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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Solve $y'' + 4y = e^{-x^2}$ using Fourier transforms

I need to solve the equation $y'' + 4y = e^{-x^2}$ using Fourier transforms. I was able to take the Fourier transform of both sides and solve for $\hat y$. I have $\hat y = ...
2
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1answer
783 views

Matlab FFT-algorithm example, one simple question

This is a question about an example for the software Matlab, but I still chose to ask it here, since I suspect that the question is more about the math involved than the software itself. I want to ...
7
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1answer
171 views

Interpretation of the Laplace transform

Here's my intuitive understanding of the Fourier transform of $f:{\mathbb R}\rightarrow{\mathbb C}$, defined by $$\mathcal{F}(f)(\omega) = \int_{-\infty}^{\infty}e^{-2 \pi i \, \omega \,x}f(x)dx$$ I ...
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1answer
31 views

sup-Inequality for function

Hoi, I want to show that for $\phi\in C_0^{\infty}(\mathbb{R}^n)$ where supp $\phi = \overline{B}(0,r):=B$ we have $$\sup_{x\in B}|\phi(x)|\leq 2R \sup|\partial_{x_1}\phi(x)| $$ I dont quite ...
2
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1answer
102 views

Norm of this linear operator

Let $X$ be the Banach space of all continuous functions $f(x)$ on $[0,2\pi]$, provided by the uniform norm $$ \|f\|=\max_{x\in [0,2\pi]}|f(x)|. $$ Let $$ ...
4
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1answer
435 views

proof of Poisson formula by T. Tao

I do not understand one thing in an article on the blog of Terence Tao: For instance, restricting a function $f: G \rightarrow \mathbb{C}$ to a subgroup $H$ causes the Fourier transform $\hat f$ ...
2
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1answer
77 views

What is the degree of the fourier expansion

Let $ f:\{-1,1\}^3 \rightarrow \{-1,1\} $ , $f(x)= \operatorname{sgn}(x_1+x_2+x_3)$; (Majority function), then Fourier expansion of $f$ is $f(x)= \frac{1}{2} ...
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0answers
27 views

$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...
2
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1answer
123 views

More on the generalized integral

Refer to my previous topic: A generalized integral need help I think we get this : $$\frac{\sin \theta}{1-2\cos \theta x+x^2}=\sum_{k=1}^{\infty}\sin (k\theta )x^{k-1}$$ Then $$\int_0^1\frac{\ln ...
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0answers
276 views

Fourier Coefficients of Complex Measure

For my homework I am trying to prove the following: Suppose $\mu$ is a complex Borel measure on $[0,2\pi)$, and define the Fourier coefficients of $\mu$ by $\hat{\mu}(n)=\displaystyle\int ...
2
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2answers
410 views

the fourier transform of a “double convolution”

Suppose I have a function $$ m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x) $$ I want to find the Fourier transform of m(x) in terms of the Fourier transforms of $f,h,g$ but for the ...
2
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0answers
126 views

Do we have closed form for these series?

Continuous on the previous question, can we get a closed form for these? $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( ...
4
votes
2answers
191 views

Fourier transform of $|x|^{-t}$

In $\mathbb{R}^d$, let $f(x)=|x|^{-t}$, its Fourier tranform $F(f)(ξ):=(2\pi)^{-\frac{d}{2}}∫_{\mathbb{R}^d} e^{ix\cdot ξ}f(x)dx$, is there any fast way to see that this integral converge at $ξ \neq0$ ...
2
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1answer
2k views

3D Fourier Transform

I'm trying to calculate the inverse of the following 3D Fourier transform. $$ \widetilde{f}= \frac{1}{(k^6-\alpha*k^2-\alpha*k_3^2)} $$ where $k = (k_1^2+k_2^2+k_3^2)^{1/2}$ the fourier transform is ...
0
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2answers
237 views

Explain 2D ellipse function in terms of Circ

Suppose we have a two dimensional continuous linear shift invariant system has impulse response: $h(x,y)=\left\{ \begin{array}{ll} \frac{1}{2\pi a^2 b^2}, & \mbox{if } ...
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0answers
281 views

Calculation of the Fourier series of $f(t)=e^{it\alpha}$ and $f(t)=|t|$

I have to compute the fourier series of these 2 functions: 1 . $$f(t)=e^{ita}\text{ for }-\pi < t < \pi ;\qquad a \in \mathbb{R}\backslash \mathbb{Z}$$ 2. $$f(t)=|t| \text{ for }-\pi < t ...
5
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2answers
283 views

Is my Fourier Series computation done correctly?

