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
0answers
164 views

Discete Fourier Transform-Additive Combinatorics

Although I am not completely familiar with the subject but I have met two 'dual' definitions of the Discrete Fourier Transform of a function $ f: \mathbb{Z} / N \mathbb{Z} \rightarrow \mathbb{C} $ , ...
0
votes
1answer
175 views

Show $\exists f \in C \ni \|f\|_\infty \le 1$, but it's Fourier series diverges

Show $\exists f \in C \ni \|f\|_\infty \le 1$, but it's Fourier series diverges. The proof is in our textbook (Katznelson, Harmonic analysis). It uses this argument. Let ...
2
votes
0answers
129 views

Intuition behind the proof for Wiener's theorem?

I am reading his proof for Wiener's theorem in Chp9 of Rudin's functional analysis. The theorems (9.4, 9.5 and 9.7) themselves are quite clear and Rudin did a good job explaining the intuition behind ...
1
vote
0answers
98 views

Filter/Removal of periodic 'delta-peaks'

Currently I am measuring data (Counts over Time). Due to measurement problems I have some nasty peaks in this data. These peaks are periodical, very sharp (~3 datapoints over a range of 10000) and ...
1
vote
3answers
521 views

convergence of autocorrelation function and existence of Fourier transform

I am studying the Wiener-Khinchin Theorem. But, I am wondering why the Dirichlet condition, which says that the autocorrelation function of WSS should be absolutely integrable, is sufficient for the ...
3
votes
1answer
180 views

Poisson integral on $\mathbb{H}$ for boundary data which is orientation-preserving homeomorphism of $\mathbb{R}$

Let $f$ be a real-valued function (in my case, an orientation-preserving homeomorphims of $\mathbb{R}$) on the real line $\mathbb{R}$ which is not in any $L^p$ -space. Let us take the simplest example ...
1
vote
1answer
90 views

Fourier transform, why this gives incorrect answer?

Let $f(x) = \begin{cases}e^{-x} & ,0<x<1\\0 & ,\text{Otherwise}\end{cases}$ I'm trying to calculate the fourier transform of $xf(x)$, by using the fact that $xf(x) = -\frac{d}{da}f(a ...
0
votes
1answer
324 views

Cesaro summable implies that $c_{n}/n$ goes to $0$

Theorem. If $\sum_{n=1}^{\infty}c_{n}$ is Cesaro summable, then $c_{n}/n$ tends to $0$. How to prove it?
1
vote
2answers
188 views

Finding spectrum using the convolution property

Using the convolution property, find the spectrum for $$w(t)= \sin(2\pi f_1 t) \cos(2\pi f_2 t).$$ I'm confused on how to solve this question. Can you give me any aproach?
2
votes
1answer
361 views

A question on a solution of an inhomogeneous heat equation.

I am now working on the following PDE equation (Evan's PDE textbook Section 2.5 No.14) \begin{align} u_{t}-\Delta u + cu=f \ \ & on \ \ \mathbb{R}^n\times (0,\infty) \\ u=g \ \ & on \ \ ...
6
votes
2answers
359 views

Paley-Wiener type theorems for distributions?

In general a theorem of Paley-Wiener type gives a relation between the decay of a function and the smoothness of its Fourier transformation, and there are plenty of them since there are many kinds of ...
2
votes
1answer
944 views

Smoothness and decay property of Fourier transformation

If my memory serves I have heard something like "the less smooth your function $f$ is, the worse its Fourier transform $\hat{f}$ decay because its Fourier transform $\hat{f}$ needs more waves of high ...
0
votes
1answer
161 views

Calculate by hand fourier transform of this sort of.

$$x(t)= A+ A_1\sin (2 \pi f t + \theta ) + A_2\cos (2 \pi f_1 t + \theta )$$ I want to find the Fourier transform $|\mathcal{F}[x(t)]|^2$ . Is this possible by hand? I can find the Fourier transform ...
1
vote
1answer
163 views

If $f$ is a bounded tempered distribution and $g \in L^1$ is then $\int_{\Bbb R^n}(f\ast\tilde\varphi)(x)\tilde g(x)\,dx$ a tempered distribution?

Let $f$ be a bounded tempered distribution, that is, $f\ast\varphi \in L^\infty(\mathbb R^n) $ for every Schwartz function $\varphi$. If $g \in L^1(\mathbb R^n)$, does the following definition define ...
2
votes
1answer
127 views

Transient vs Steadystate

Let $$s(t) = \cos(wt)\cdot u(t)$$ with $u(t)$ being the unit step. Suppose we can represent such a signal as the sum of a transient and of a "steady state". A transient is a short-time wide-band ...
1
vote
1answer
91 views

Strong Law of Large Numbers Under a Transformation

I have some random variable, $x$, distributed according to a probability density function (pdf), $f\left(x\right)$. The Strong Law of Large Numbers (SLLN) implies that, for an expected value, given ...
0
votes
1answer
99 views

