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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51 views

Heat equation with heat source in form of delta function

Consider the problem \begin{equation} \left\{\begin{array}{cc}u_t-u_{xx}=\delta_0,&0<x<1,\ t>0\\ u_x(0,t)=u_x(1,t)=0,&t>0,\\ u(x,0)=0,& 0\leq x\leq 1.\end{array}\right. ...
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58 views

Fourier Transform in $L^2(\mathbb{R})$

I have found a proof to the following theorem which is a fair bit shorter than the proof in my notes. I would be very grateful if someone could tell me whether this way works or whether I've made an ...
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33 views

Fourier transform from Laplace transform

So what I did was Laplace transform $f(t)$ to $F(s)$ and then plug in $s=jw$. However, when I tried this for $cos(3t)$, $$F(jw)={jw\over9-w^2}$$ was the answer. I don't know if this is correct, and ...
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32 views

What is the Fourier transform of $1/f(x)$?

Given $F(t) = \mathcal{F}\{f(x)\}$ is the Fourier transform of $f$, how can one express $\mathcal{F}\left\{\dfrac{1}{f(x)}\right\}$ in terms of $F(t)$? EDIT: To be more concrete, I want to compute ...
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3D Fourier Transform - Angle between $\mathbf{k}$ and $\mathbf{r}$

The definition of the Fourier transform for three dimensions is $$\mathcal{F}[f(\mathbf{r})](\mathbf{k})=\int e^{-i\mathbf{k}\cdot \mathbf{r}}f(\mathbf{r})\,d^3 r$$ If the function $f(\mathbf{r})$ ...
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123 views

Fourier transforms intuitive explanation

I have read on wikipedia that: The Fourier transform of a function of time itself is a complex-valued function of frequency, whose absolute value represents the amount of that frequency present in ...
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41 views

How to find fourier transform of $e^{-x^2}$?

I want to find the fourier transform of $e^{-x^2} = \int_{-\infty}^{\infty}e^{ikx-x^2}\,dx$ using contour integration. I consider the rectangular contour $C$ with verticies $\pm R, \pm R + ik$ Then ...
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20 views

Complex exponential argument to a function

In many texts on signal processing, the following notation is used to describe the Fourier transform of a discrete time signal $x$: $$ \hat{X}\left(e^{j\omega}\right) = ...
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53 views

Create periodic function from combining non-periodic functions

I'm studying recurrent neural networks which often use tanh as an activator function which is not periodic. However in research and papers it's shown that these recurrent neural nets can exhibit ...
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78 views

Discrete fractional fourier transform matrix

I am trying to write a matlab code for some calculations based on Discrete fractional fourier transform. in this article: Optimal filtering in fractional Fourier. after equation (7) a notation Fa is ...
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29 views

Parametrization of arbitrary objects to display on an x-y-scope

I am trying to find an approach for general parametrization of an arbitrary geometric object or closed curve. Though I am not sure if I am on the right path with that. Basically I have an geometric ...
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28 views

Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
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23 views

FFT of k*k matrix from FFT of a j*j matrix

FFT of matrix a j by j matrix, A $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ = $\begin{bmatrix}10 & -2\\-4 & ...
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20 views

Calderón–Zygmund lemma and Lebesgue measure

Can someone please explain the relationship between the Calderón–Zygmund lemma and Lebesgue measure, thanks.
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71 views

Fourier transform of $\frac{\sin2\pi t-\sin\pi t}{\pi t}$

I'm helping a student out with determining the transform above. His instructor apparently offered a hint about applying the convolution theorem, but I can't seem to get anywhere with that suggestion. ...
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23 views

How can I make the mean of samples be approximately equal to the mean of actual continuous signal?

Suppose there is signal f(t) that is continuous and periodic. It is known that this f is T-periodic. (but it's not necessarily a single cosine f(t).( I'd like to make the mean of samples be ...
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61 views

Negative index Sobolev spaces, properties

I was hoping to find some references for the following facts: Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. (i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then ...
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29 views

How to check periodicity of $f(t)$ using samples

Suppose that we know that signal $f(t)$ is $T_1$-periodic. Let $f_1 = 1/T_1$. But we want to know whether signal is $T_2$-periodic also. Let $f_2 = 1/T_2$, and $f_2$ is positive integer multiples of ...
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66 views

Smoothness-and-decay relationship of the Fourier transform

Recall that a function $f\colon \mathbb{R} \to \mathbb{C}$ is said to be rapidly decreasing if $$\sup_{x \in \mathbb{R}} \big|x^k f(x)\big| < \infty \quad \text{for all} \quad k = 0, 1, 2\dotsc$$ ...
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25 views

Prove $\int_0^\infty f(t) \frac{1}{t+x} dt$ is its own Fourier cos transform if $f(t)$ is its own Fourier cos transform

The problem says to use the fact that $g(x) = \int_0^\infty f(t) e^{-xt}$ is its own Fourier sine transform if $f(x)$ is its own cos transform. My working so far: $F_c(\int_0^\infty f(t) ...
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12 views

Result obtained on deletion of finite number of Fourier Coefficients

I want to know the answer to the following question. If a finite (but fixed) number of Fourier coefficients (of any choice) of a Fourier series are made $0$, then will the new series be a Fourier ...
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46 views

one parameter divergence formulation

I was reading this article: http://www.cims.nyu.edu/~trogdon/index_files/publications/periodic.pdf screenshot: And it is not clear to me how to write a pde in a divergence form. How can I transform ...
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32 views

PCA Vs Fourier Transform

As a general rule when trying to deconstruct a noisy signal into its components. When is it better to use Principal Component Analysis and when is it better to use a Fourier Transform?
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44 views

If the signal's frequency is multiples of the first harmonic frequency, transform method similar to DFT but use less number of samples?

