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 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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1answer
67 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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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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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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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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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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20 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 ...
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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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1answer
50 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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2answers
34 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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1answer
33 views

Tricky sum inequality coming from Fourier Series

Show $\sum\limits_{m=N+1}^\infty\frac{1}{m^{2k}}\leq \frac{N-1}{(N+1)^{2k}}\left(1+(1+\frac{1}{N})^{2k}\sum\limits_{m=2}^\infty \frac{1}{m^{2k}}\right)$. I tried making the substitution $m\rightarrow ...
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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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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 ...
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1answer
123 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 ...
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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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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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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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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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47 views

Heat flow equation via Fourier Series

I know how to solve heat equations and wave equations defined on $\mathbb{R}^n\times(0,\infty)$ using Fourier transform. But I am having trouble solving similar equations defined on finite intervals ...
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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
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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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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7 views

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 ...
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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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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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33 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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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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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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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 ...
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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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24 views

Complex Exponential Euler's Identity Simplification

I am working on a problem in a signals and systems class where I am being asked to find the inverse Fourier transform of a function. I am given the equation: $$X(j \omega)= \begin{cases} -\omega ...
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1answer
15 views

Show that there is $C_1,C_2$ s.t. $C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$ where $D_N$ is the dirichelet kernel.

Let $D_N=\frac{\sin(\pi(2N+1)x)}{\sin(\pi x)}$ the dirichlet kernel. Show that there is $C_1,C_2$ s.t. $$C_1\log(N)\leq\|D_N\|_{L^1(\mathbb S^1)}\leq C_2\log(N)$$ where $\mathbb S^1=\mathbb R/\mathbb ...
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9 views

Relation between support of a function and that of its DFT

Let $f : \mathbb{Z}_N \to \mathbb{C}$, let $\zeta_N$ be a primitive $N$-th root of unity and let $\hat{f} : \mathbb{Z}_N \to \mathbb{C}$ be the DFT of $f$ given by $\hat{f}(m) = \sum_{n \in ...
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40 views

Integration similaring to Fourier transform of Gaussian function

I would like to calculate the integral: $$\int^{\infty}_{0}x\cdot \exp(-x^2)\cdot \exp(-ikx)dx$$ Are there some tricks to solve it? Many thanks.
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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 ...
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1answer
47 views

how to integral $\int_{-\infty}^{\infty}{\frac{e^{iyx}}{1+y^2}}dy$ , $x\ge 0$

How to integral $$\int_{-\infty}^{\infty}{\frac{e^{iyx}}{1+y^2}}dy,x\ge 0$$ without using Fourier transform? In fact, this integral is the fourier inverse transform of $\frac{1}{1+y^2}$. and ...
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1answer
17 views

Convolution bounds

For $t\geq0$, let $g_\beta(t)=e^{-t}\sin(\beta t)$, where $\beta$ is a real number, and for $t<0$, $g_\beta(t)=0$. Find $h*g_\beta(t)$ for all $t\geq0$, where ...
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9 views

Fourier transform of real part part of a complex signal

Let $$g(t) = g_r(t) + j g_i(t)$$ be a complex signal with $g_r(t)$ and $g_i(t)$ real signals. We can write the Fourier transform $$G(f) = G_r(f) + jG_i(f),$$ where $G_r(f)$, $G_i(f)$ and $G(f)$ are ...
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1answer
49 views

Applying the Riesz-Thorin Interpolation theorem

Consider a linear operator $T$ given by $$T(f)(x)=\int_{Y}K(x,y)f(y)d\nu(y),\qquad x\in X.$$ Let $1\le p_{1}$, $q_{1}\le\infty$, $p_{0}=1$, $q_{1}=\infty$, $\frac{1}{p_{1}}+\frac{1}{p_{1}'}=\infty$ ...
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1answer
29 views

What is the meaning of ifft2(H)(l)

I want to calculate $$f(l): = \frac{1}{{{N^2}}}\sum\limits_{{k_1},{k_2} = 0}^{N - 1} {{e^{2\pi ikl'/N}}H(k)}, $$ where $l=(l_1,l_2)\in\{0,\ldots,N-1\}^2$. ifft2 in MATLAB can be used to calculate ...
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1answer
72 views

The series $\sum_{k=2}^\infty \frac{ \cos(kx)}{k \ln k}$ is bounded below by $c \log\log x$ near 0, thus fails to be uniformly convergent

The convergent series $$ \sum_{k=2}^\infty \frac{ \cos(kx)}{k \ln k} $$ defines a function in $H^{1/2}([0,2\pi])$. This is an example of such a series for which convergence on is not uniform. I want ...
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1answer
37 views

An equality about the Sobolev space $W^{1,2}$

From the Plancherel identity, we know that $$\int_{\mathbb{R}} |f(x)|^2\,dx=\int_{\mathbb{R}}|\widehat{f}(\xi)|^2\,d\xi$$ is valid for all $L^2(\mathbb{R})$ functions and in particular for Schwartz ...
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1answer
30 views

Prove that inverse fourier cosine transform of $\exp(-tkw^2)=\frac{1}{\sqrt{2kt}}\exp(-\frac{x^{2}}{4kt}) $

In the process of solution of a PDE via Fourier cosine transform the author assumes at one step $$F_{c}^{-1}\exp(-tkw^2)=\frac{1}{\sqrt{2kt}}\exp(-\frac{x^{2}}{4kt}) $$ where Fc^{-1} is fourier ...
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9 views

Transform and domain such that phase shift of the original function results in shift in transformed domain variable

It is known that the Fourier transform of a phase-shifted function results in a constant shift of the dependent variable of the phase spectrum: If $ F(x(t)) = X(w) = |A(w)| \cdot e^{-i \cdot ...
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21 views

Unitary transformation with eigenvectors of fourier basis

I'm trying to understand this statement made in https://users.cs.duke.edu/~reif/courses/randlectures/UVnotes/lec18.pdf in the last paragraph: "Since U is multiplication by a group element, the ...
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0answers
34 views

Fourier transform identity on $L^{p}(\mathbb{T})$

Let $p\in (1,2]$. I want to prove there exists $C$ such that for $f\in L^{p}(\mathbb{T})$ we have $$\sum_{n\in\mathbb{Z}}|\hat{f}(n)|^{p}|n|^{p-2}\le C\|f\|^{p}_{L^{p}(\mathbb{T})}$$ If $p=1$, it's ...
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0answers
27 views

Fixed points of convolution

Let $k\in L^2(0,1)\times(0,1)$. Assume that there exists $f\in L^2(0,1)$, such that $$f^{(n)}\ast k=f^{(n)}, \,\,\,n=0,1,\cdots ,$$ where the convolution is taken over interval $(0,1)$ and $(n)$ ...