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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120 views

Show $\lim_{N\to\infty}\int_0^\pi\left(\frac1{\sin\frac{x}2}-\frac2x\right)\sin\left((N+\frac12)x\right)dx=0$

Prove that the function $\csc(x/2)-2/x$ is integrable on $(0,\pi)$. In fact, prove that it is bounded. In fact, prove that it tends to zero as $x\to0$. Use this to show that ...
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2answers
124 views

Laplace's equation in polar coords

Question: Suppose that the function u(r, $\phi$) satisfies Laplace’s equation for plane polar co-ordinates (r, $\phi$) i.e. $$ ∇^2u = \frac{1}{r} \frac{∂}{∂r}(\frac{r∂u}{∂r}) + ...
3
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1answer
110 views

Does anyone know a solution to this PDE?

I ran into this PDE: $$\frac{\partial y(x,t)}{\partial t} = A\,x^{\gamma-1} \left(\frac{\partial y(x,t)}{\partial x} + x \, \frac{\partial^2 y(x,t)}{\partial x^2}\right)$$ If it helps in any way, ...
0
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1answer
32 views

von Neumann stability analysis for irregular meshes

All the litterature I have come across about the von Neumann stability analysis is performned on regular grids. Can the analysis be performed analytically on irregular grids, or does it have to be ...
7
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207 views

Function with product of sine kernel

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Also, $f$ is continuous and goes to $0$ at $\pm \infty$. Let ...
3
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1answer
103 views

Fourier Integral evaluation

We're doing fourier integrals in class, but unfortunately I have no idea how to even begin to tackle this one. The examples we have done in class were way simpler than this one: $$ \int_0^\infty ...
0
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1answer
41 views

Find the Fourier Transform of $f(x)$, given that $f(x) = |x|$ if $-1<x<1$ and $f(x)=0$ otherwise

I'm doing some general practice questions on Fourier Transform from my book. Came across this one and don't know from where to begin. How do I put the absolute value of $x$ in the integral for the ...
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1answer
62 views

What is the Fourier transform of an M like function

Given the function $$ f(x)= \begin{cases} \vert x \vert& \text{, for }\;\vert x\vert\le M \\ 0 & \text{, otherwise} \end{cases} $$ for some constant $M$. What would be the form for the ...
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0answers
62 views

Monotonically decreasing Fourier transform

What would be the conditions on $f(x)$ such that it's Fourier transform $F(k)$ would be monotonically decreasing from $k=0$ to half range ($F(0)$ would be the maximum, and it would "fall" on both ...
2
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2answers
294 views

Integral through Fourier Transform and Parseval's Identity

$$ \int_{-\infty}^{\infty}{\rm sinc}^{4}\left(\pi t\right)\,{\rm d}t\,. $$ Can you help me evaluate this integral with the help of Fourier Transform and Parseval Identity. I could not see how it is ...
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1answer
62 views

Understanding a step in the proof of the Inverse Fourier transform theorem

I'm trying to understand the proof of the Inverse Fourier Transform theorem in Stéphane Mallat's "A wavelet tour of signal processing". Near the end of the proof, we have: $ \lim_{\epsilon ...
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2answers
189 views

Writing function as infinite Fourier sum with sine kernel

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Also, $f$ is continuous and goes to $0$ at $\pm \infty$. Let $K(y)=\dfrac{1}{\pi y}\sin(\pi y)$. Show that ...
0
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1answer
207 views

Inner product of function of period $2\pi$ with exponential

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous with period $2\pi$. Prove that $$\lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{j=1}^Nf\left(\dfrac{2\pi j}{N}\right)e^{-2\pi ...
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136 views

Is the range of the Fourier transform $L^1(\mathbb{R}) \to C_0(\mathbb{R})$ closed under “quasi-inversion”?

Definitions: A function $f \in C_0(\mathbb{R})$ is quasi-invertible if $1 \notin \operatorname{ran}(f)$. The quasi-inverse of such an $f$ is $\frac{f}{f-1}$. Some discussion: Let $C_1(\mathbb{R})$ ...
2
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1answer
143 views

Fourier transform supported on compact set

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Show that $$\hat{f}(y)=1_{[-\pi,\pi]}(y)\sum_{n=-\infty}^\infty f(n)e^{-iny}$$ in the sense of $L^2(\mathbb{R})$-norm ...
2
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1answer
83 views

Fourier series formula with finite sums

Let $f\in C(\mathbb{R}/2\pi\mathbb{Z})$, meaning that $f$ is continuous with period $2\pi$. Let $x_N(j)=2\pi j/N$. Define $$c_N(n)=\dfrac1N\sum_{j=1}^Nf(x_N(j))e^{-ix_N(j)n}.$$ Show that for any ...
4
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1answer
108 views