See my fourier series calculation of this function if you please! $ f(t)=\left\{\begin{array}{ll} 0, & \text{for } \ -\pi<t<0 \\ 1, & \text{for } \ 0 < t < ...
0
votes
1answer
114 views

A small question on fourier series

Why the series is divergent, but the equation holds? $$\sum\limits_{i=1}^{\infty }{\sin kx}=\frac{1}{2}\cot \left( \frac{x}{2} \right)$$
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0answers
50 views

Variation on a completeness relation

The completeness relation for the spherical harmonics is: $$\sum_{l=0}^{\infty} \sum_{m=-l}^{l} Y_{lm}^*\left(\theta_1,\phi_1\right)Y_{lm}\left(\theta_2,\phi_2\right) = ...
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0answers
72 views

Probability measures on $\mathbb{T}$ whose Fourier coefficients tend to 1

Let $\mu$ be a probability measure on the complex unit circle $\mathbb{T}$. Are the following two assertions equivalent? $\limsup_{n\to\infty}|\hat{\mu}(n)|=1$. There exists an increasing sequence ...
3
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458 views

inverse fourier transform an exponential function

I try to find double inverse Fourier transform of $\;\exp\left({A \large\frac{\varepsilon^2 \xi^2+\eta^2}{\xi^2+\eta^2}}\right)$ where A is constant, $\varepsilon$ is possitive number and $\xi$ and ...
1
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1answer
42 views

An equivalent condition for a measure on $\mathbb{T}$ to be symmetric

Is it correct that a Borel probability measure $\sigma$ on the complex unit circle $\mathbb{T}$ is symmetric (i.e. $\sigma(A)=\sigma(\overline{A})$ for every Borel set) iff ...
0
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1answer
93 views

Sum of series & sequences

I dont know how to evaluate the first one, for second one I can only show the sum is less than 2. $$\begin{align} & \prod\limits_{k=4}^{\infty }{\left( 1-{{\left( \frac{3}{k} \right)}^{3}} ...
0
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1answer
347 views

Power spectrum for discrete signals.

If $x(t)$ is a real (aperiodic) power signal, i.e. \begin{equation} 0<\lim_{T\rightarrow\infty} \frac{1}{T}\int_{-T/2}^{T/2}|x(t)|^2 dt<\infty \end{equation} $x_T (t)$ is a truncated version of ...
5
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0answers
412 views

How to find the inverse Fourier Transform of the product of two bessel functions of the first kind and a complex exponential function?

I am attempting to find a closed form or symbolic expression of the inverse Fourier transform of the product of two Bessel functions of the first kind and a complex exponential, e.g. $P(t) = IFT_w \{ ...
2
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1answer
322 views

Fourier transform - Poisson Equation - Exercise

I have some troubles concerning the following question: Let $\phi \in C_c^\infty(\mathbb{R^n})$. Prove using the Fourier transform that the Poisson equation $\Delta f=\phi$ has at most one solution ...
6
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2answers
463 views

log sin and log cos integral, maybe relate to fourier series

I try to use the method of differentiation under integral sign for the first one And integrate it back, but I failed to find the constant $c$ .... Anyone hav other method? $$\begin{align} & ...
2
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1answer
146 views

A complex square root in the Schrödinger kernel

Consider the initial-value problem for the Schrödinger equation $$\tag{IVP} \begin{cases} i\frac{\partial u}{\partial t}+\Delta u=0 & x\in \mathbb{R}^n,\ t\in \mathbb{R}\setminus \{0\} \\ &\\ ...
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1answer
189 views

PDE - (homogeneous) Heat equation - Solution?

today I have a question in PDE. It concerns the heat equation: Formulate the (homogeneous) heat equation for functions $f:(0,\infty)\times\mathbb{R^n} \longrightarrow\mathbb{C}$. Derive an equation ...
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1answer
51 views

there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
2
votes
2answers
248 views

Integral of $f(x) \exp(ikx)$ with finite bounds calculated using Fourier transform, and its derivative

I have an integral which I need to calculate numerically along the lines of $$ I(k)=\int_0^{L} \exp(i k x)f(x) dx $$ where $x$ and $L$ are real. $f(x)$ is not necessarily periodic and differentiable ...
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1answer
115 views

Approximation of the Fourier Transform of General Functions in a Box

I'm trying to get a general approach for the Fourier Transform of functions $f$, only in a restricted area $-\frac M2\le x \le \frac M2$, where ${\frak F}_{f(x)}(\omega)$ exists. My idea was the ...
2
votes
3answers
233 views

Understanding Fourier Transfomation of $\sin(ax)$

I'm currently trying to understand time continuous Fourier transformations for my Signal Analysis class (Electrical Engineering major). In my Textbook we have a set of "useful FT-pairs", one of which ...
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0answers
47 views

Short-time Fourier transform of sequence of functions

Suppose I have a sequence of functions $f_N(t)= a(\frac t N ) e^{i \omega t}$ where $\omega$ real and $a$ is a compactly supported smooth function. I would like to determine $a$ and $\omega$ by using ...
2
votes
2answers
470 views

How to realise N-point FFT?

