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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79 views

Is Plancherel's theorem true for tempered distribution?

Let $f, g\in L^{2},$ by Plancherel's theorem, we have $$\langle f, g \rangle= \langle \hat{f}, \hat{g} \rangle.$$ My Question is: Is it true that: $$\langle f, g \rangle= \langle \hat{f}, ...
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1answer
55 views

Poisson summation in analytic number theory, an example

I've read a theorem of a course of analytical methods in theory of numbers, and I want to ask to clarify it, since I know this theorem from other context. The theorem is Poisson summation formula (I ...
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1answer
62 views

Prove that the DFT Matrix is Unitary

We have that the DFT Matrix is: $$ W = \frac{1}{\sqrt{N}} \begin{bmatrix} 1&1&1&1&\cdots &1 \\ 1&\omega&\omega^2&\omega^3&\cdots&\omega^{N-1} \\ ...
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0answers
54 views

A very interesting Fouriertransform

$$ \int_{-\infty}^{\infty}d\omega\frac{e^{i \omega t}e^{i \omega R}}{{((\omega-\Omega-iA^2(1+e^{i \omega R}))((\omega-\Omega-iA^2(1-e^{i \omega R}))}} $$ All constants are positve real numbers. ...
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1answer
62 views

Soft question: Can one learn Fourier Analysis without a working knowledge of Integration Theory

As the title indicated, I am wondering if one (probably as an undergraduate math major) can learn much of Fourier Analysis, without taking a course in integration theory. I am taking a very light ...
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39 views

Prove that taking the inverse Fourier transform of frequency returns time.

If we evaluate the inverse Fourier transform of X(w) how do we know we get x(t) back? Link to X(w) and x(t) equations I know that integrating in the frequency domain results in getting information ...
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1answer
9 views

FT of a tempered distribution is a function

I have a question. When is the Fourier transform of a tempered distribution a function? I guess if the FT is a function, itself must also be a function. But I don't know how to go further. Thanks for ...
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0answers
28 views

Spectral Characterization of Reed-Solomon Codes

I am having trouble understanding the spectral characterization of Reed-Solomon codes. My script states the following: An evaluation codes is defined as: $$C = \{(c_0, \ldots, c_n) : c_l = ...
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0answers
14 views

Phase of sinc function

After calculating the Fourier coefficients of the signal $x(t)=a*rect(\frac{t-nT_0}{T})$ we get the Fourier coefficient to be $X_k=a*\delta *sinc(k*\delta)$ $T_0$ is the period $f_0$ is the base ...
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0answers
43 views

Verifying work on Fourier series

I'm learning about Fourier series and need some help with this following problem: Consider the function $f(x) = \frac{\pi - x}{2}, \ x \in [0, 2\pi)$ extended periodically with period $2\pi$. Find ...
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1answer
30 views

is there a sequence of compact support functions(who are $ C^\infty ) $ limiting to $ f(x)$=1

Is there a sequence of compact support functions (who are $ C^\infty\ ) $ limiting to $ f(x)=1$ in condition that there is no compact $K \in R$ such that cond{ $m: supp(g_m)$ ⋂ $K \ne ...
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1answer
45 views

Statistics of the product of two white noise Fourier amplitudes

Consider two sequences of random numbers \begin{align} A &= \{a_0, a_1, \ldots a_N\} \\ B &= \{b_0, b_1, \ldots b_N\} \, . \end{align} where each $a$ and $b$ value is independently drawn ...
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2answers
33 views

Proving Fourier transform of $\int_0^\infty e^{-x}x^{a-1}dx = \Gamma(a)(1+i\omega)^{-a}$

Given $a > 0$, $$f(x) = \begin{cases} e^{-x}x^{a-1}, &\mbox{if } x > 0 \\ 0, & \mbox{if } x \leq 0 \end{cases}$$ Prove the the Fourier transform of $f$, $\hat{f}(\omega)$ ...
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0answers
27 views

eigenvalues and eigenfunctions of the laplacian

I have a question. If we want to solve heat equation on torus or any bounded domain in $\mathbb{R}^n$, we can use the method of separation of variables and the question of getting the eigenvalue and ...
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3answers
67 views

