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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6
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3answers
119 views

Proving $\sum_{n =1,3,5..}^{\infty }\frac{4k \times \sin^2(\frac{n}{k})}{n^2}=\pi$

Proving $$\sum_{n =1,3,5..}^{\infty }\frac{4k \times \sin^2(\frac{n}{k})}{n^2}=\pi$$ Where $k$ any number greater than $0$ I tried to prove it by using the Fourier series but I couldnt find ...
4
votes
0answers
35 views

How to find Green's function using Fourier-Bessel expansion

The Green's function satisfies the non homogeneous Bessel equation can be written as $xg''+g'+\left(k^2x-\frac{m^2}{x}\right)g=-\delta(x-\xi)$ where $m\geq0$ and an integer. The boundary conditions ...
1
vote
1answer
22 views

Fourier sine series of cosine?

In the middle of a PDE I'm trying to solve, I've gotten $$\sum_{n=1}^\infty T_n(0) \sin(nx) = \cos(3x)$$ Is this even possible? How can you expand a cosine (even) in terms of sines (odd)? Did I ...
2
votes
1answer
41 views

Fourier transform and dual vector space

In Serre's A Course In Arithmetic, it says the following: I don't know what it is talking about, I know the definition of $f'$, but what is This is in the last sentence refered to? $f'$ is a ...
0
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2answers
31 views

Prove the relation related to this Fourier Transformate

I have this Fourier trasformate $$\hat{f}(p)=\int_{-1}^{+1}(1-x^2)e^{-ipx}dx$$ and I have to prove this relation $$\hat{f}(p)=\int_{-1}^{+1}(1-x^2)e^{-ipx}dx =2 \int_{0}^{1}(1-x^2)\cos(px) dx$$ ...
0
votes
1answer
20 views

Positivity of a Sine transform of a positive function

Consider a function $f(t)$ with $f(t>0)>0$ and $f(-t)=-f(t)$. Can I make any statement about the positivity of the Sine transform $$\hat{f}(\omega) = \int_{0}^{\infty} \sin(\omega t) f(t) ...
2
votes
0answers
27 views

A proposition to prove the real interpolation of positive exponent

Let $p,A\in(0,\infty)$ $\|f\|_{L^{p,\infty}(X,\mu)} := \sup \{\lambda\mu(\{|f|\geq\lambda\})^{\frac{1}{p}}\}$ Show that the following are equivalent (1) $\|f\|_{L^{p,\infty}(X,\mu)}\leq C_pA$ for ...
0
votes
0answers
13 views

How to calculate the frequency component of a Hermite function?

So I've read here that the Hermite functions are eigenfunctions of the fourier operator. I've written some code to map a function to a hermite function basis. $$ f(x) = \sum_n a_n H_n( x ) ...
0
votes
2answers
27 views

Solving differential equation using Fourier

I'd like to solve the following equation using Fourier $$ y(t)+ {\sqrt 2\over 2\pi5} {dy(t)\over dt}+({1\over2\pi5})^2 {d^2y(t)\over dt^2}=x(t) $$ where $x(t) = u(t)$ (step function) So far i've got ...
0
votes
0answers
31 views

Removable singularity of $|z|^\alpha$ and its fourier transform

The idea that $|z|^\alpha$ for $\alpha \in (-1,0)$ has a removable singularity seems somewhat draconian. But by the proof of Riemann's theorem, since we have $\lim_{|z| \to 0} |z \times ...
1
vote
1answer
27 views

Multiple self-convolution of rectangular function - integral evaluation

I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise. I define $$ g_2 (y) = \int_{-\infty}^{\infty} ...
1
vote
2answers
68 views

Evalutation of an integral

I am trying to evaluate the integral $$ \int_{-\infty}^{\infty} \mathrm{d} w \frac{1}{w^n} \left (e^{iwx} -1 \right )^n e^{-iwx} $$ for some positive integer $n$ without much success. It doesn't ...
0
votes
1answer
19 views

Why discrete fourier transform consider only complex coefficients with non-negative indices

This is surely a silly question. $f(x)$ for $x \in [0, 2\pi]$ $f(x)=\sum_{k=-\infty}^{\infty} c_k e^{i kx}$ However, when we consider the discrete fourier transform, we seem to estimate $f(x)$ with ...
3
votes
1answer
37 views

$\int_0^\infty f^k (t)\cdot t^j \,dt$

Compute the $$\int_0^\infty f^{(k)}(t)\cdot t^j \,dt$$ for $$0\le j\le k$$ function values of f $$(f\in C_0 ^{\infty}[0,\infty))$$ Any ideas how to compute this integral?
2
votes
1answer
29 views

Is there way to choose $R$, $r$ so that the Lebesgue measure $m(A_{R,r}) \to 0$ as $\varepsilon\to 0$?

