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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laplace transform of $t^nf(t)$

I have: $$\mathcal{L}(t^nf(t)) = \int_0^\infty t^nf(t)e^{-st}\ dt = \left(-\dfrac{d}{ds}\right)^n \int_0^\infty f(t) e^{-st}$$ I don't understand where the derivative came from
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If a sequence of function converges in $L^2$ sense, then its Fourier series converges in $l^2$ sense?

I have the following material from my lecture notes and I am trying to prove it, but I am not sure how to do the second part. Suppose we have a sequence of continuous functions $\{q_n (x)\}$ ...
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23 views

Periodic Foricing Terms

The question asks to find the solution for the initial value problem: $ y''+\omega^2y=sin(nt),\quad y(0)=0,\quad y'(0)=0 $ where $n$ is a positive integer when a) $\omega^2\neq n^2$ and b) ...
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show that $\sum P(x,y) e^{x^2 + y^2}$ is a modular form over $\Gamma_0(4)$ where $P(x,y) = x^4 - 6 x^2 y^2 + y^4$

In these lecture notes of Zagier, I read that generalized theta functions are still modular forms. Let $q = e^{2\pi i z}$ $$\theta(z) = \sum_{(x,y) \in \mathbb{Z}} \Big[ x^4 - 6 x^2 y^2 + y^4 \Big] ...
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Show that f is constant by using fourier coefficient

Let f be a 2$\pi$ periodic, Riemann integrable function and let $\alpha$ be an irrational number. Suppose that $f(x+2\pi\alpha)=f(x)$ for all x. Show that f is constant almost everywhere. I know that ...
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How to prove the absolute value of this Fourier coefficient is bounded?

The question is: Let $f:[0,1)\to C $ be a step function. Prove that there exists a constant C such that for all integers $k\neq 0$, $$ \mid \,\hat{f}(k)\mid\le C\,\big/\mid k \,\mid.$$( $\hat{f}(k)$ ...
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51 views

What does the overline symbol mean?

So I have got a question in an old exam paper for Fourier Analysis. Let $f:I\to C$ be an integrable function. Prove that$\int_I \overline{f(x)}= \overline{\int_I f(x)}$. The problem is that I ...
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32 views

How to prove the Fourier coefficient of a convergent sequence converges as well

So the question is, $\,f_1, f_2,...,f:T\to C$ are integrable functions with $ \,f_n\to f$ in $\parallel \cdot \parallel_1 $ as $n\to \infty$. Let $k\in Z$. Prove that, the Fourier coefficient, $ ...
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15 views

Easy Fourier Transform

I am asked to find the fourier transform of $f(x)= a-|x|$ when $x<|a|$ and $f(x)=0$ otherwise. I have done the calculation and end up with $\dfrac{1-ik-e^{-ika}}{k^2}$, the answer shown is ...
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58 views

Show that $\sup_{x \in \mathbb{R}}|\sigma_nf(x) - f(x)| \leq C\frac{\log n}{n}$, for $f$ $2\pi$-periodic and Lipschitz.

I'm learning about Fourier analysis and need help with the following 2 problems: $(1)$ Show that $\forall t \in (0, \pi], K_n(t) \leq \min\{n +1, \frac{\pi^2}{(n + 1)t^2}\}$. $(2)$ For the ...
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33 views

Is there a general way to prove this Fourier transform property?

We know that one of the important Fourier transform properties is that, the Fourier transform of a narrow function has a broad spectrum, and vice versa, We can easily see this in this example, the ...
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23 views

An exponential sum problem.

If $q$ is a prime how do we compute $$\sum_{a,b,c\mod q} I_{1/4,\epsilon}(a)I_{1/4,\epsilon}(b)I_{1/2,\epsilon}(c)I_{1/2,\epsilon}(ca^{-1}b)I_{1/2,\epsilon}(cab^{-1})$$ where $I_{a,\epsilon}(x)=1$ if ...
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134 views

Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
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1answer
446 views

Wavelet or FFT for Transient signal analysis?

For now I use FFT to analyze the response of an electrical system to some transient signal. The transient signal is $x(t)$, which translates to $X(w)$ in the frenquency domain. On the other hand I ...
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116 views

Intuition behind Fourier and Hilbert transform

In these days, I am studying a little bit of Fourier analysis and in particular Fourier series and Fourier/Hilbert transforms. Now, I am confident with the mathematical definitions and all the ...
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1answer
31 views

Some confusion about the application of fourier transforms to derivatives

Assuming we denote Fourier transforms as follows: $\mathcal{F}[f(t)](\omega)=\tilde{f}(\omega)$, then we have the following identity: $\mathcal{F}[\frac{df}{dt}](\omega)=i\omega \tilde{f}(\omega)$ ...
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26 views

Integration by parts (convolution)

I am trying to evaluate the integral $$F(\nu) = \frac{jT}{2 \pi} \int^{\infty}_{- \infty} \delta ' (\nu ') \ \text{sinc} (T (\nu-\nu')) e^{-j 2 \pi (\nu-\nu') (T/2)} \ d \nu'$$ Where $\delta$ ...
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24 views

Fourier transform of sampling function

Calculate the Fourier transform of $f_{ZOH}$ (the zero-order hold reconstruction of a sampled signal). Where $f_{ZOH} (t)= f(kT), \ \ kT \leq t < (k+1)T,$ and the sampled signal is $$f_s = f(t) ...
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What is $\int^{\infty}_{-\infty} e^{(ik+l)x} d x$?

