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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How to understand the mapping between a periodic function to its Fourier coefficients?

For a periodic function $f(x)=f(x+T)$, its Fourier transform can be written as an infinite sum: $$ f(x)=\sum_{-\infty}^{\infty}c_n e^{2\pi i x/T}. $$ This seems to suggest that the information ...
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108 views

Several questions about integral operators.

I have been fumbling with expressions of the form \begin{equation} A\{f\}(s) = \int A(s,t)f(t)\operatorname{dt} \tag{$\star$} \end{equation} as a generalization of the matrix product. When looking ...
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17 views

Fast Fourier Transform: How is the roots of unity matrix divided?

For an example for input size N=8, how is the roots of unity matrix divided for a divide and conquer approach? My understanding is that it's divided into four quadrants, Ma with J&K evens; Mb ...
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1answer
29 views

Inverse Fourier Transform Proof

I am aware of how Fourier Transformation and Fast Fourier Transformation works, however I do not understand the logic of the inverse of FFT. Could someone explain why the inverse fourier ...
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3answers
2k views

Heaviside step function fourier transform and principal values

I found the following answer on SE: Fourier transform of unit step? However, it is still not clear to me and maybe somebody could explain it clearer. Problem I have the following in my notes of ...
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1answer
47 views

Computing the Fourier transform of $e^{-|x|}$

I'm trying to compute the Fourier Transform of $e^{-|x|}$ however I receive a different answer than wolfram alpha. Wolfram Alpha gets $\sqrt{\frac{2}{\pi}}\frac{1}{1+\varepsilon^2}$ and I get ...
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1answer
32 views

Problem with finding a fourier transform

Help me to find a 2D fourier transform: $$\int dx dy\ \frac{e^{-ik_x x}e^{-ik_y y}}{\sqrt{x^2 + y^2}}.$$ All I've done so far is $$\int dx dy\ \frac{e^{-ik_x x}e^{-ik_y y}}{\sqrt{x^2 + y^2}} ...
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1answer
65 views

Gibbs Phenomenon and Fourier Series

a) Show the partial sum $$S = \frac{4}{\pi} \sum_{n=1}^N \frac{\sin((2n-1)t)}{2n-1}$$ which may also be written as $$ \frac{2}{\pi}\int_0^x\frac{\sin(2Nt)}{\sin(t)}dt$$ has extrema at $x= ...
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1answer
46 views

convolution and associativity

Ok Let talk about this,... I am now so confused. 1-$$\mathcal{F}\Big\{c(x-x_0)b(x-x_0)\Big\}=\mathcal{F}\Big\{c(x-x_0)\Big\}\circ\mathcal{F}\Big\{b(x-x_0)\Big\}\\=\Bigg[e^{-2ix_0y}C(y) ...
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48 views

Convolution of a function with itself

Function $\phi (x)$ is defined as: $$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$ How do I find the convolution of $\phi(x)$ with itself? I ...
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1answer
60 views

Compact set of measure zero and sequence of Harmonic Functions with nice properties.

I was studying John B. Garnett's book Bounded Analytic Functions, and then I decided to try the following problem: Let $E \subset \mathbb{R}$ be a compact set, with $|E|=0$. Prove that there ...
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3answers
111 views

Klein Bottle discrete harmonics?

Studying discrete representations of a function, on a $2$ Dimensional compact surface, brings to the use of spherical harmonics for the 2-sphere, and discrete Fourier transformations for the 2-torus. ...
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30 views

Number Theoretic Transform (NTT) to speed up multiplications

I recently heard that the Number Theoretic Transform (NTT), which is a specialization of Fast Fourier Transformation (FFT) over the ring modulo an integer, can be used to speed up certain ...
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1answer
45 views

Partial Sum Fourier Series

Show that the partial sum $$f_N(x)=\frac{4}{π}\sum^N_{n=1}\frac{\sin((2n-1)x)}{2n-1}$$ may be written as $$f_N(x)=\frac{2}{π}\int_0^x\frac{\sin(2Nt)}{\sin(t)}\,dt$$ The original question is 'Sketch ...
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3answers
5k views

Compare Fourier and Laplace transform

I would like to clarify main difference between Fourier and Laplace transforms and also understand if exponential factor is main difference between this two method. So Fourier transform is ...
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35 views

Intuition behind the proof of the Inverse Fourier Transform?

