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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Non periodic Fourier Series Point Convergence

If $f$ is a real-valued non-periodic continuous function that is differentiable at the point $x_0$, is it true that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
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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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21 views

Compute the solutions of the following equation in Fourier space:

$$\frac{d^3u}{dx^3} − αxu = 0, x ∈ R, $$ where $ α > 0$ is some constant and $u(x)$ is assumed to satisfy $\int_R u(x) dx = π.$ I know this is a ODE so this is what I came up with so far: ...
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22 views

Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
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The maximum value (peak) of multiple self-convolution of rectangular function

In Multiple self-convolution of rectangular function - integral evaluation, formula for self-rectangular function of rectangular function seems to have been derived. How do we prove that this formula ...
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21 views

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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Help establishing a bound on the Fourier coefficients of a bounded $2\pi$ periodic function that is discontinous at the end points?

This is from a practice midterm, and I'm having trouble with the first part. Suppose $f$ is a $2\pi$-periodic function that is continuous and differentiable on the interval $[-\pi, \pi]$, but has jump ...
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1answer
20 views

Diagonal of multidimensional DFT

If $X$ is a $n\times n$ square matrix and $F$ its Discrete Fourier Transform, is there a way to compute the diagonal $(F_{1,1},\ldots,F_{n,n})$ without explicitly computing the full DFT? How about ...
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13 views

Fourier transform on fractional Sobolev spaces

We say that a tempered distribution $f$ satisfies $f \in H^s(\mathbb R)$ for some $s \in \mathbb R$ if $(1+|\xi|^2)^{s/2} \hat f \in L^2(\mathbb R)$. Here, $\hat f$ denotes the Fourier transform of ...
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Puzzle to puzzle you:image [on hold]

Suppose 3 1D Signals x(t), y1(t) and y2(t) are given as x(t)=sin(40*pi*t); y1(t)=.5*sin(40*pi*t) and y2(t)=x(t)+y1(t). Left side =Right side Here,values of x(t)and y1(t)i.e.(Right side) are given ...
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14 views

How this result in archived in Fourier series

I was reading some notes about functions of symmetry in Fourier series and came across the following result for a function with symmetry of an odd quarter wave $$\begin{align} ...
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7 views

Substitution in complex-valued Fourier integral

In Knapp (Representation theory of semisimple groups, 86'), on page 34 it is shown by means of Euclidean Fourier transform that the principal series representation of $SL(2, \mathbb C)$ is irreducible ...
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Intuition Behind the Riesz Transform

Define the Riesz transform in singular integral form \begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. ...
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1answer
32 views

Fourier Transform of $\delta(t-nt)$

Given the discrete signal $x(n)=\begin{bmatrix} \alpha ^n, n\geq 0 \\0, n<0 \end{bmatrix}$ where $\alpha \in (-1,1)$ and some natural number $N$, we know that the discrete signal $y(n)$ (where $0 ...
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1answer
29 views

Find the inverse fourier transform of simple function

Suppose that the fourier transform of a signal $x(t)$ is $\hat x(u)=\frac{1}{2u_m}(1+\cos (\frac{\pi u}{u_m}))$ where $u_m \geq |u|$.$t$ here stands for time so $t \geq 0$ We sample the original ...
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Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ ...
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Obtaining the density of a Compound Poisson Process using Fourier Inversion Formula [on hold]

If $X_t=\sum_{i=1}^{N_t}J_i$ and $E(e^{itX_t})=e^{\lambda t (E(e^{itJ_1})-1)}$ Using the Fourier Inversion Formula, $f(x)=(1/2 \pi))\int_{-\infty}^{\infty}e^{-itx}e^{\lambda t ...
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18 views

Fourier transform and series

Let $f \in L^2(\mathbb{R})$ and $F(f|_{[m,m+1]})$ be the Fourier transform of a restriction of $f$. Does this imply that $$\sum_{m,n \in \mathbb{Z}} |F(f|_{[m,m+1]})(2 \pi n)|^2 $$ exists and is ...
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8 views

Sampling in the time domain vs. sampling in the frequency domain

If I have a sample rate of 2 seconds on a 128 second time window (64 samples total) and then I do a Fourier transform, what is my sample "rate" in the frequency domain? Will I end up with 64 ...
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Solving the Wave Equation using Fourier Transforms

The problem is: \begin{equation} u_{tt} -c^2u_{xx} - a^2 u = 0 \end{equation} with $\hspace{2mm}-\infty < x < \infty $, $ \hspace{2mm} u(x,t) \hspace{2mm}$ bounded as $ x \rightarrow \pm ...
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properties of orthonormal systems and hilbert spaces [on hold]

I need to show (a) $\implies$ (b) For an orthonormal system $\{\phi_i\}_{i=1}^\infty$, and a Hilbert space $H$, the following are equivalent: (a) If $\langle f,\phi_i\rangle=0$ $\forall i$, ...
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Fourier Decompositon

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier ...
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28 views

Fourier Series Convergence

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function that is differentiable at the point $x_0$. Prove that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
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1answer
19 views

Fourier series phasor form and sin/cos form

can anyone give me a link on how to convert the forms (from phasor to sine/cos and vice versa)? I am new to this and I can't find the convertion table with a valid explaination.
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20 views