About fourier transform in the PDE

In the PDE as below$$ \partial_t u - \frac{i}{\rho} ( - \partial_x^2 )^{\rho /2} u = 0 \;\;\;(t,x) \in \Bbb R^2 $$ How can I prove that $$ (- \partial_x ^2 )^{\rho/ 2} = \scr F ^{-1} | \xi |^\rho ...
2
votes
1answer
73 views

Laplacian in $\Bbb R^2$ acting on compact test-function

I am trying to follow an argument in Strichartz's "A Guide to Distribution Theory and Fourier Transforms" We consider $\langle \Delta u, \rho \rangle$ where $\Delta u$ is the two dimensional ...
0
votes
1answer
99 views

Convergence of approximations to coefficients of harmonic series'

Assume that I have some probability density function, $f\left(x\right)$. If I want to approximate it using a Fourier series I can use the identity: $$c_{k}=\frac{1}{2\pi}\int_{-\pi}^{\pi} ...
9
votes
0answers
1k views

Is this Fourier like transform equal to the Riemann zeta function?

This question builds upon the answer to this question. This new question has only minor changes compared to the previous question, but the scale factor of the output from the Fourier like transform is ...
2
votes
1answer
291 views

Path integrals using Fourier transformation

While going through a book named Mirror Symmetry, I came across a path integral, $$Z(\beta) = \int\limits_{X(t+\beta) = X(t)} DX(t) \exp\left(-\int\frac{1}{2}( \dot{X}^2 + X^2)dt\right)dt $$ where ...
1
vote
1answer
174 views

Fourier transform of a sum of chirps

A chirp signal is defined as follow: $$x(t)=\sin(\omega t^2)$$ I have the following modified chirp: $$y(t,N)=\sum_{k=1}^{N}\sin(\omega t+k\beta t^2)$$ My problem is to find the Fourier transform ...
2
votes
0answers
78 views

A question on algorithm complexity

It is well-known that the evaluating the Discrete Fourier Transform definition directly has a complexity $O(N^{2})$ for a signal with bandwidth $N$. How to see or show that the fast Fourier transform ...
2
votes
1answer
113 views

Show translation is not continuous in $\text{Lip}_\alpha(T)$

Let $f=\sqrt{|x|} \in \text{Lip}_\alpha(T)$, where $\text{Lip}_\alpha(T)$ is the set of Lipschitz function with Lipschitz constant $\alpha=1/2$ on the unit circle $T$. What is $$ \|f\|=\sup_{t\in T,h ...
3
votes
1answer
228 views

Fourier series to integral - Joseph Fourier's explanation

I do not understand the way that Joseph Fourier took from fourier series to integral Following is from his book The Analytic Theory of Heat, 1822, translated by Freeman, 1878 Page.345, Last ...
4
votes
1answer
297 views

A condition on Fourier transforms that implies absolute continuity

Is there any condition on the Fourier transforms of 2 positive measures $\sigma , \mu$ on the complex unit circle $\mathbb{T}$ that implies absolute continuity ( $\sigma\ll\mu$)?
0
votes
0answers
270 views

fourier transform of positive function

I am having trouble with this question: Show that there exists a compactly supported $C^\infty$ function $\phi$ on $\mathbf{R}$ such that $\phi \ge 0, \phi(0) >0$, and $\hat{\phi} \ge 0$. I ...
2
votes
1answer
167 views

In-place inverse of DFT?

I'm trying to understand (by implementing) the Cooley Tukey algorithm for an array $[x_0, \dotsc, x_{2^N-1}]$ of real valued data. Since the input data is real valued, the spectrum will have ...
1
vote
1answer
195 views

Is there a Jordan-Dirichlet theorem for Fourier transform?

Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be $L^1$ and $x \in \mathbb{R}$. Suppose that $\int_{-1}^0 |(f(x+t)-a)/t| dt < \infty$ and $\int_0^1 |(f(x+t)-b)/t| dt < \infty$ for some $a,b \in ...
3
votes
2answers
192 views

Does convergence of Fourier transforms imply convergence of measures?

Let $\{\sigma_n\}$ be a sequence of measures on the complex unit circle $\mathbb{T}$ and let $\sigma$ also be such a measure. Suppose that $\hat{\sigma_n}(k) \rightarrow \hat{\sigma}(k)$ as ...
2
votes
1answer
359 views

Finding the 'inhomogeneous' plane wave solutions of the wave equation via Fourier analysis

When one solves the wave equation $$ ( \partial_t^2 - v^2 \nabla^2) \mathbf{E}(\mathbf{x},t) = 0 $$ in $\mathbb{R}^3 \times \mathbb{R} $ using the Fourier transform method, the general solution is ...
1
vote
0answers
255 views

Projection Slice theorem getting rid of artifacts?

I have employed the fourier(projection) slice theorem in matlab. I have a 3D image, P(x,y,z) defines their pixel intensities at a given location int he image volume, it is discrete and uniform. I ...
3
votes
3answers
469 views

Need a crash course in fourier analysis, recommend resources

I need to be able to understand everything about fourier analysis asap. Could you recommend one or two references or books that are considered 'the book' to learn this subject?
1
vote
1answer
70 views

FFT processor which can only be used once

Given an 8 point FFT processor, which can be used only once, compute the DFTs of the sequence. $$x_1(n)=[1,8,6,7,4,2,3,1]$$ $$x_2(n)=[1,4,3,2,8,7,6,1]$$
3
votes
1answer
291 views

How to show that $\frac{1}{\pi} \int_{-\pi}^{\pi} \cos(mx)\cos(nx) \,dx = \delta_{mn}$?