Let $k$ and $l$ be two real numbers. We know that $\int^{\infty}_{-\infty} e^{ikx} d x = 2 \pi \delta(k)$. Here $i$ is the imaginary unit and $\delta(\cdot)$ is the Dirac delta function. What is ...
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Why are differential equations with sinusoidal source terms easier to solve than others?

I am a software engineer trying to wrap my tiny human brain around Fourier Transforms for a project I'm currently working on. Although I will ultimately use an open source Math library to do all the ...
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22 views

Fourier coefficients of a triangle function

I'm trying to find the Fourier coefficients ($c_n$), of the following function : for $x$ in $[-\pi/2;\pi/2[$ $f(x)=x$ for $x$ in $[\pi/2;3\pi/2[$ $f(x)=\pi-x$ I think its not that hard, but I keep ...
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28 views

Fast Fourier Transform for non-trigoniometric bases

The fast fourier transform allows decomposition into a sin/cos basis in $N \log(N)$ complexity. Can one generalize the algorithm (or the ideas used) to other basis, e.g. orthogonal polynomial bases ...
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Value Proposition of Fourier Analysis?

I am a software engineer trying to wrap his head around Fast Fourier Transform (FFT). Specifically, I need to implement it as part of some software I am writing. Now I can handle the implementation of ...
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23 views

Fourier Transforms of exponential functions

I wanted to find the Fourier Transforms of the following exponential functions: 1. $e^{iax^2}$ 2.Fourier inverse transform of: $\cosh(xa)/\sinh(\pi x)$ , a is a constant in both cases.
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357 views

Recovering the spatial Fourier transform from the space-time Fourier transform

This CW question is aimed at developing some intuition (grokking) about a certain formula of Fourier analysis. Any kind of explanation (physical, geometrical, analytical ...) is welcome. If we ...
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1answer
47 views

Given $n$ points in the plane, prove that less than $2n^{\frac{3}{2}}$ pairs of points are a distance 1 apart.

Given $n$ points in the plane, prove that less than $2n^{\frac{3}{2}}$ pairs of points are a distance 1 apart. It seems like Piegeon Hole Principle but I don't know how to proceed.
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1answer
34 views

If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$

I'm learning about Fourier analysis and need help with the following problem: Suppose $f \in L^1[-\pi, \pi]$ and $\alpha_n, \beta_n$ are the Fourier coefficients of $f$. Show that if $f$ is odd ...
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454 views

Fourier Series Reduced Form: Phase Angle and Spectra

Im very confused regarding how to determine the angle on the reduced or harmonic form representation of the Fourier series. Some books state the following: $$f(t)=F_0+\sum_{n=1}^\infty |F_n ...
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557 views

Fourier Transform of Schwartz Space

I am trying to read through Corollary 8.23 in Folland, p. 250, which is a proof that the Fourier transform maps the Schwartz space into itself. I do not see why the following is true $$\|x^\alpha ...
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1answer
14 views

Laplace equation with boundary conditions in polar coordinates

Show that the problem with this boundary conditions $u_{rr}+1/ru_{r}+1/r^2u_{\theta\theta}=0$, $\quad 0 < r < 1, \quad 0 < \theta < \pi$ $u(r,0)=0$ $u(r,\pi) =T_0$ $u(1,\theta) =T_0 $ ...
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51 views

Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots = \frac{1}{2}\cot\frac{x}{2}$ for $x \neq 2k\pi, k \in \mathbb{Z}$

I'm learning about Fourier series (specifically Cesàro summation) and need help with the following problem: Show that the Cesàro sum of the series $\sin x + \sin 2x + \sin 3x + \ldots$ is equal to ...
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51 views

Why is $A(\mathbb{T})\subset C(\mathbb{T})$?

where $A(\mathbb{T})$ is the space of Absolutely Converging Fourier Series and $C(\mathbb{T})$ is the space of Continuous Functions, both over $\mathbb{T} = [0,1)$. If $ f\in A(\mathbb{T})$ is $f\in ...
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squared bicoherence matlab parameters

I would like to use a modified version of the squared bicoherence formula in Matlab. HOSA does not contain the squared bicoherence formula. The bispectrum is given by the following equation $$\ ...
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99 views

Is the regularization of a Fourier transform unique?