Linear span of poisson kernels dense in $L^1(\mathbb{T})$

A paper I am reading ("Schur's Algorithm, Orthogonal Polynomials, and Convergence of Wall's Continued Fractions in $L^2(\mathbb{T})$" by Sergei Khrushchev...really a great paper) repeatedly mentions ...
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28 views

proving a claim using discrete fourier transform of sequence

I get why the answer to 1 is: but I dont have a clue about b. Please help!
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27 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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24 views

Parseval's identity does not hold for constucted basis

As part of an exercise, I was asked to show that given an orthonoraml basis $(\varphi_1,\varphi_2,\varphi_3,...)$ in $L_2[-\pi,\pi]$, we can construct an orthonormal basis $(\psi_1,\psi_2,\psi_3,...)$ ...
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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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how manipulating the coefficients vector effects on the result of DFT?

given: calculate: note that the given DFT is from n order and we want to compute DFT's from 2n order. edit: this is my try of B. i don't see where the given DFT is used and how to proceed: ...
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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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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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1answer
20 views

Fourier Transform for Boundary Value Problems

I am trying to understand the problem defined by $$\phi_{xx} + \phi_{yy} = 0, in -\infty \lt x \lt \infty, y \gt 0$$ $$\phi = f(x) \space as \space y \to 0, \phi = 0 \space as \space y \to \infty$$ ...
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1answer
38 views

Show $\int_{-\pi}^{\pi}[f'(t)]^2 dt \geq \int_{-\pi}^{\pi}[f(t)]^2 dt$

Given that $f$ is $C^1$ on $\mathbb{R}$ with period $2\pi$ and $\int_{-\pi}^{\pi}f(t)dt =0$, show $$\int_{-\pi}^{\pi}[f'(t)]^2 dt \geq \int_{-\pi}^{\pi}[f(t)]^2 dt$$ I've tried using Bessel's ...
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35 views

Calculating Fourier Transform of $\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$

This question deals with finding the Nyquist Frequency of a given signal. Suppose you have the signal $x(t)=\sum_{n=1}^{3}\sin(2\pi \frac{n}{8}\frac{t}{T})$ in the time domain where $T>0$ is some ...
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30 views

Pontryagin Dual of the Unit Circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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Fourier transform of $e^{-4\pi ^2 x^2}$

How do you prove $$\int_{-\infty}^{\infty}e^{-(2\pi x + i\xi/2)^2}dx=\int_{-\infty}^{\infty}e^{-(2\pi x)^2}dx$$ for $\xi \in \mathbb{R}$. The Question arises from calculating the Fourier Transform ...
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How to calculate full width half max of this curve

I'm trying to calculate the full width half max of this function and I keep receiving non-sense answers. The function is $F(ω)=A_0 (\frac{\frac{1}{τ}}{(\frac{1}{τ^2})+(ω_0 - ω)^2})$ and I then have ...
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1answer
18 views

What is a good source for learning fourier transformations as an application

I'm an undergraduate physics major and for my research I need to start learning and understanding fourier transformations for my research. Does anyone know good resources for doing this. I don't need ...
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22 views

Fourier transform of the error function, erf (x)

I define $\text{erf}(x):=\frac{2}{\sqrt{\pi}}\int_0^xe^{-\xi^2}d\xi$. What is its Fourier transform (unitary, ordinary frequency)? That is, simplify ...
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Periodic convolution of functions

Define the periodic convolution of functions in L2([0; 1]). What theorem of convolution do I use to define this and how do I solve this?
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A Fourier Transform Which Is Cartesian Separable

We say that the Fourier transform of a complex-valued function $f\in L^{1}(\mathbb{R}^{n})$ is separable if there exist single-variable functions $g_{1},\ldots,g_{n}$ such that $$g_{1}(\xi_{1})\cdots ...
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31 views

Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
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Complex Fourier Coefficients by Inspection?

This is the solution to a fourier series problem, of the function $sin(\omega_0t)$: I understand how the author has used Euler's formula to split this function into two exponential terms. However, ...
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Elementary properties of Fourier transform on the space of tempered distributions.

I'm trying to prove that some of the basic facts of Fourier transform on $L^1(\mathbb{R}^n)$ also holds on the space of tempered distributions. For example: Suppose that $F\in \mathcal{S}^\prime$, ...
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Where does the imaginary unit dissapear in the Fourier transform of $f(t)= \exp(iat)$?

So I make the Fourier transform of$ f(t)= e^{iat} $on $[- \pi, \pi]$ for some real $a$ and i get: $$a_n=\frac{2a \sin(a \pi)(-1)^n}{\pi(a^2-n^2)}$$ $$b_n=\frac{2i(n\sin(a \pi) (-1)^n)}{\pi(a^2 - ...
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Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
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31 views

Getting fourier coefficient by integrating over half the period?

In the book Schaum's Outlines of Analog and Digital Communications solved problem 1.2, the author calculates the fourier coeffecient $C_0$ for the rectangular pulse train: where $a$ is assumed to be ...
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Can $\sum_{a = -\infty}^{\infty} e^{i\omega aT_0}$ be represented by dirac delta functions?

The usual definition of dirac delta function says that $\delta(x-\alpha)=\frac{1}{2\pi} \int_{-\infty}^\infty e^{ip(x-\alpha)}\ dp $. The appearance similarity makes me think that it may be possible ...
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Approximation of one function by other using a smooth multiplier function

This problem is from the Book, Harmonic Analysis by Katznelson (Problem 2, Page 160). Suppose $f$, $g\in L^2(\mathbb{R})$ such that $f(x) = 0$ implies $g(x)=0$ for almost all $x\in\mathbb{R}$. Then ...