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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Fourier Transform of Heaviside Function

I'm trying to find the Fourier transform of $H(k - |x|)$, where $H$ is the Heaviside step function. I've solved a few Fourier transforms recently, but this one is giving me a bit of trouble. I'd ...
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33 views

exponential term evaluation doesn't make sense in this example

I am studying for my final and doing some practice questions, but I am confused by something: Here the solution says k at 0 we get N/2, but there is no way that answer is correct. If k is at 0 the ...
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27 views

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why?

Is it Possible to represent $f(x) =\arctan(x)$ as a fourier series ? Why ?
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37 views

Stuck trying to solve wave equation in $n$-dimensions.

Solving the wave equation $u_{tt} = c^{2} \Delta{u}$ subject to $u(0,x) = f(x)$ and $u_{t}(0,x) = g(x)$ gives us d'Alembert's formula. I'm looking to solve the wave equation, subject to these same ...
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1answer
29 views

Expression for characteristic function of a truncated RV

Let $\langle\Omega,\mathscr{F},\mathbb{P}\rangle$ be a probability space, let $X$ be a random variable defined thereon with density $f$ and $\phi$ be its characteristic function. Then if $A \in ...
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41 views

Using Fourier Analysis to determine Green's Function of Laplace's equation

I have previously seen the Green's function for Laplace's equation in two spatial dimensions determined using the method of images. Since then, I have learned some more Fourier analysis and have ...
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25 views

Fourier Transform of a function with sinusoidal sampling

What is the relation between the Fourier Transform (FT) of $f(x)$ with regular sampling and the FT of $f(x)$ with sinusoidal sampling? In other words, it's a FT of a function composition $f\circ ...
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24 views

Support of polynomial distributions

Assume $u\in\mathcal{S}'(\mathbb{R}^n)$ is a tempered distribution such that $\widehat{u}$ is compactly supported and $u^k$ defines a distribution for each $k=1,\cdots,m$. Let $p_1,\cdots,p_m$ be ...
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1answer
22 views

Fourier transformation of h(-t)

Ask a simple question: we know $F[h(t)] = H(f)$, where $h(t)$ is the impulse response. How to show $F[h(t)] = H^*(f)$? My answer is just $H(-f)$.
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1answer
39 views

Fourier Transform of $1/(\pi\cdot t)$ by Duality

I'm asked to prove using "duality property" the Fourier transform of $$\frac{1}{\pi t} = -j sgn(f)$$ I have the proof steps but I'm quit not understanding it: multiply by $j = \frac{j}{j(\pi t)}$ ...
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1answer
27 views

Fourier transform Excercise

I am stuck on an excerise which says that prove the fourier transform $f(k)$ of a real function satisfied the condition $f(-k)=f*(-k)$. Where the astericks denotes the complex congugate. I am ...
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24 views

Find the complex Fourier series

Find the complex Fourier series representation of the function $$ f(t) = \begin{cases} 1,\quad\text{if}\quad 0 < t < 2 \\ 0,\quad\text{if}\quad 2 < t < 4 \end{cases} $$ with the period ...
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1answer
36 views

Rewriting $e^{-a|t|}$

here I have to prove the fourier transform of $e^{-a|t|}$ , the beginning of the proof is to rewrite $e^{-a|t|}$ as: $e^{-at} U(t) + e^{at} U(-t)$, I know how to continue the proof starting from this ...
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23 views

Hankel transform of shifted Gaussian function

I'm trying to find Hankel transform of the function $$e^{-(r-r_0)^2}$$ Could please anyone confirm that it doesn't exist ?
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31 views

A highly oscillatory integral

I am considering the following integral $$ \int_{-\infty}^{\infty} \text{d} z' e^{-i\alpha(z-z')}e^{iV(z')(z-z')}\text{sign}(z-z'), $$ where $\alpha\in\mathbb{R}$ is a (large constant) and ...
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23 views

how to show that Kirchhoff's formular solves wave equation.

There is one exercise in text book, Fourier Analysis: An Introduction, Stein p.211 Ex# 11. I have no idea how to handle that formula. please help me!! this is Theorem 3.6 The definition of ...
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1answer
46 views

Proving the Riemann-Lebesgue Lemma in $L^1(\mathbb{R}^n)$

$\mathbf{Riemann-Lebesgue \ Lemma \ in \ L^1(\mathbb{R}^n)}$. Suppose that $f \in L^1(\mathbb{R}^n)$. Then $\hat{f}(k) \rightarrow 0$ as $|k| \rightarrow \infty$. I cannot understand any of the ...
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1answer
25 views

Showing that $||\hat{f}||_{\infty} \leq ||f||_1$ in $L^1$

Let $f \in L^1(\mathbb{R}^n)$ then $\hat{f} \in L^{\infty}(\mathbb{R}^n)$ and $||\hat{f}||_{\infty} \leq ||f||_1$ How do you prove this or where can I find a proof of this fact?
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40 views

Should I start with complex variables before fourier series?