Fourier analysis — Proving an equality given $f, g \in L^1[0, 2\pi]$ and $g$ bounded

We were given a challenge by our Real Analysis professor and I've been stuck on it for a while now. Here's the problem: Consider the $2\pi$-periodic functions $f, g \in L^1[0, 2\pi]$. If $g$ is ...
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588 views

What is a spherical Gaussian kernel?

In this paper (page 8, Example 3), a spherical Gaussian kernel is defined by the formula $$K(\mathrm x,\mathrm y)=e^{-2\epsilon(1- \mathrm x\cdot\mathrm y)}$$ where $\mathrm{x,y}\in ...
2
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1answer
66 views

Find the Fourier series of the trigonometric polynomial $f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx})$

I'm learning about Fourier series and need help with this problem: Given the trigonometric polynomial $$ f(x) = \frac{a_0}{2} + \sum_{k = 1}^{n}(a_k\cos{kx} + b_k\sin{kx}) $$ find the Fourier ...
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0answers
31 views

Making a density argument work

A homework problem I have is to show that $1/N \sum_{n=1}^N f(n\alpha) \to \int_\mathbb{T} f dx$ for any $\alpha$ irrational, and $f$ lebesgue integrable. I understand why this is true for $f$ ...
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0answers
56 views

The Fourier-Stieltjes transform is uniformly continuous

Let $G$ be a locally compact Abelian group and $\hat{G}$ be its dual group, that is the group of all complex functions $\gamma:G\to\mathbb C$ such that ...
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29 views

Power spectral density of a Poisson process

Poisson processes can be used to model, for instance, shot noise, and are ubiquitous in many engineering, physical or biological problems. What can be said about it's power spectral density? I ...
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0answers
21 views

A formula for Fourier transform

While doing some computations I arrive at the expression $$ P\left(x\right) = \int_{-\infty}^{\infty}\frac{d\phi}{2\,\pi}e^{i\,\phi\,x}\,\frac{f\left(\phi\right)}{i\,\phi},\quad(1) $$ where I know ...
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1answer
21 views

Conservation law for Benjamin Ono equation

Consider the Bejamin Ono equation \begin{equation} \partial_t u + H\partial_{xx}u = u\,\partial_x u, \end{equation} where $u=u(x,t): \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is a real scalar field, ...
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1answer
42 views

How to prove this function is integrable??

Let $f(x)=0$ when $x<0$, and $f(x)=1$ if $x\geq 0$. Choose a countable dense sequence $\{r_n\}$ in [0,1]. Then, show that the function $F(x)=\sum_{n=1}^\infty 1/n^2 f(x-r_n)$ is integrable and has ...
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0answers
18 views

How to understand the weak - star convergence of $ \phi_N \ast \mu$

I am currently reading a book called classical and multilinear harmonic analysis. In section 1.2.2, I found that I couldn't understand the third statement of proposition 1.5 says that for any $\mu \in ...
0
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1answer
48 views

Is Riemann–Lebesgue lemma valuble in $L2(\mathbb{R})$

If $f\in L_1$ on $\mathbb{R}$, that is to say, if the Lebesgue integral of $|f|$ is finite, then the Fourier transform of $f$ satisfies $$\hat{f}(z):= \int_{\mathbb{R}} f(x)e^{-izx} dx \rightarrow 0, ...
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22 views

Why is the Hadamard transformation considered the Fourier transform on $\mathbb Z_2$?