Suppose that a continuous signal $f(t)$ has the first harmonic frequency $f_1$. $f(t)$'s frequencies that are not integer multiples of $f_1$ are known to have zero signal magnitude $|F(\omega)|$. This ...
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56 views

About a generalization of the Riemann-Lebesgue lemma

We have that for $f$ $\in$ $L^{1}$$(\mathbb{R}^n)$, $g$ measurable and bounded on $\mathbb{R}^n$ and, for any rectangle $R$, $$ \lim_{m(R)\rightarrow\infty} \frac{1}{m(R)}\int_Rg(x)dx = 0 $$ Then: ...
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35 views

If f is continuous and f $\in$ L1 then $\lim_{\tau\to \tau_0} \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $0$?

Where $ \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $\int_\mathbb{T} \vert f(t-\tau)-f(t-\tau_0)\vert \ dt$ I find it easy to see when f is uniformly continuous, since we would have $\vert ...
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62 views

Calculate Distance between Fourier Transforms

I'm working with signal data (specifically data from accelerators and gyroscopes), and I take their Fourier transforms to get a better idea of the dominant frequencies. I'd like to compare the ...
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32 views

Fourier transform of a real white noise on a 3D cubic lattice

I'm facing the following problem: I have a cubic domain of side $L=2\pi$; this domain is divided in a cubic grid, each side is divided in $N$ points, where $N$ is an even integer number. the ...
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52 views

Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the nth order partial derivative of a function with respect to x, using the transform variable k is: (i*k)^n * [F(k)] so for the 1st order derivative ...
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37 views

Significance of orthonormal basis in wavelet analysis

I've recently been looking into wavelet analysis and I have the question: What is the importance of wavelets having an orthogonal basis, say as opposed to a bi-orthogonal basis or otherwise? I'm ...
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44 views

Overly Regular PDE Solution - where's my mistake?

I was looking at this question and attempting to come up with my own answer when I "proved" a statement which seems far too strong: if $f\in L^2(\mathbb R^n)$, then the unique solution to $u-\Delta ...
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29 views

Polar Coordinate for Heat Equation

I'm currently working on how to derive the Laplacian of the Steady-state Heat Equation from the polar coordinates. In other words, I tried to show that: $\frac{\partial^{2}{u}}{{\partial r^{2}}} + ...
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48 views

two-dimensional wave equation

I have a question about "the two-dimensional wave equation", I have solved it but I wanted to know if I've done right. Because the solution is long, I just write the answer but I can write the ...
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37 views

Fourier transform of integral function

A function $s(t)$ is defined by $s(t)=\int_x p(t-cx)dx$ where $\tau = cx$ is a time variable and $t\neq \tau$. What is the Fourier transform, $S(\omega)$, of the function $s(t)$? I know that for a ...
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What does it mean for the fourier transform to map $L_2$ to $H_2$?

I am in a very introductory fourier transform class. When I looked up online about fourier transforming as a linear mapping, there was this reference that the fourier transform maps $L_2$ into $H_2$ ...
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Existence of compactly supported Fourier transforms on LCA groups

I'm trying to prove the following theorem: The following are equivalent for a locally compact abelian group: $G$ has an open compact subgroup. There exists a nonzero $f\in C_c(G)$ such that ...
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37 views

Is it feasible to think of laplace transform and z transform as projections?

For Fourier transform, it has been ingrained in my head that all we are doing is projecting a function onto its Fourier basis, namely $(1, cos(t), sin(t),...cos(nt), sin(nt) ...)$ Can anyone comment ...
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31 views

Sum of unitary complex numbers

Let us define: $$\varphi(x,n,t):=\frac{1}{n}\sum_{y=1}^n \left| \sum_{k=1}^n e^{2ik\pi (x-y)/n + 2i \sin(2k\pi/n) t} \right|$$ Does somebody have an idea how to prove that $$ \sup_{x=1,...,n} ...
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Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
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68 views

Using FFT to compute DFT of a polynomial

i currently studying about FFT and DFT and we were given simple question: Use the recursive FFT to compute the DFT of this polynomial of '3' degree: $$-1\:+\:4x\:+\:3x^2$$. So, i go to this ...
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84 views

The Heisenberg uncertainty principle in the time-frequency plane

The Heisenberg uncertainty principle says that it is impossible to have a signal with finite support on the time axis which is at the same time band limited. Is the following reasoning correct: When ...
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29 views

Does the convolution property suffice to show $\hat \chi*\hat \chi=\hat \chi$?

Given a compact and sufficiently regular set $\Omega\subset\mathbb{R^n}$ and its characteristic function $\chi=\chi_\Omega$, I would like to conclude that the (inverse) Fourier transform ...
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70 views

How to calculate the $(3)$ and $(4)$?

In Gérald Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory" Cambridge University Press 1995, On the page of 97-98, I Can calculate the $(1)$ and $(2)$, but I do not know how ...
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23 views

two dimensional extension of the Fourier transform of $|x|^\alpha$?

I know the Fourier transform pair \begin{equation} |x|^\alpha \leftrightarrow -2\sin(\pi\alpha/2)\Gamma(1+\alpha)|\omega|^{-1-\alpha} \end{equation} Can this formula be extended to two dimensional ...
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22 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 ...
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99 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 ...
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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
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54 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 ...
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24 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 ...
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48 views

Transforming 1D Burger's Equation into infinitely many coupled ODE's

I've been working on the following problem but I can't justify my steps, would a savvy mathematician kindly tell me what, if any, violations I've made. Problem: Show Burger's equation can be written ...