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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Can we choose $g$ so that $\|(g\widehat{(f^{3})})^{\vee}\|_{L^{p}} \leq C \|g_{1}f\|_{L^{2}}^{r} \|(g_{2}\hat{f})^{\vee}\|_{L^{s}}$?

Let $f, f^{2}, f^{3}\in L^{q}(\mathbb R)\cap C_{0}(\mathbb R)$ where $ q\geq p, \ \text{and}$ and $C_{0}(\mathbb R)$ is the class of continuous functions vanishing at infinity. My Questions: ...
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342 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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23 views

Proving a fourier transform expression with green's formuls

Using Green's formula, show that: $${\cal F}\left[\frac{d^2f}{dx^2}\right]= -w^2F(w) + \frac{e^{iwx}}{2\pi}\left(\frac{df}{dx} - iwf\right) \\(evaluated\ from\ -\infty\ to\ \infty)$$ last part is ...
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1answer
41 views

Proof of the discrete Fourier transform of a discrete convolution

Let the discrete Fourier transform be $$ \mathcal{F}_N\mathbf{a}=\hat{\mathbf{a}},\quad \hat{a}_m=\sum_{n=0}^{N-1}e^{-2\pi i m n/N}a_n $$ and let the discrete convolution be $$ ...
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1answer
53 views

Smoothness of inverse Fourier transform

Let $\hat{f}(\xi)$ be a smooth function on $\mathbb{R}^n$ that decays like $|D^\alpha_\xi \hat{f}(\xi)| \lesssim (1 + |\xi|^2)^{-\frac{1}{4}(1 + |\alpha|)}$, where $\alpha$ is a multi-index such that ...
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1answer
73 views

Estimating an integral using the Poisson summation formula

Consider a continuous $L^1$ function $f$ : $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ such that $supp$ $\widehat{f}$ $\subset$ $[-1,1]$ and $f(n)$ $\geq$ $0$ if $n$ $\in$ $\mathbb{Z}$. The problem is to ...
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1answer
48 views

Fourier Series Expansion, error in coefficients?

After reworking the problem many times I keep getting the same (incorrect?) answer. So the problem as stated is Find the Fourier expansion of : $$ f(x) = \begin{cases} x &\text{ if }0 ...
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19 views

Norm convergence of approximations to the identity

Let $\varphi \in L^1(\mathbb{R}^d)$ be such that $\int_{\mathbb{R}^d} \varphi(x) \, dx = 1$. For each $\varepsilon>0$, let $\varphi_\varepsilon:= \varepsilon^{-d} \varphi\left( \dfrac x ...
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1answer
26 views

Bound on the sup norm for derivatives of a particular $C^\infty$ function

I'm reading textbook "A Primer of Real Analytic Functions" and on page 86 the following "obvious" claim is made: Let $|| \cdot ||$ be the sup norm on $[0, 2 \pi]$ and define function $f$ to be ...
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2answers
44 views

Fourier inverse of a function to get dirac

I'm trying to get the dirac function from a fourier inverse tranform: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iw(x-x_0)}dw$$ It is this last step I am stuck on to get the conclusion. Original ...
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56 views

inserting absolute value in Hilbert transform and a discrete version of Hilbert transform

It is well known that the Hilbert transform $H(f)(x)=p.v. \int\frac{f(x-y)}{y}dy$ is bounded on $L^p(\mathbb{R})$ for $p\in(1,\infty)$. I want to consider some variants of $H$. 1) What happens if we ...
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1answer
39 views

Integral of $\int_{-\infty}^{+\infty}\left |{\frac{\sin{x}}{x(1+x^2)}}\right|^2\,dx $

So the first part of the questions asks us to find the Fourier Transform of $$ f(x) = \left\{ \begin{array}{ll} e^{y} & \quad {-\infty}<x < 0 \\ e^{-y} & ...
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1answer
479 views

Is Fourier transform characterized by its diagonalization properties?

Let us fix the following convention for the Fourier transform in $L^1(\mathbb{R})$ space: $$\hat{f}(\xi)=\int_{-\infty}^\infty f(x)\, e^{-2\pi i x\xi}\, dx.$$ We then have the following properties: ...
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1answer
43 views

Why the sum of the list is 4?