My math professor has said that it would be useful to start with complex analysis before learning fourier series in the signals and systems course(I'm an undergrad EE). Do you agree with that and why ...
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46 views

Does $|f\sin (x)|$ integrable on $\mathbb{R}$ imply that $|f|$ integrable on $\mathbb{R}$?

I guess not. Because we usually require $|f|$ to be integrable on ℝ so that it has the fourier transform. Can anyone give me an counterexample for the statement in the title? I have searched for ...
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1answer
20 views

What happens to fourier transform of the sampled output of pure sinusoidal input of 26kHz if sampled with 44.1kHz sample frequency?

Because pure sinusoidal signal only contains impulses, I was wondering what happens to the fourier transform of the sample output from the sinusoidal input of $26$kHz if the sampling is done with ...
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23 views

bessel function with Fourier transform

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
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1answer
32 views

Odd or Even for Fourier Series?

I have the function $f(x) = -x^2 + x\pi$ and $0\le x\le \pi$ and without seeing the graph I want to show if it is odd or even, but of course $f(x) = f(-x)$ doesn't show that it is even because I can't ...
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39 views

$f\in M(\mathbb{R})$ but $\hat{f}$ is not

I am studying Fourier analysis. I noted some problems state $f,\hat{f}\in M(\mathbb{R})$ as assumption, where $M(\mathbb{R})$ denote the collection of all continuous and of moderate decrease functions ...
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38 views

Riemann-Lebesgue Lemma for Spherical Harmonics expansion

Here is my question: A basic result of classical Fourier analysis is that the fourier coefficients of an $L^1$ function must tend to zero (Riemann-Lebesgue Lemma). Is there analogous result to the ...
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1answer
43 views

proving Riemann-Lebesgue lemma

I have looked at proofs of the Riemann-Lebesgue lemma on the internet; all of these proofs use the technique of Riemann integration and making step functions. ...
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1answer
34 views

Sharpening a curve

I have a frequency domain graph as shown. I need to "sharpen" the curve to get a better response, and computing large butterworth orders is not possible on my machine. Hence, I would like to know if ...
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2answers
39 views

Fourier Transform of Sine

I'm having trouble calculating the Fourier Transform of the sin function. Specifically, the function $ G(\omega)=\int _{-\infty}^{\infty} g(t)\ e^{-i \omega t} dt $ For the fourier transform of $ ...
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12 views

estimate on a convolution

Let $\psi$ be a non-negative Schwartz function on $\mathbb{R}$ such that supp$\hat{\psi}$ is contained in $[-0.1, 0.1]$ and $\hat{\psi}(0)=1$. Define $\psi_k(x)=2^k\psi(2^kx)$ for any integer $k$. Let ...
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1answer
7 views

Support of Auto-correlation

Suppose $f\in C_0^{\infty}(\mathbb{R}^n ),$ then clearly we have supp$(f\ast f)\subseteq$ supp$(f)+$ supp$(f)$. The question is whether supp$(f\ast f)\subseteq 2$ supp$(f)$ holds? Any counterexample?
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40 views

$L^2$ and uniform norm of $\text{sinc}\, x$ and its derivatives

Looking at the graphs of the derivatives of $\mathrm{sinc}\,x$, it appears that they all are bounded by $1/x$, with $[\mathrm{sinc}\,(x)]'$ the sole exception: A few questions: 1) With the ...
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1answer
16 views

Solving convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside step function

How does one solve convolution $f(t)*g(t)$ where $f(t) = u(t) - u(t-2)$ and $g(t) = e^{-2t}u(t)$ where $u(t)$ is heaviside (unit) step function? I tried using Fourier transform of both functions to ...
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1answer
42 views

convolution with $C^{\infty}$ produces $C^{\infty}$

Problem: So I have the following function in ...
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61 views

Basic Fourier analysis explanation needed wrt a function $f$ and a finite Borel measure $\mu$

An extract from Chapter 12 of Matilla's Geometry of Sets and Measure on Euclidean Spaces I do not believe that formulas (12.1-12.3) are easily seen to be valid. I do not understand what ...
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34 views

Integration by parts with Bessel function $j_0$

I need to prove this: $$ \mathcal F{\frac{1}{r^2}}\frac{d}{dr}r^2 \frac{dC}{dr}$$ $$= (\frac{2}{\pi})^{1/2} \int_0^\infty\frac{1}{r^2}\frac{d}{dr}r^2\frac{dC}{dr}j_0(kr)r^2dr$$ $$ =-k^2 ...
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6 views

Power law in power spectrum and memory.