The Fourier transform of the Coulomb potential $1/\vert \mathbf r \vert$ of an electric charge doesn't converge because one obtains $$F(k)=\frac {4\pi}{k} \int_0^\infty \sin(kr) dr.$$ The standard ...
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Laplace equation with Boundary value conditions by parts

I don´t know how to procced in this problem by parts $u_{xx}+u_{yy}=0$, $\quad 0 < x < \pi, \quad 0 < y < \pi$ $u(x,0)=0$ $u(0,y) = \begin{cases} y, & \text{for } 0 < y < ...
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Does $ \int_a^b |f(x) - f_1(x)| = 0$ imply $ \int_a^b |f(x) - f_1(x)|^2 = 0$?

Context:I'm trying to solve this problem: Suppose $f, f_1, g, g_1$ all Riemann integrable complex valued functions on $[a, b]$ such that $f \sim f_1$ and $g \sim g_1$. Prove $\langle f, g \rangle ...
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1answer
139 views

The proof of the Plancherel Theorem

I am reading the proof of the Plancherel Theorem in Folland. But I am quite confused about one of his claims. Suppose $f,g \in L^2(T)$ and $\hat{f}\in L^1$ ($\hat{f}$ is the Fourier transform). then ...
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A Hilbert transform that takes several functions

While playing with some PDE I came across a singular integral that looks something like ...
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28 views

How to solve this Laplace boundary value problem by Fourier series

can someone help me?, I don't know how to proceed in the last boundary condition $u_{y}(x,1)=x(1-x)\ $ $u_{xx}+u_{yy}=0\ $, $\ 0<x<1,\ 0<y<1$ $u(0,y)=0$ $u(1,y)=0$ $u_{y}(x,0)=0$ ...
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Mean-Square Fourier Convergence

Let $ \left \{X_n\right \} ^{\infty}_{n=1}$ be any orthogonal (in the $L^2$ sense) set of functions. Let $$S_N(f) = \sum^{N}_{n=1} \frac{(f, X_n)}{ \left \|X_n\right \|^2} X_n$$ be the “Fourier ...
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Taking inverse Fourier transform of $\frac{\sin^2(\pi s)}{(\pi s)^2}$ [duplicate]

How do I show that $$\int_{-\infty}^\infty \frac{\sin^2(\pi s)}{(\pi s)^2} e^{2\pi isx} \, ds = \begin{cases} 1+x & \text{if }-1 \le x \le 0 \\ 1-x & \text{if }0 \le x \le 1 \\ 0 & ...
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Fourier transformed multiplication operator leaves $L^2([-C,C])$ invariant?

Let $C > 0$ be some constant and $L^2([-C,C])$ the square integrable functions on $[-C,C]$. Let $\delta > 0$ and let $M_{|\cdot |^\delta}$ denote the multiplication operator on $L^2(\mathbb R)$ ...
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16 views

Computing eigenfunctions, difference between beta and beta prime.

I am trying to implement the method described in the following paper. I am just kind of confused as to the difference between beta and beta prime. I am much more a computer programmer than ...
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Relations between Laplacian and Fourier transform

I have been thinking this question for a couple of days. To be more specific, I want to derive an explicit expression of Fourier transform under some coordinate chart of a manifold. What I do have ...
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17 views

How to find the Fourier series of the following function?

I know how to calculate a Fourier series for odd and even functions but I wish to evaluate the $2\pi$ periodic function given by $$f(x)=\begin{Bmatrix} 0~~~-\pi \leq x \leq 0 \\ 1~~~0 < x \leq ...
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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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22 views

$f(t)=\sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}$ for $f\in L^{2}(\mathbb T)$?

Let $f\in L^{2}(\mathbb T).$ Define $g(t):= \sum_{n\in \mathbb Z} \hat{f}(n) e^{2\pi i n t}, (t\in \mathbb T).$ Since $\hat{f} \in \ell^{2}(\mathbb Z),$ we note that $g\in L^{2}(\mathbb T).$ My ...
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1answer
41 views

Fourier Analysis and its applications [duplicate]

My question has two parts: $1)$ Could anyone explain in simple terms what a Fourier Transform is? $2)$ What are some of the applications of Fourier Analysis in the field of high school mathematics? ...
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1answer
32 views

When is the Fourier Transform of a function periodic?

Using the duality property, I guess it happens whenever the original signal is composed of a sum of dirac delta functions spaced at equal intervals of time. I conclude this as the Fourier transform ...
2
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1answer
124 views

Tempered distributions and convolution

I remember that if $f,g \in \mathcal{S}(\mathbb{R}^n)$ , then it is well-defined \begin{align*} \displaystyle (f \ast g)(x)= \int_{\mathbb{R}^n} g(x-y)f(y)dy=\int_{\mathbb{R}^n} (\tau_x ...