Wolfram Alpha says Sum[Sin(Pi*n/4)]/(Pi*n/4),{n,-Infinity,Infinity}] is equal to 4 but I don't know how to resolve it... In my signal and system homework,this ...
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9 views

Errors of approximating continuous Fourier transform by discrete Fourier transform

In http://planetmath.org/approximatingfourierintegralswithdiscretefouriertransforms some error analysis of using DFT to approximate continuous Fourier transform is indeed done, but there are things I ...
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1answer
17 views

$h = \sum_{n=0}^\infty (ae^{j\omega})^n$ , how is the approximation of this equal to $\frac {1}{1-h}$

the question in the title. im working on a z- transform problem. to find the Z - transform of $x(n) = a^ncos(\omega n)u(n)$, u(n) being the step unit function essentially i come down to the answer ...
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2answers
41 views

Application of Plancherel/Parseval

Assuming $u,v\in L^1\cap L^2$, then how do you show that $$\int uv=\int \hat{u}\hat{v}$$ I tried using Plancherel, but didnt give any nice result. Any ideas/hints? Thanks
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26 views

find the fourier transform of $xf(x)$ appended

I've seen the method in which you prove this fourier transform, but what if you don't recognize that $$xf(x) e^{i k x} = \frac{1}{i} \frac{\partial}{\partial k} \Big[ f(x) e^{i k x} \Big] $$ would I ...
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1answer
22 views

Applying Fourier transform to a gaussian

Let $$G_\beta(w) = e^{\beta w^2}$$ Now I get the process of applying a fourier transform (or inverse) to get a new gaussian: $$G_\beta(x) = G_\beta(0) e^{\frac{-x^2}{4\beta}}$$ but in doing the ...
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1answer
11 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
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10 views

Fourier transformation of Principal value distribution [duplicate]

I have the principal value distribution defined as $pv(\frac{1}{x})(\phi)=\int^\infty_0\frac{1}{x}(\phi(x)-\phi(-x))dx$ and I want to show that the fourier transform is given by $-\pi i\cdot ...
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24 views

Fast Fourier Transform of a function: WolframAlpha vs calculated result

I have the following function: $$\frac{3}{\sqrt{12}\cdot\cosh(x)}$$ I want to calculate the Fourier transform of this function. When calculated with WolframAlpha, I get as result: ...
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1answer
37 views

How to calculate convolution integral?

I know the formula for a convolution integral but how would you actually carry out one when you have two piece-wise defined functions? If you had $$ f(x) = \left\{ \begin{array}{ll} ...
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21 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
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2answers
49 views

Integral using Parseval's Theorem

How would I integrate $$\int_{-\infty}^{+\infty} \frac{\sin^{2}(x)}{x^{2}}\,dx$$ using Fourier Transform methods, i.e. using Parseval's Theorem ? How would I then use that to calculate: ...
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2answers
50 views

$\|(g\widehat{(f|f|^{2})})^{\vee}\|_{L^{2}} \leq C \|f\|_{L^{2}}^{r} \|(g\hat{f})^{\vee}\|_{L^{2}}$ for some $r\geq 1$?

Let $g\in C_{c}^{\infty} (\mathbb R)$, and $f, |f|^{2}f\in L^{2}(\mathbb R)\cap C_{0}(\mathbb R)$ (where $C_{c}(\mathbb R)$ is the class of smooth functions with compact support and $C_{0}(\mathbb R)$ ...
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34 views

How do I show $\int_{-\infty}^\infty \frac 1{(a^2+s^2)(b^2+s^2)} ds=\frac {\pi}{ab(a+b)}$ using the solution to the following Fourier transform?

For a function $f_a(x)=e^{-a|x|}$ , where $a>0$ I have found that the fourier transform of it is as follows, i know this is correct. $\def\F{\mathcal F}$ \begin{align*} \F(f_a)(s) &= ...
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34 views

Asymptotic expansion of a Fourier Transform as $\omega\rightarrow 0$

First of all, I do apologise if the question is not formulated in precise mathematical terms, but as a physics student I lack a formal background on rigorous functional analysis. Suppose we have a ...
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28 views

Exact formula for alias of Discrete Fourier transform for periodic sigals

Suppose that $f(t): \mathbb{R} \to \mathbb{C}$ is a $T$-periodic signal, with highest frequency $f_h$. Now suppose that our sampling rate frequency is lower than $f_h$, and is not any multiples of ...
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1answer
14 views

ODE with finite Fourier expansion periodic coefficients

Regard the ordinary differential equation $$ \dot a(t) = z(t) a(t) $$ where $a(t)$ and $z(t)$ are matrix valued such that $z$ is periodic ($z(t+2\pi)=z(t)$). Then it is well-known (Floquet theory), ...
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3answers
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3D Fourier transform

I don't know how to evaluate an integral of the form $$\int d^3 r \exp(-i \vec r\cdot\vec q)\exp(-a^2 r^2)$$ where $a\in \mathbb R$. Could anyone please teach me how to do this integral? Many ...
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1answer
23 views

how do i find the fourier transform of the function $f_a(x)=e^{a|x|}$

Having trouble finding the fourier transform of the function $f_a(x)=e^{-a|x|}$ , where $a>0$ I currently have that $$ \mathcal{F}(f_ax) = \int_{0}^{\infty} e^{-ax}\, e^{ i s x} \,ds= ...
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1answer
30 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
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1answer
34 views

