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).

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Calculating a Poisson probability from the chacteristic function?

In a previous homework assignment we were given a function that corresponds to an arbitrary angular distribution $A_{FB}=(F-B)/(F+B)=(F-B)/N$, where F = # of events in the forward hemisphere, B = # of ...
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185 views

How is study of fractals related to fourier/spectral/harmonic analysis?

In chap. 3 of "Fractal Geometry of Nature" Mandelbrot mentions that "part of the study of fractals is the geometric face of harmonic analysis" (spectral or Fourier, he specifies). But to my dismay ...
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308 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 ...
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49 views

how to prove the convolution formular?

let $\overset{\backsim} {g}(x)=g(-x)$; suppose $u,\phi,\psi$ always make the integral significant,$E_n$ is the n-dimensional euclidean space. Then how to prove ...
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1answer
30 views

variables of Fourier analysis - how to prove their relations

Fourier transform: $\hat{f}(\xi) = \int_{-\infty}^{\infty} f(t)\ e^{- 2\pi i t \xi}\,dx$ where $t$ can be time and $\xi$ can be frequency. So, the question is how do we prove that $t$ and $\xi$ can in ...
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134 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} $ , ...
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1answer
166 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 ...
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116 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 ...
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160 views

how to correctly implement my own version of fft in matlab?

Currently I'm developing my own version of fft/ifft on matlab (homework) based on algorithm of Cooley/Tukey (radix 2). My code works fine for zero-padding images, but, how can I handle non-zero ...
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87 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 ...
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1answer
85 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 ...
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1answer
289 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?
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1answer
515 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 ...
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1answer
354 views

heat equation solution

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
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1answer
319 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 \ \ ...
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42 views

Is there meaning behind a sum of Derichlet kernels?

I have arrived at the following expression: $$\sum_{n=0}^{\infty} \sqrt{n+1}D_{n}\left(x\right)$$ Where $D_n\left(x\right) = ...
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1answer
131 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 ...
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1answer
158 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 ...
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1answer
110 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 ...
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1answer
79 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 ...
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849 views

Instructive proofs in functional analysis

I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
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1answer
68 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 ...
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277 views

Fundamental role of the Fourier Transform

I am currently learning about the Fourier Transform and the associated Fourier Analysis. So far I realize that it has a number of applications, but more than that it seems to be central to Functional ...
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90 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} ...
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1answer
187 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 ...
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1answer
168 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 ...
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61 views

Power Spectral Density excercise

Compute the power spectral density of $x(t) = \operatorname{sgn}(t)$ Hint: $$\lim_{t\to\infty} t(\operatorname{sinc}(ft))^2 = \delta(t).$$ Please help me I have to solve this exercise urgently. ...
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75 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 ...
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446 views

Fourier transform independent of kernel?

I've tried computing a windowed Fourier transform using various kernels that were all made from periodic signals of the form a + bi with b being a shifted version of a. I used square waves, sin waves, ...
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1answer
107 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 ...
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160 views

Using Khinchin's inequality

At the end of page 5 of the Tao's lectures notes, he sets $\psi$ a Schwartz function supported on the unit cube $[0,1]^n$ and choose $f(x)=\sum_{k=1}^N\epsilon_k\psi(x-ke_1)$, where $e_1$ is one of ...
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1answer
133 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 ...
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1answer
352 views

fancy about inverse discrete Fourier sine and cosine transform (i.e. Fourier sine and cosine series)

In order to find $f(x)$ so that $F(u)=\sum\limits_{x=0}^\infty f(x)\sin\dfrac{\pi ux}{L}$ and $F(u)=\sum\limits_{x=0}^\infty f(x)\cos\dfrac{\pi ux}{L}$ , we can borrow the idea from Fourier sine ...
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198 views

Are the amplitudes of these frequency spikes equal to 1 when the real part of the complex number “s” is equal to one half?

Over at stack overflow I asked a question about how to plot the Riemann zeta zero spectrum from the von Mangoldt function. Then I asked a question about calculating the Riemann zeta function at the ...
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1answer
157 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 ...
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1answer
69 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. $$x_1(n)=[1,8,6,7,4,2,3,1]$$ $$x_2(n)=[1,4,3,2,8,7,6,1]$$
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247 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 ...
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1answer
904 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 ...
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148 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 ...
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210 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 ...
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3answers
417 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?
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131 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 ...
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275 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} ...
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1answer
65 views

Question regarding the function $R_X(t)=\frac{1}{\pi} \sum_{p\leq x} \frac{\sin(t\log p)}{\sqrt{p}}$

I want to show that the expected value $\mathbb{E}_{\omega ,T}(R_x(t)^{2k})$ behaves asymptotically as: $$\frac{(2k)!}{k!\cdot 2^k} \left(\frac{\log(\log T)}{2\pi^2}\right)^k$$ for $T^\epsilon < ...
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247 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} = ...
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2answers
84 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 ...
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361 views

convolution of signals

I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals: $$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$ where $u(t)$ is ...
2
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1answer
137 views

Smoothness of Fourier series

In a book from differential equations I found the following theorem, without proof and references: Let functions $f, g: R \rightarrow R$ be continuous and $2\pi$-periodic and let $m\in N$. Assume ...
2
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1answer
172 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 ...
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1answer
112 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 ...