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 $\sin(2\pi fm t) \sin(2\pi fc t) $

How do I find the Fourier transform of $\sin(2\pi f_m t)\sin(2\pi f_c t) $? My main confusion comes from $fm$ and $fc$. If I had the same frequency I could of used a trig identity and then Fourier ...
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36 views

About Hardy-Littlewood maximal function from Grafakos' “Classical Fourier Analysis”

The Hardy-Littlewood maximal operator $M$ is defined by $$ M f(x)=\sup_{r>0}|B(x,r)|^{-1}\int_{B(x,r)} |f(y)|dy. $$ There is the following example in Grafakos's "Classical Fourier Analysis" book ...
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Intuition behind Fourier and Hilbert transform

In these days, I am studying a little bit of Fourier analysis and in particular Fourier series and Fourier/Hilbert transforms. Now, I am confident with the mathematical definitions and all the ...
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38 views

Error in the statement of Wirtinger's inequality?

Theorem. Suppose that $f(x)$ has a continuous derivative on the interval $[0, 1]$, and that $\int_0^1 f(x)\, dx=0$. Then $$\int_0^1 |f'(x)|^2\, dx\ge 4\pi^2\int_0^1 |f(x)|^2\, dx.$$ Proof. ...
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35 views

Inverse Fourier transforms with Heaviside step function

I want to find the inverse Fourier transforms of: $$u(\nu + 1) \ \exp(-\nu)$$ Attempt: So the inverse Fourier transform is given by: $$\int^\infty_{-\infty} u(\nu + 1) \ e^{-\nu} e^{j2 \pi t} \ ...
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Using Paley-Wiener theorem and Fourier inversion formula to get this result

I want to solve the problem #8 of Stein's book Complex analysis in Chapter 4, for the first part I've got the following: We know that the coefficients of a series are: ...
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35 views

Proof that fixed points of a null field are zero

In this context, a null field means a field constructed of planar waves $e^{I k_{\mu} x^{\mu}}$ that satisfy the null condition $k_{\mu} k^{\mu} = 0 \implies c^2 k^2 = \omega^2 $ Suppose we have a ...
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29 views

Inverse Fourier Transforms

Find the inverse Fourier transform of the following: $$\sin(2 \pi \nu T) \cos (10 \pi \nu T) / (\nu T)$$ Attempt: I was told it was easier if we rewrite this in terms of a $sinc$ function. I think ...
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48 views

How can I write this in a more convenient way?

This question is part of a question deriving Fourier coefficients, so one of them is $$b_n = \frac{1-\cos(\frac{n\pi}{2})}{n\pi}$$ I think this is ugly, so $$ b_n = \begin{cases} ...
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28 views

Using Fourier Series to find the sum of a numerical series

I have to use a Fourier series to compute the sum of the series $$\frac{1}{2} + \sum_{n=1}^\infty (-1)^n \frac{1}{n^2 + a^2}$$ My guesses are the Fourier series $$e^{ax} = \frac{e^{a\pi} - ...
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17 views

Prove the following about the integral of Fourier coefficients

I'm having a difficult time going from $$\sum_{n=1}^\infty (\cos nx \int_{-\pi}^\pi f(t) \cos nt dt + sin nx \int_{-\pi}^\pi f(t) \sin nt dt)$$ to $$\sum_{n=1}^{\infty}(\int_{-\pi}^\pi f(t) \cos ...
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17 views

Application of Abel's Method to Summation of Fourier Series Question

The series $$f(x, r) = \frac{a_0}{2} + \sum_{n=1}^\infty r^n (a_n \cos nx + b_n \sin nx)$$ where $0 \le r \lt 1$ clearly converges, as the terms monotonically decrease. My question: My textbook ...
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27 views

Periodic version is constant implies f is constant

Let $f:\mathbb R\to \mathbb R$ be a uniformly continuous function with finite measure support. Let $g$ be the periodic version of $f$ defined on [0,1), that is: $$ g(x)=\sum_{k\in \mathbb Z}f(x+k) $$ ...
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39 views

Is $|x|^{-r}$ tempered distribution?

The Schwartz space, $S(\mathbb R): = \{f\in C^{\infty}(\mathbb R): \sup_{x\in \mathbb R} |(1+|x|)^{\alpha} D^{\beta}f(x)|< \infty , \forall \alpha, \beta \in \mathbb N \cup \{0\} \}$ ...
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28 views

Is there is notion of Fourier transform of distribution?

We note that every tempered distribution is a distribution. Can we find a example of distribution which is NOT a tempered distribution? Can we talk of Fourier transform of that ...
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13 views

Determining wave algorithm based on sine wave

I have some data that I've noticed conforms to a sine wave and I want to approximate it as closely as I can. In the graph, the blue line is the data I want to model as closely as possible. From ...
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32 views

Derive the Fourier Transform

I have been asked to derive the Fourier Transform for $$f(x)=\frac{1}{x^2+a^2}$$ where $a>0$. I know the Fourier Transform is equal to ...
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18 views

Time-Shifted Trigonometric Fourier Series Coefficients

I'm trying to find the Trigonometric Fourier series coefficients for a particular periodic function. Given $$f(t) = 2 - \frac9\pi \sum_{n=1}^\infty\frac1{2n-1}\sin\left(\frac{2n-1}2 \pi t\right)$$ ...
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33 views

Let $\hat{g}=f.$ Is $g$ continuous?

Let $f: \mathbb R \to \mathbb C$ such that there exists $g\in L^{1}(\mathbb R)$ with $\hat{g}=f.$ Then by Riemann-Lebesgue Lemma, we have $f$ is continuous and vanishing at infinity. My ...
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25 views

Polar form of the Fourier transform of $\sin(t)$

I'm studying signal processing, and I came across the Fourier transform of sin(t). It ends up being a purely imaginary (dirac delta) impulse pair. But when considering the frequency domain ...
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17 views

What is the Permutation Matrix in FFT DFT Factorization?

Given: $$F_N = \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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63 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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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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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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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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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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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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80 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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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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140 views

If the Fourier serie $S_nf\longrightarrow g$ in $L^p$ then $f=g$.

Suppose that $f\in L^1(\mathbb S^1)$ where $\mathbb S^1=\mathbb R/\mathbb Z$. Suppose that the sequence of partial Fourier sums $\{S_nf\}_{n\geq 1}$ converge in $L^p(\mathbb S^1)$ toward some $g\in ...
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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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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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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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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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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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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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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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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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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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49 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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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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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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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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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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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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121 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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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} ...