How to show that $\displaystyle\frac{1}{\pi} \int_{-\pi}^{\pi} \cos(mx)\cos(nx) \,dx = \delta_{mn}$? If you use $\cos(x)\cos(y)=\cos(x-y)+\cos(x-y)$, you get that $$\begin{align} \frac{1}{\pi} ...
2
votes
0answers
325 views

Fourier transform for Neumann boundary condition

I need to solve system of two coupled partial differential equations numerically. $\frac{\partial x_1}{\partial t} = c_1\nabla ^2 x_1 + f_1(x_1,x_2) \\$ $\frac{\partial x_2}{\partial t} = ...
0
votes
2answers
86 views

How to show that function $g(x)=f'(\lambda x)$ is periodic?

Let $\lambda > 0$ and let $f(x)$ be a periodic function that has period $a$. How to show that function $g(x)=f'(\lambda x)$ is periodic and determine its period. Just some hints, please. I have ...
2
votes
2answers
344 views

A positive “Fourier transform” is integrable

Let $f\in L^1_\mathbb C(\mathbb R^n)$. I once read, in one of my old exam, that if $\hat{f}(\mathbb R^n)\subset\mathbb R_+$, then $\hat{f}\in L^1(\mathbb R^n)$. As far as I remember, the professor ...
2
votes
1answer
188 views

What is wrong with this IDFT trick?

In this section from Wikipedia about IDFT, three methods are given for expressing the Inverse Discrete Fourier Transform in terms of the direct transform. Being curious, I implemented the three ...
3
votes
1answer
115 views

Convergence of the Real analysis

The question is find the Fourier series of "|cost| for all t". I already found the fourier series But now the question asks " At which values of $x$, does the series fail to converge to ? To what ...
5
votes
1answer
98 views

Inequality for Fourier transform of measure

I am having trouble with the following question. Let $\mu$ be finite measure on $\mathbb{R}$ and let $\hat{\mu}(\xi) = \int_{-\infty}^\infty e^{-ix \xi} d\mu(x)$ be its Fourier transform. Prove that ...
2
votes
1answer
118 views

Implementing Discrete Fourier Transform

I am trying to implement Discrete Fourier Transform (by definition, in quadratic time). I wrote this http://jsfiddle.net/uunsm/12/ My result function really goes through discrete points, but it is ...
2
votes
1answer
1k views

Fourier transform of inverse rectangular pulse

Given the inverse rectangular function: $p(t) = \begin{cases}1&\mbox{ if }|t| > a,\\ 0 &\mbox{ if } |t| < a,\end{cases}$ where $a>0$ is real. And using the Fourier transform defined ...
2
votes
1answer
155 views

Relation on fourier coefficients implies smoothness for a periodic continuous function

I just came across with the following question.. suppose we are given a periodic function of period $2\pi$. We define $a_n$ and $b_n$ to be the Fourier coefficients of $f$. To be precise, we have ...
3
votes
1answer
253 views

Fourier Inversion formula on $L^2$

The Fourier transform is defined by $ \mathcal{F}f(\xi)=\int_{\mathbb{R}^n} e^{-ix\cdot \xi} f(x) dx $ If we restrict to Schwartz functions on $\mathbb{R}^n$, then the Fourier transform has an ...
2
votes
1answer
464 views

Using the discrete fourier transform to approximate the regular fourier transform

This may be an elementary question, but I'm not sure how DFT/FFT is used to approximate regular Fourier transforms. Consider the radial distribution function $g(r)$. The structure factor is defined ...
2
votes
1answer
163 views

Fourier transform of an exponentially attenuated function

Suppose I have a function $f(t) \in L^2(\mathcal{R})$ and it is specified by: $$f(t) = \int_{-\infty}^{\infty} H(\omega) \exp(-\beta t \omega) \exp(i \omega t)\,d\omega$$ Suppose $H(\omega)\in ...
1
vote
0answers
138 views

Order of partial sums in the derivatives of the Fourier series

Given periodic function $f\in C^{w}[0,1]$ with its Fourier series $f(x)=\sum\limits_{s=-\infty}^{\infty}f_{s}\exp(2\pi isx)$. What can one say about the asymptotic order of ...
0
votes
1answer
134 views

Reading a DFT plot - did I get this right?

I am simulating the evolution of a liquid film through the solution of a 4th order nonlinear partial differential equations. Of late, I began experimenting with DFT of the result that I have. My ...
1
vote
2answers
614 views

Fourier transform of heat equation

I need to solve following partial differential equation with Fourier transform numerically. $ \frac{\partial T}{\partial t} = \nabla(c\nabla T) $ where T is temperature, c heat conductivity and t is ...