For example, if I need to calculate 15 point fft, I can use DFT. But it is a long process. As far as I know, FFT can be used when the size is 2^n. What are the efficient ways to perform a 15 point ...
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2answers
277 views

Exercise - Fourier Transform

I have got another question concerning the Fourier transform. I hope somebody can help me. Let $f \in L^1(\mathbb{R}^n)$. Prove that $\hat{f}$ is continous. (ok, I was able to show it) If ...
5
votes
2answers
111 views

Exercise - Fourier transform in two variables?

I need your help for the following problem: Compute the fourier transform of the functions $$\chi_{[0,+\infty[}e^{-x} \quad \text{ and } \quad \frac{e^{(-\frac{x^2}{2})}}{1+iy}$$ The second function ...
5
votes
2answers
973 views

Integration (Fourier transform)

$\mathrm{Re}(i) = 0$, but the fourier transform of $f(x) = e^{-ix^2}$ is $g(\alpha) = \sqrt{\pi\over i}\times e^{i\alpha^2 \over 4}$, is it not? Is there an easy to show that it is so, knowing the ...
1
vote
1answer
373 views

Intuition behind the convolution of two functions

Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation ...
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3answers
120 views

Recover filter coefficients from filtered noise

I have a digital signal which may be represented as noise filtered with an FIR (finite impulse response) filter. Let us suppose that the noise consists of pulses (nonzero samples on a zero ...
0
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0answers
48 views

Is it even possible to “visualize” this in the time domain?

I'm trying to understand single sideband modulation. If you want to conserve bandwidth and you don't mind complicated math, then SSB is for you. So far the only thing I could find online to help me ...
2
votes
1answer
97 views

Fourier transform of product

I would like to know the fourier transform of the product of the Cauchy probability distribution $f(x)=\frac{1}{\pi (1+x^2)}, -\infty<x<\infty$ with itself. I know that the fourier transform of ...
0
votes
1answer
100 views

Filter signal through convolution

I am a little bit unsure if I've set up the following problem correctly: Consider the signal $$f(t) = e^{-t}(\sin(5t) + \sin(3t) + \sin(t) + \sin(40t)) \quad 0 \leq t \leq \pi$$ Filter this signal ...
2
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1answer
211 views

Convergence of the Fourier Transform of the Prime $\zeta$ Functions

I think I found a way to write the truncated Prime $\zeta$ function like this: $$ P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n} \sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}} ...
-4
votes
1answer
150 views

Laplace transform of $y'' + 3y' + 2y = f(t), \; y(0) = y'(0) = 0$ [closed]

Can you do this? This is part of my final year EE work. I need to solve this in order to figure out how my sensor is behaving. $$y'' + 3y' + 2y = f(t), \; y(0) = y'(0) = 0$$ where $f(t)$ is a ...
5
votes
1answer
1k views

Fourier transform of Cauchy principal value

I try to understand the direct computation of the Fourier transform of the distribution `Cauchy principal value' $v.p \frac{1}{x}$. I don't understand the following change of order of integration: $$ ...
2
votes
1answer
303 views

The Sobolev norm for vector-valued functions

For a compactly supported function $f: \mathbb{R}^n \to \mathbb{C}$, the Sobolev norm is defined by $$\|f\|_s^2 = \int |\hat{f}(y)|^2(1+|y|^2)^sdy.$$ Here $\hat{f}$ is the Fourier transform of $f$, ...
0
votes
1answer
115 views

Laplace Transform of this function

Find $L\{F(t)\}$ if $$F(t) = \begin{cases} \sin t & \text{between }0 < t < \pi \\ 0 & \text{between } \pi < t < 2\pi \end{cases}$$ Really stumped by this one. Please can you ...
1
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3answers
453 views

Is this function - characteristic function of a random variable?

$\phi(t)= \begin{cases} 1,&\text{if $|x|< 1$;}\\ e^{-(|x|-1)^2} ,&\text{if $|x|\geq1$.} \end{cases} $ Can anyone help?
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3answers
2k views

Fourier transform of $\frac{\sin{x}}{x}$

can you help me with this question find $\sin$ Fourier transform of $\frac{\sin{x}}{x}$