Integral equals sum for $L^2$ function with compact-support transform

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Show that $$\int_\mathbb{R}|f(x)|^2dx=\sum_{-\infty}^\infty|f(n)|^2$$ I know that $f$ must be continuous and going ...
4
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1answer
227 views

Weak convergence and Fourier transform

For positive integer $k$, let $\mu_k=\dfrac{1}{2}\left(\delta(x)+\delta\left(x-\dfrac{2}{3^k}\right)\right)$. Let $dC_k=\mu_1\ast\cdots\ast\mu_k$. We have that $dC_k$ converges weakly to $\mu_C$, ...
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1answer
89 views

Derivative of the Fourier integrals calculated in a point

I have considered the following definitions for the Fourier integral pairs: $\Phi(\omega) = \frac{1}{(2\pi)^3} \int_{\mathbb{R}^3} e^{ix\omega \cos\theta}R(x)x^2 dx \sin\theta d\theta d\phi$ $R(x) = ...
4
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2answers
75 views

Fourier tranform supported on interval yields continuity?

Let $f\in L^2(\mathbb{R})$ be a function such that the Fourier transform $\hat{f}$ is supported on $[-\pi,\pi]$. That is, $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ is supported on ...
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1answer
54 views

Why is $\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}\cos\left[ a\xi \right]\hat{f}(\xi)d \xi = f(a)$?

Background: We are looking at the wave equation on $\mathbb{R}^n$ via the Fourier transform. If $u(x,t)$ solves $\Delta u = u_{tt}$ in $\mathbb{R}^n$, with $u(x,t) = f(x)$ at $t=0$ and $u_t(x,t) = ...
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1answer
4k views

Fourier series coefficients proof

Can somebody help me understanding the fouries series coefficients? I know that if we have: $$f(n) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$ And because of the ...
4
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1answer
102 views

Easy question about fourier transform

I think this is an easy one ... but I just can't find an answer. Assume $f : \mathbb{R} \to \mathbb{R}$ is a $n+2$ times differentiable function and and all drivatives up to the order $n+2$ are in ...
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1answer
291 views

Fourier transformation: Determining the axis

I need some help with the Fourier transformation of my data. My original data is a Distance VS Time: upon doing a Fourier Transform, I get the following: I understand that normally after a ...
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1answer
217 views

Evaluating $\int\,\cos(x)\cos(\omega x)\,dx$ using trigonometric addition formulas

I'm looking to solve the integral $$\frac{1}{\pi}\int_{-\frac{\pi}{2}}^\frac{\pi}{2}\, \cos(x)\cos(\omega x)\,dx$$ by rewriting the terms using the trigonometric addition formulaes. It should end ...
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1answer
84 views

convolution of two functions and relations with their p-norms

Let $f\in L^p(\mathbb{R})$, $g\in L^1(\mathbb{R})$, $1\leq p< \infty$. Then I have proved the convolution $f\ast g\in L^p(\mathbb{R})$ and $||f\ast g||_p\leq ||f||_p||g||_1$. Does $f\ast g$ ...
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1answer
114 views

Should I learn Fourier Analysis or Complex Analysis first?

Are the two subjects highly interrelated? Which draws more heavily from the other? Which do you recommend I learn first? Thank you.
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245 views

Closed form for integral $ \int_0^{\pi} \frac{\sin (m \phi)}{(1 + r \cos \phi)^n} d\phi$

Is there a closed form for $n>0$ integer, $m\neq 0$ integer, and $|r|<1$ real?
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1answer
103 views

When is it ok to Fourier Transform functions that are not in L^2?

I was recently reading a paper that used a Fourier decomposition to solve a differential equation. The equation looked like this: $$ \frac{V(t)'}{V(t)} = \Phi_0\,x_0\,(P(t)-Y)\,\exp(-k_0t)+\text{some ...
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3answers
627 views

Morlet's wavelet reconstruction formula

The CWT (continuous wavelet transform) of a signal $x(t)$ is $$X_w(a,b)=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt$$ In order to reconstruct the ...
2
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1answer
139 views

Calculate the Fourier transform on $\mathbb{R}$ of $\frac1{(1+x^2)^2}$ and $\frac{x}{(1+x^2)^2}$.

Calculate the Fourier transform on $\mathbb{R}$ of $\frac1{(1+x^2)^2}$ and $\frac{x}{(1+x^2)^2}$. Also calculate the Fourier transforms of $\frac1{(1+ix)^2}$ and $\frac{\cos(\pi x/2)}{1-x^2}$. ...
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52 views

Seeking better understanding of Fourier transform?