I am interested in the proof of the Inverse Fourier Transform for absolutely integrable real valued functions. The proof I have read asks you to consider an auxiliary function $g_{a}(x)$ defined as ...
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1answer
33 views

Computing inverse Fourier transform

To compute the inverse Fourier transform I need to evaluate the following integral ...
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51 views

Relation between Dirichlet convolution and Bell series and convolution of functions and the Fourier transform?

We define the Dirichlet convolution of two arithmetic functions $a,b:\mathbb{N}\to\mathbb{C}$ to be $$ (a*b)(n)=\sum_{d\mid n}a(d)b\left(\frac{n}{d}\right). $$ Given a prime $p$, we define the Bell ...
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2answers
57 views

Fourier Series Representation $e^{ax}$

a) Compute the full Fourier series representation of $f(x) = e^{ax}, −π ≤ x < π.$ b) By using the result of a) or otherwise determine the full Fourier series expansion for the function ...
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1answer
35 views

Using dummy variable to derive Nth partial Fourier sum

I am working on a problem$^{(1)}$ as follow: Using the standard formulas for the Fourier coefficients, show $$F_N(x) = \frac1{2\pi} \int_{-\pi}^{\pi} \left( 1+2 \sum_{n=1}^{N} ...
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29 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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1answer
28 views

Help with Homework Problem for Img Processing class

I have this homework about Image Processing: Give the general equation of a complex or real-valued digital image that produces a delta function in the frequency domain. Demonstrate this function for ...
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0answers
24 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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1answer
37 views

Discrete Laplace transform. Analogy to change of basis

Assume $$f=\sum_{k=0}^{N-1} c_k\cdot E^{k}$$ where the vector $E^k$ is $$E^k = (e^{2 \pi i k\cdot M}(0),e^{2 \pi i k\cdot M}(1),\cdots,e^{2 \pi i k\cdot M}(N-1))$$ (M is a constant and e represents ...
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140 views

State-of-art of the Discrete Fourier Transform

I would like to know what is the state-of-art in the research of the discrete Fourier transform. I have listed some questions to help answering, please add your own to make the list more ...
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1answer
28 views

Fourier transform doubting factorization

I have to find the fourier transform for $$ {1\over 1+16t^4} $$ I guess going there is a better way to solve it than going throug the integral but I'm not even sure if the factorization i made is ...
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1answer
19 views

DFT by $n$ samples of a continuous periodic signal with more than $n$ frequencies

It is known that if we only have $n$ samples and take DFT, we only get at most $n$ distinct frequency data. But let's say that there is a continuous periodic signal with more than $n$ frequencies, ...
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1answer
42 views

Sign mistake in Fourier transform of $\frac{x}{1+x^2}$.

I want to calculate the distributional Fourier transform of $u(x) = \frac{x}{1+x^2}$ in one dimension in the distributional sense as $u\notin L^1$. I use the distributional definition of the Fourier ...
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9 views

Find distortion exponent from Fourier fitting

I'm facing this problem in my master thesis: we are measuring the signal from a sensor which is, physically, a $\sin^2$ (or $\cos^2$). Some non idealities distort the signal by introducing an exponent ...
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29 views

Fourier Coefficients

So (r, θ) denotes polar coordinates in the plane, let a > 0 be a constant. $\nabla^2u=0$, in r < a $\frac{\partial u}{\partial r}+\gamma u=h(\theta)$ on r = a h is an arbitrary periodic ...
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27 views

If a signal is periodic, can the error of approximation by Discrete Fourier Transform be avoided when using finite number of samples?

As title says, if a signal $f(t)$ is periodic, can approximation errors of approximation by discrete Fourier transform (DFT) be avoided when only finite number of samples are used?
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Using Discrete Fourier trasform of the samples of a continuous/periodic signal to obtain frequency data similar to FT of the original signal

Suppose we have a continuous and periodic real-valued 1D signal $f(t)$. Let us say we obtain finite number of samples $f(n)$ from $f(t)$. Is there a way to take discrete Fourier transform of $f(n)$ ...
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1answer
40 views

Proving that the Gaussian is the minimizer of Heisenberg's uncertainty product via variational calculus techniques

The version of Heisenberg's uncertainty product I am interested in is $$\left(\int_{-\infty}^{\infty} t^2|f(t)|^2\,dt\right)\left(\int_{-\infty}^{\infty}t^2|\mathcal{F}f(t)|^2\,dt\right),\tag{1}$$ ...
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1answer
24 views

Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$. But why is it the case that ...
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41 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
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1answer
45 views

Fourier Series estimation

I know that the Fourier coefficient of $t\mapsto \frac{1}{\sqrt{\vert t\vert}}$ are given by some Fresnel integral, and behave like $O(n^\frac{-1}{2})$. Reciprocally, if I get a Fourier Series whose ...
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11 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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1answer
17 views

Discrete fourier transform of random noise interpretation

Let's say i have a real valued random noise $\eta(t)$, for which I took $N$ samples with $N$ even, i have therefore a vector of sampled values. I want to compute the Discrete Fourier transform of the ...
2
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1answer
55 views

What is the theory behind Fourier transform of “bad” (e.g. unbounded) functions?

When I was first introduced to Fourier transform, its core was a formula for it, something like: $$\tilde f(k)=\int_{-\infty}^{\infty} e^{-2\pi i kx}f(x)\text{d}x.\tag1$$ It works nice for good ...
5
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1answer
28 views

Is f necessarily continuously differentiable under given conditions?

$f_m:\mathbb{R} \rightarrow \mathbb{R}$ is function series that are integrable on any compact segment , $2\pi$ periodic and $f_m \overset{u}{\rightarrow} f$ converging uniformly. Moreover for all $n ...
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2answers
65 views

Prove that $\int_{-\infty}^\infty P_n(x) \, dx = \pi /n$

Let $P_n(x) = \frac{n}{1+n^2x^2}$. Prove that for every $n\in\mathbb{N}$ $$\int_{-\infty}^\infty P_n(x) \, dx = \frac{\pi}{n}$$ And for every $\delta > 0$: $$\lim_{n\to\infty} ...
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1answer
22 views

Can we obtain Fourier transform of a continuous signal using finite number of samples of the signal with known frequency cutoff?

Suppose that there is a continuous signal with highest frequency known. Is there a way so that we only sample the signal finite times and obtain the Fourier transform of the original signal (which ...
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1answer
53 views

Show that the distribution is of the form $C \delta + f$

I'm trying to solve this problem: Let $ u = p.v.(1/x)$, $\phi$, $\psi \in C^{\infty}_c$. I want to show that the distribution $(\phi u )* (\psi u)$ is of the form $C \delta + f$ for some constant C ...
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Fourier-transform of fermionic model: book recommendation

I am currently interested in the mathematics of fermionic fock spaces and especially its Fourier-transform $$ \mathcal F(g)(y)=\int \exp \left(-\sum_{i=1}^n y_ix_i\right)\cdot g(x) dx $$ with the ...
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2answers
42 views

Show that $|\sum _{j=1}^n \frac{\cos (2\pi jx)}{j}| \leq C -\log |\sin (\pi x)| $

Show that there exists a constant $C\geq 0$ such that for all $n\geq 1$ : $$|S_n(x)|=|\sum _{j=1}^n \frac{\cos (2\pi jx)}{j}| \leq C -\log |\sin (\pi x)|, \quad \forall x\in ]0,1]. $$ Since $S_n(x)$ ...
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22 views

Fourier Transforms of hyperspherical harmonics

I am trying to compute the Fourier Transform of a function on a 3-sphere, $f(\hat{Q})$, where $\hat{Q}$ is a unit vector in four-dimensional space. The function $f(\hat{Q})$ is expressed as a series ...
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1answer
156 views

Cooley-Tukey FFT with arbitrary radices

The radix-2 FFT using Cooley-Tukey utilises two interleaved transforms of length $N/2$, and you can see near the bottom of that section that we can find the second half of the original transform by ...
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21 views

Frequency spectrum of signal and is it real?

$x(t) = 2 + 5 cos(-t + \pi/4) - 2sin(3t + 5) + 3(cos(5 t + \pi/2).cos(4t) - e^je^t $ a) To find Fourier series coefficients of the following signal I need to use inverse Euler formula. But I need ...
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1answer
47 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 ...
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25 views

Discontinuous periodic waveforms

There 3 types of periodic discontinuous/ bounded waveforms in Fourier analysis. Discontinuity in y(t) ( square and sawtooth waves); discontinuity in y'{t} (triangular waves); and discontinuity in ...