Fourier transform, circular symmetry

I need to compute the two-dimensional Fourier transform of a function with circular symmetry: $$ \int dx dy\, \frac{e^{i (k_x x+k_y y)}}{((z'-z)^2-t^2+x^2+y^2)((z'+z)^2-t^2+x^2+y^2)} $$ For ...
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1answer
55 views

Lipschitz continuity of complex valued functions (Fourier basis)

Apologies if this is a rather stupid question, but as a student with no prior knowledge on complex analysis, I was wondering if the following function of $x$ \begin{equation} f(x)=e^{-iqx} ...
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1answer
30 views

Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
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94 views

Fourier series for convex plane curves.

The following problem is from Stein's Fourier analysis. This problem explores another relationship between the geometry of a curve and Fourier series. The diameter of a closed curve $\Gamma$ ...
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150 views

System of first order ODEs with coherent sinusoidal time varying coefficient

I have encountered equations of the form $$\frac{{d{\bf{y}}(t)}}{{dt}} = \left( {{A_0} + {A_1}\cos (\omega t)} \right){\bf{y}}(t)$$where ${\bf{y}}$ is a vector and ${{A_0}}$ and ${{A_1}}$ are square ...
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1answer
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Solving $\int_{-\infty}^\infty f(\tau) {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau$.

We know that, for any real numbers $\lambda$ and $\nu$, it has \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= ...
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42 views

Negative index Sobolev spaces, properties

I was hoping to find some references for the following facts: Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^n$. (i) If $\nabla p$ is a distribution in $H^{-1}(\Omega)$ then ...
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2answers
129 views

An analogous definition of Fourier transform $\hat f(u) = \int_{-\infty}^{+\infty} f(t) \exp(- i u t) dt$ for sinc-function.

We know the definition of Fourier transform $$\hat f(u) = \int_{-\infty}^{+\infty} f(t) \exp(- i u t) dt \ \ \ (*)$$ It is widely used in the analysis in the frequency of dynamical systems, in the ...
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1answer
38 views

$\|fg\|_{A (\mathbb T)} \leq C \|f\|_{L^{2}} \|g\|_{A (\mathbb T)}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
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17 views

FFT of k*k matrix from FFT of a j*j matrix

FFT of matrix a j by j matrix, A $\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ = $\begin{bmatrix}10 & -2\\-4 & ...
3
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1answer
79 views

Decay of the Fourier transform of the surface measure of the sphere via uncertainty

I'm working through Tao's Recent Progress on the Restriction Conjecture notes (http://arxiv.org/abs/math/0311181). Currently, I'm working on problem 2.4, which will eventually allow us to compute the ...
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42 views

Represent 1/f(x) as integral

Suppose I have a function $f(x)=\int_a^b p(t)g(x,t)dt $ where $p(t), g(x,t)$ are known. I was wondering if there was a way of representing the reciprocal as an integral as well? i.e. ...
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17 views

Calderón–Zygmund lemma and Lebesgue measure

Can someone please explain the relationship between the Calderón–Zygmund lemma and Lebesgue measure, thanks.
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1answer
45 views

Prove $\int {\operatorname{sinc}}\big({\tau}-\lambda\big) {\operatorname{sinc}}\big({\tau}-\nu\big)d\tau= {\operatorname{sinc}}(\lambda-\nu ).$

I want to prove the following relation. For any real numbers $\lambda$ and $\nu$, we have \begin{equation} \int_{-\infty}^\infty {\operatorname{sinc}}\big({\tau}-\lambda\big) ...
0
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1answer
25 views

Sign with Fourier transformation, convolution, periodicity

Let $x(t)$ be the sign with Fourier transformation $$X(\omega)=\delta(\omega)+ \delta(\omega-\pi)+\delta(\omega-5)$$ and let $h(t)=u(t)-u(t-2)$. Is $x(t)$ periodic? Is the convolution of $x(t)$ ...
2
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3answers
107 views

A Fourier Analysis Question I am stuck at

If $f,g\in C[-\pi,\pi]$,and $f,g$ are $2\pi$ periodic, prove that $$\lim_{n\to\infty}\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)g(nt)\mathbb dt=\big(\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)\mathbb ...
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0answers
11 views

Fourier transform of spherical harmonics divided by $|\vec{x}|^{3}$

I need a formula for Fourier transform of spherical harmonics divided by third order polynomial of the form $$F^{m}_{l}(\vec{x}) = \frac{Y^{m}_{l}(\vec{x})}{|\vec{x}|^{3}}$$ Spherical harmonics are ...