I'm quite confused on the one part of the Fourier transform. I don't understand what is the term $\left(u*x + v*y \right)$ mean. I mean $u$ and $v$ are the axis for frequency domain and $x$, $y$ are ...
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1answer
11 views

Find occurrences of sequences in sound wave

I have quite basic knowledge of mathematics. Consider two axis, time and dB (sound levels). My question is if it's possible, using fourier or any other method, to find how many occurrences of ...
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44 views

Estimative in Hs spaces

Consider $W(t)$ defined by $\widehat{W(t)} = (2 \pi)^{\frac{-n}{2}} \cos (t|\xi|)$. Consider $g \in H^{s-1} (R^{n-1})$. Show that: $$ \| W(t ) *g\|_{H^s} \leq c \sqrt2 (1 + |t|)\| g\|_{H^{s-1}}$$ ...
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133 views

Convolution theorem for other transforms

The Fourier transform is an integral transform with turns any function into a superposition of sinusoidal waves. The convolution theorem states the astonishing property that if you convolve two ...
5
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1answer
171 views

Fourier Transform of $\frac{1}{(1+x^2)^2}$

I need to find the Fourier Transform of $f(x) = \frac{1}{(1+x^2)^2}$ Where the Fourier Tranform is of $f$ is denoted as $\hat{f}$, where $\hat{f}$ is defined as ...
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0answers
128 views

Discrete Fourier Transform on a shifted frequency grid

I use the discrete Fourier transform in 3D to solve my model partially in real space and partially in Fourier space. The DFT pair is defined as \begin{equation} ...
0
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1answer
54 views

Fourier transform of powers of a function

Assume one has real valued functions $f(x)$ and $g(x)$ that belong to the Schwartz space. I know that the Fourier transforms of $f^3(x)$ and $f^2(x)g(x)$ can be expressed straightforward in terms of ...
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2answers
553 views

Fourier transform convention: $\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{\pm ikx}dx $?

I've come across the Fourier transform being defined as: $$\tilde{f}(k)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} f(x)e^{ikx}dx$$ But this convention is not present in the Wikipedia article. The ...
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1answer
65 views

Fourier Series and Summation

$\sum_{n=1}^\infty \frac{1}{n^2}$ can be computed in straight-forward way by computing the Fourier co-efficients of $f(x)=x$ and applying Parseval's identity. Likewise, $\sum_{n=1}^\infty ...
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2answers
85 views

Relationship between Fourier coefficients of $f\left(x\right)$ and $f^{-1}\left(x\right)$

Say I have a function $f\left(x\right)$, which can be expressed as a Fourier Series: $$f\left(x\right)=\sum_{k=-\infty}^{\infty} c_k e^{ikx}$$ Define the inverse of $f\left(x\right)$ as, ...
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0answers
247 views

Parseval's theorem, DFT to FT

So I am working with DFTs for the first time and I have some problems. I have a discrete signal to this signal i want to apply a DFT, then I want to use the output in an integral. So $S(t)$ is my ...
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1answer
89 views

Does $f(0) = +\infty$ when $\hat f \geq 0$ and $\int \hat f (s) \ ds = +\infty$?

Throughout, $f \in L^1(\mathbb{R})$ and $\hat f \in C_0(\mathbb{R})$ is its Fourier transform $s \mapsto \int e^{its} f(t) \ dt$. Motivation: If $\hat f \in L^1(\mathbb{R})$ too, then, by Fourier ...
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1answer
100 views

Convergence of convolution

I have a vector $x = (..., x_{-1}, x_0, x_1, ...)$ and a vector $w = (..., 0, 0, 1, 1, .. , 1, 1, 0, 0, ..)$ (with $2M + 1$ components equals to 1) such that $y = x \cdot w = (0, 0, .., x_{-M},x_{-M ...
2
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1answer
93 views

Fourier transform of $f[n]$ and $f[-n]$

Hi I am just wondering, If I have a signal $f[n]\in \mathbb{C}^L$, i.e. $f$ is $L$-periodic, i can also define $h[n]=f[-n]$. Is it true that the Fourier transform of $f$, say $\hat{F}$, and the ...
2
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1answer
33 views

Fourier and differentiation operators

For a function $f:\mathbb{R}\rightarrow\mathbb{R}$ in the Schwartz class, define $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$ We can show that $T^2f(y)=f(-y)$, and $T^4f(y)=f(y)$. ...
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1answer
34 views

How do you compute the Fourier Transform of this Unit-Impulse Function?

I have been given this problem from a textbook (not homework, trying to study for an exam. The goal is to find the Fourier transform of this function. $\sum_{k=0}^\infty a^k*\delta(t-kT), |a|<1$ ...
2
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0answers
67 views

Inverse Fourier transform to get convolution

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $u(x,0)=f(x).$ Let ...
0
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2answers
54 views

Finding Fourier transform of initial condition

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
5
votes
2answers
149 views

If $f \in L^1(\mathbb{R})$ and $\hat f \geq 0$, is $f$ continuous?

Suppose $f \in L^1(\mathbb{R})$. I am wondering what conditions on $\hat f = \left[ s \mapsto \int e^{its} f(t) \ dt \right] \in C_0(\mathbb{R})$ suffice to make $f$ continuous (or, more accurately, ...