Wikipedia says about the two-element Hadamard transformation $H_1$: This $H_1$ is precisely the size-2 DFT. It can also be regarded as the Fourier transform on the two-element additive group of ...
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1answer
28 views

Fourier transfrom of $\cos(\frac{x}{2})$ truncated to $[-\pi,\pi]$

I cant seem to get this right; I end up with $\dfrac{\cos(2 \pi^2 \xi )}{\frac{1}{4}-4\pi^2 \xi^2}$ after ; $$\frac{1}{2} ...
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0answers
21 views

Fourier analysis and Set theory problem

Let A be a finite, nonempty subset of the integers, and $a(x)$ defined on $[0,1]$: $a(x) = \displaystyle\sum_{a \in A} {e^{2\pi i ax}}$ Show that $\int_{0}^{1} |a(x)|^4 dx$ is the number of ...
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0answers
31 views

Phase Noise & Jitter: Understanding Cyclostationary Processes

My question relates to trying to understand the ways to characterize cyclostationary processes. Reference to the literature would be helpful! My question has the following parts: Part I: Phase ...
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1answer
25 views

Prove that if $f\in L^p(\mathbb{R_d})$ and $\phi\in\mathbb{S^d}$ then $f*\phi\in\mathbb{C^\infty}$

Show that if $f\in L^p(\mathbb{R^d})$ and $\phi\in\ S(\mathbb{R^d})$ then $f*\phi\in\mathbb{C^\infty}$, where $S(\mathbb{R^d})$ is the Schwartz class. How does one prove this rigorously? I have ...
3
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1answer
105 views

Negacyclic FFT multiplication

I am using an FFT to multiply polynomials. Because I want the program to be as fast as possible I am leaving away the zero padding. For example, if we want to calculate: $(58 + 37x + 238x^2 + ...
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0answers
7 views

Same sample points but get the different result of Fourier transform

Fourier Transform formula:$G(f) = \int_{-\infty}^\infty g(t)e^{(-i2 \pi ft)}dt$ I want to transform the following two equations: $ cos(2\pi ...
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0answers
42 views

Find a function that's convolution with itself is a given function

I would like to solve this equation for $f(x)$: $$ \int_{-\infty}^{\infty} f(z)f(t-z) dz = g(t). $$ Are there any standard ways to solve such problems? $g(t)$ can be assumed to be continuous, but may ...
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0answers
19 views

Solve Fourier transform to sovle integral equation

I want to use Fourier transform to solve $$\int_{-\infty}^\infty e^{-|x-y|}u(y) dy = f(x).$$ I notice that $$e^{-|x|}* u(x) = f(x)$$ so $$\mathscr{F}(e^{-|x|})(t) \mathscr{F}(u)(t) = ...
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1answer
36 views

$\partial^2_t u(x,t)=\partial^2_x u(x,t)$ - periodic BC

Hi I am looking for a complete solution to the pde given below, it is a hyperbolic pde and I specify the initial conditions and boundary conditions (periodic). Thanks for your help. I show what I do ...
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0answers
32 views

PDE Separation of Variables

I'm trying to find a solution to: $v_t = kv_{x x} , 0 < x < l, 0 < t < ∞$ $v(0, t) = 0$ $v_x(l, t) = 0$ $v(x, 0) = −U$ I have: $v(x,t) = X(x)T(t)$ $X(x)T'(t) - kT(t)X''(x) = 0$ ...
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1answer
35 views

PDE Using Fourier Series

I'm trying to find the solution to(I don't need to find the coefficient): $v_t = kv_{x x} , 0 < x < l, 0 < t < ∞$ $v(0, t) = 0$ $v_x(l, t) = 0$ $v(x, 0) = −U$ Where U is a constant ...
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2answers
30 views

Fourier series time shift proof?

Prove that if $f(x) \sim \sum c_k e^{ikx}$, then $f(x+t) \sim \sum c_k e^{ikt} e^{ikx}$. Replacing the instance of $x$ with $x + t$, we have that $$f(x + t) \sim \sum c_k e^{ik(x+t)} = \sum c_k ...
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0answers
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Prove that $g(x)=(1+|x|^4)^{-a}$, where $a>0$ is in $\mathcal{C}^{\infty}$ but not Schwartz function.

