pi] \to \mathbb{C}$ is analytic (infinite differentiable) and periodic. Define $a_n:= \frac{1}{2 \pi}\int_{-\pi}^\pi f(x) e^{-inx}dx$ (the Fourier coefficient of $f$). Show ...
2
votes
0answers
37 views

Exercise 22, Chapter 5 of Stein and Shakarchi's Fourier Analysis

I am working through Stein and Shakarchi's Fourier Analysis and am stuck on this exercise. I would appreciate any hints. The statement of the exercise follows. Notation: $\mathcal{S}$ is the Schwartz ...
1
vote
1answer
80 views

$\lim_{s\to 0^+}\int_0^\infty a(t) e^{-st} dt $

$$\int_0^\infty a(t) e^{-st} dt = f(s)$$ What is the meaning of the limit of this integral as $s\to 0^+.$
2
votes
0answers
48 views

using Paley-Wiener to get support and then estimate inf sup

Define the function $$ \tilde{f}_n(\omega)=\frac1{\sqrt{2\pi}} \frac{\sin R\omega/2}{R\omega/2} s_n(R\omega/2\pi),$$ where (using the Weierstrass product representation for $\sin$) $$ s_n(w) = ...
2
votes
1answer
42 views

Periodicity of an infinitely differentiable function

Consider $f:[-\pi,\pi] \to \mathbb{C}$ be an infinitely differentiable function with $f^{(n)}(-\pi) = f^{(n)}(\pi)$ for all $n \in \mathbb{Z}^+$. Is this a periodic function ? I think it is a ...
0
votes
1answer
31 views

Schwartz space on $\mathbb T^{n}$

For the definition of Schwartz space space on $\mathbb R^{n},$ see this. My Questions: (1)Is it make sense to talk of Schwartz space on torus $\mathbb T^{n}$ ? If yes, what can be the analogous ...
1
vote
0answers
39 views

How to integrate this 1D Fourier transform?

\begin{equation} \int _{-\infty} ^\infty |s|^{-5}~e^{-(s-s_0)^{-4}} e^{\imath st}ds \end{equation} where $s_0$ is a positive real number
2
votes
0answers
96 views

how to use Matlab ifft to calculate the following integral? [duplicate]

$$R(t)=\int_{-\infty}^\infty\dfrac{\omega e^{i\omega t}}{(3-\omega^2)^{2}+4\omega^2}\,d\omega$$ where t is a integer and $t>0$ I used to calculate this integral by numerical integral,but it seems ...
1
vote
0answers
16 views

Calculate FFT of 1/r green's function

I am trying to write the Poisson equation solver in C, using FFTW library. For given density of charge I need to calculate potential assuming periodic boundaries. My idea is to use convolution, simply ...
2
votes
0answers
51 views

A deep understanding of the Fourier transform

I feel like i don't understand the Fourier transform. I've seen what it does and its properties but even after reviewing various proofs i don't understand why we end up explicitly with a relation ...
2
votes
1answer
34 views

The Fourier transform of $ e^{-|x|^\alpha}, \alpha>0. $

Do you know the Fourier transform of $$ e^{-|x|^\alpha}, \alpha>0. $$ Does it have an implicit formula. In the spacial cases $$ (e^{-|x|})^\hat{\,}(\xi)=\frac{2}{1+4\pi^2\xi^2},\ ...
1
vote
0answers
12 views

Inverse Fourier Transform of $1/k^2$ in $\mathbb{R}^N $

This comes up in the context of finding the Green's function of Poisson's equation for $\mathbf{x} \in \mathbb{R}^n $ $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Attempt by using Fourier ...
10
votes
3answers
801 views

Recursive Integration over Piecewise Polynomials: Closed form?

Is there a closed form to the following recursive integration? $$ f_0(x) = \begin{cases} 1/2 & |x|<1 \\ 0 & |x|\geq1 \end{cases} \\ f_n(x) = ...
1
vote
1answer
21 views

control of an integral using maximal function

Let $I$ be a compact interval with center $c(I)$ and N be a large positive integer. It seems to me that there exists a constant $C$ such that for any good function $f$ (e.g. Schwartz function) we have ...
1
vote
1answer
31 views

Expression for characteristic function of a truncated RV

Let $\langle\Omega,\mathscr{F},\mathbb{P}\rangle$ be a probability space, let $X$ be a random variable defined thereon with density $f$ and $\phi$ be its characteristic function. Then if $A \in ...
1
vote
0answers
21 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
4
votes
0answers
39 views

Estimating the (double) Riesz transform.

I'm trying to verify the following estimate, which appears in a paper I'm reading. It seems I'm missing something easy, I just can't figure this out. $\textbf{Background}:$ For a function $f \in ...
0
votes
0answers
32 views

How to prove $\hat f$ is uniformly continuous in $R^n$?

Let the Fourier transform be defined by $\hat f(\xi)=\int_{R^n}f(x)e^{-ix\xi}dx$. Suppose $f\in L^1(R^n)$. How to prove $\hat f$ is uniformly continuous in $R^n$?
0
votes
1answer
48 views

proving Riemann-Lebesgue lemma

I have looked at proofs of the Riemann-Lebesgue lemma on the internet; all of these proofs use the technique of Riemann integration and making step functions. ...
4
votes
1answer
77 views

Is there an intuitive way to understand why a frequency cannot be writen as a sum of other frequencies?

Let's say we have a sequence of functions $(f_n)$ and for $n$, $f_n$ is a cosine at frequency $k$, i.e. $f_n(t)=\cos(2\pi kt)$ for some $k$ which depends of $n$. Let $n_1,\dots n_k$ be natural ...
1
vote
1answer
20 views

Using partial fraction decomposition to find inverse Fourier transform

I've reduced my problem to $H(w) = \dfrac{1}{(1-\frac{1}{4}e^{-jw})(1-\frac{1}{3}e^{-jw})}$. I need its inverse discrete Fourier transform. My thinking is that I could use partial fraction ...
4
votes
1answer
47 views

When is an oscillating integral small?

I hope, the title is not too confusing. My question is the following: We all know the Riemann-Lebesgue-Lemma stating that for $f\in L^1(\mathbb R)$, one has $$ \lim_{k\to\infty} \int ...
2
votes
1answer
592 views

A generalized version of the Riemann-Lebesgue lemma

Let $f \in L^1 [0,2\pi ] $ and let $ g $ be bounded and $2\pi$-periodic. Prove that $$ \hat{f}(0)\cdot\hat{g}(0)=\lim_{n\to\infty}\frac{1}{2\pi}\intop_0^{2\pi}f(t)g(nt)dt $$ where $\hat{f}(0)$ ...