If we generate white noise and do the FFT of it, we get the same amplitude for each of the frequencies. Therefore, the output of the FFT of the noise follows approximately the power law ...
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1answer
42 views

Find the Fourier transform of $\frac1{1+t^2}$

Find the Fourier transform of $$f(t)=\frac1{1+t^2}$$ using contour integration that $$F\{f(t)\}=\int^\infty_{-\infty}\frac1{1+t^2}e^{2\pi ft}dt$$ How can I do this?
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Strategies for approximating fourier transform of $k$-th power of the $n$-th derivative of a function

For a function $f(x)$ with Fourier transform $\hat{F}(q)$, I'm interested in understanding the relationship of the Fourier transform of a power of a derivative of $f$ to $\hat{F}(q)$. Explicitly, I ...
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1answer
43 views

Young's inequality for convolutions

Let's assume that the convolution $f * g$ is continuous with $\lim_{|x| \to \infty}(f*g)(x) = 0$ and that $f, g \in L^2$. Then the following inequality holds $$ \| f * g \|_{\infty} \leq \| f \|_2 ...
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1answer
62 views

Fourier transform of compactly supported differentiable function

Let $K$ be the space of infinitely differentiable functions $\mathbb{R}\to\mathbb{C}$ with compact support. I read the unproved statement in Kolmogorov-Fomin's Элементы теории функций и ...
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Fast Fourier Transform and its example

I read the wikipedia and my textbook, but I can't understand the whole process of Fast Fourier Transform. Especially the book uses the Cooley-Tukey algorithm and it gives an example of 4X4 matrix like ...
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29 views

Fourier transform of distribution

Let $f\in S_{\infty}$ be a Schwartz function and let us define a linear functional,for any $\varphi\in S_{\infty}$, $S_{\infty}\to\mathbb{C}$, $\varphi\mapsto (f,\varphi)$ ...
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1answer
19 views

Support of tempered distribution under exponetiation and differentiation

Suppose $u$ is a tempered distibution in $\mathbb{R}^n$. How are supp$(\widehat{u})$ and support of $\sum_{|\alpha|\leq k}\frac{\partial^{\alpha}\widehat{u^n}}{\partial x^{\alpha}}$ compared , where ...
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1answer
51 views

The inverse Fourier transform of $\widehat{\varphi}(\xi)e^{-4\pi^2 i|\xi|^2 t}$

I need help to compute the following integral $$\int_{\mathbb{R}^n}\widehat{\varphi}(\xi)e^{-4\pi^2 i|\xi|^2 t}e^{2\pi i\xi \cdot x} \mathrm{d}\xi $$ where $\widehat{\varphi}$ is the Fourier ...
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43 views

using fourier method to compute this integral

Use the method of Fourier analysis to calculate the following integral: $$ \int_{0}^{\infty} \frac{\cos x}{1+4x^2} \operatorname{d} x .$$ Could someone help about this question? what ...
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18 views

Hoe can I find the Inverse FourierTransform for 1/(1+w^4)?

I have the expression $S(w)=\frac{1}{1+w^4}$. I am trying to find its inverse FourierTransform. I know that I have to get a sin-cos expression, but I haven´t found the way to do it. On the tables that ...
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22 views

Product of Fourier transform

I'm supposed to calculate : $ F[\delta(x-a)](\nu)F[e^{i2\pi \nu x}](\nu)$ Since $F[\delta(x-a)] = e^{i 2\pi \nu a}F[\delta(x)] = e^{i 2\pi \nu a}$ it leaves : $$e^{i 2\pi \nu a}F[e^{i 2\pi \nu x}] ...
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18 views

Solve this PDE, using Fourier transforms

PDE: $v_t(x, t) = kv_{xx}(x, t) + bv_x(x, t)$, $v(x, 0) = f(x)$, $\quad-\infty < x < \infty$ I can't apply the inverse Fourier in this case. If someone could help me, because i can't find a ...
2
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1answer
58 views

Prove a trigonometric series is positive

Let $f(x)= \sum_{n=-\infty}^\infty \frac {e^{inx}}{1+n^2}$ on $[-\pi,\pi]$. Prove $f(x)>0$ for $x\in[-\pi,\pi]$. This is an review question for my Fourier course. I am not sure how to approach ...
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23 views

An upper bounded for partial Fourier sum

Let $f$ be a Riemann integrable function on $[-\pi, \pi]$ such that $|\hat{f}(n)|\le \frac{K}{|n|}$ for some constant $K > 0$ and all $n\neq 0$. Show that $$|S_N(f)(x)|\le \sup_{y\in [-\pi, ...