Prove that $g(x)=(1+|x|^4)^{-a}$, where $a>0$ is in $\mathcal{C}^{\infty}$ but not Schwartz function. But this function is goes to zero when $x$ goes to $\pm\infty$. Then Why it is not in ...
2
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1answer
65 views

Rudin's Real & Complex - Q9.11 (Fourier)

I have solved most of Question 9.11 of Big Rudin : Find conditions on $f$ and/or $\widehat{f}$ which ensure the correctness of the following formal argument : If $\varphi(t) ~=~ ...
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0answers
33 views

When does this fourier series converge?

For which $-2\pi < x < 2\pi$ does this series converge? $$1 = \sum_n^{\infty} A_n\cos\left[\left(\frac{1}{2}+n\right)x\right]$$ The cosine function is piecewise orthogonal. I found $$A_n ...
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0answers
29 views

Convergence of the Fourier series of a continuously differentiable function

I'm taking an introductory course in Fourier analysis and I'm trying to solve the following problem Prove that the Fourier series of a continuously differentiable function $f$ on the circle is ...
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0answers
33 views

Fourier transformation of piece-wise function

Let the function $ f(x)= \begin{cases} 0: &|x|>1\\ 1: &|x| \leq 1 \end{cases} $ $|x|$ is the euclidian norm of $x$. My question is how we calculate $F(f)$? (the Fourier transformed). I ...
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1answer
255 views

The discrete Fourier transform of a Dirichlet charachter

I usually work in number theory so I am not familiar with Fourier transforms, I have read up on them and know the basics but it never seems to be in number theory language. I am trying to find the ...
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0answers
14 views

Do we have $(1+|y|^{-s})(1+|y|^s)\hat{u} \in L^2(\mathbb{R}^n)$?

For $0 < s < \infty, s\in\mathbb{R}$, and $u \in L^2(\mathbb{R}^n)$. Suppose $(1+|y|^s)\hat{u} \in L^2(\mathbb{R}^n)$, do we have $(1+|y|^{-s})(1+|y|^s)\hat{u} \in L^2(\mathbb{R}^n)$? The "hat" ...
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0answers
38 views

Derive the characteristic function of the standard normal distribution N(0,1)

A: Derive the characteristic function $\phi (u)$ of the standard normal distribution N(0,1) by solving: $\int_R e^{iux} f(x) dx$ where $f(x)$ is the probability density function of $N(0,1)$ and ...
395
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30answers
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Different methods to compute $\sum\limits_{k=1}^\infty \frac{1}{k^2}$

As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}.$$ However, Euler was Euler ...
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0answers
61 views

if $f(x)$ is periodic $\left|\int_1^\infty f(x) x^{-s} dx\right| \sim C\left|\int_1^\infty \sin(x) x^{-s} dx\right|$ when $\text{Im}(s) \to \infty$

is it true that if $f(x)$ is periodic, non-constant and bounded $$\text{when } T \to \infty ,\qquad\qquad\sup_{|t| \ <\ T}\ \ \left|\ \int_1^\infty f(x) x^{-\sigma-it} dx\ \right| \ \sim \ ...
0
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1answer
54 views

Fourier transform of $\frac{1}{\|x\|} \chi_{B_1(0)}(x)$

Define $f: R^3 \rightarrow R, f(x) = \frac{1}{\|x\|} \chi_{B_1(0)}(x)$ (with $f(0) = 0$). I would like to calculate the fourier transform $g(\xi) = \int_{R^3} f(x) e^{-ix\cdot \xi}dx$. I tried ...
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0answers
49 views

Fourier Transforms and Sums

Suppose I have the following sum: $$ \sum_{x = -\infty}^{\infty} \int_{-\pi}^{\pi} f(j) \; e^{i j x} dj $$ Assuming that everything is sufficiently smooth and convergent, then exchanging the sum with ...