Let an integrable function $f:\mathbb{R}\to\mathbb{R^+}$ such that its Fourier transform $\widehat f(\lambda)\ne 0$ for almost $\lambda\in\mathbb{R}$. Define ${A_{R,r}} = \left\{ {\lambda:\left| ...
0
votes
0answers
7 views

$f\in C^{\infty}(0,\infty)$ and $f^j(x)=O(x^-a_j)$ with $0\le a_j\lt1$

If $$f\in C^{\infty}(0,\infty)$$ and $$f^j(x)=O(x^-a_j)$$ with $$0\le a_j\lt1$$ then prove that $$f\in C^{\infty}[0,\infty)$$ I do not know how to start to prove this any ideas?
0
votes
1answer
36 views

DTFT of Impulse train is equal to 0 through my equation.

Let me have an impulse train function as below, $$ x[n] = \sum_{m=-\infty}^{\infty} {\delta[n-f_0 m]} $$ where, $f_0 \in \textbf{Z}$. Now, I am trying to calculate its DTFT, so I put it into DTFT ...
0
votes
1answer
13 views

Shifted dirac delta function of DTFT is equal to 1 or not?

I am wondering which one is correct approach. Let me have an equation, $x[n] = \delta[n-m]$. If I try to calculate its DTFT(Discrete Time Fourier Transform) as below, $$ X(e^{j\omega}) = ...
1
vote
1answer
23 views

What is the Fourier transform of $e^{(-a+bi)x^2}$?

Let $a>0$. Let $f:\mathbb{R}\rightarrow\mathbb{C}$ be $f(x)= e^{(-a+bi)x^2}$. What is the Fourier transform of $f$? Here is what I have tried: The exponential decay of $e^{(-a+bi)x^2}$ means that ...
1
vote
1answer
14 views

Inverse Fourier transform gives wrong results

If I have a polynomial: $a(x)= 2 + 5x -3x^2 +x^3$ The Fourier transform for N=4 is the evaluation of this polynomial in ${\omega}^0,{\omega}^1,{\omega}^2,{\omega}^3$ with ${\omega}^h = \cos(2\pi ...
0
votes
1answer
19 views

Let $x_{0}=1$ and $x_{1}=-1$ For $n\geq0$ inductively define $x_{n+2}=x_{n+1}+6x_{n}$

I am not so sure how to do this problem and would like some help here. How would you induct a relation given this information here? I mean I know what induction means but I'm not so sure what I'm ...
1
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0answers
23 views

Prove $\hat{f}(\omega)\neq 0$ if $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete

Let $f\in L^1(\mathbb{R})$. s.t $\{f(x-t)\}_{t\in\mathbb{R}}$ is complete. Prove that $\hat{f}(\omega)\neq 0$ for all $\omega\in\mathbb{R}$ Suppose the system is complete for any $g\in ...
1
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0answers
34 views

Proving $\mu\ast K_n\to\mu$

Let $\{K_n\}$ be approximating unit and $\mu\in M(\mathbb{T})$. Show that $\mu\ast K_n\to \mu$ weakly means $$\int f(t)d(\mu\ast K_n)(t)\to\int f(t)d\mu(t)$$ Suppose $\mu\in L^1(\mathbb{T})$, I ...
0
votes
2answers
27 views

fourier transform integral, parseval's theorem?

I have a fourier transform which is $$X(jω)=\frac{\cos(2ω)}{ω^2+ω+1}$$ and I am trying to calculate the value of the integral: $$∫x(t)dt \ \ \ \ \ \ x \in (-\infty, \infty)$$. I was thinking I ...
2
votes
1answer
39 views

logarithmic integral function and asymptotic expansion

Show that Logarithmic integral function $$\int_2^x {1\over \log(t)} \, dt = Li(x)$$ has asymptotic expansion of the form $${x\over \log(x)}\cdot\sum_{j=0}^\infty a_j\cdot (\log(x))^{-j}.$$ I tried ...
1
vote
2answers
18 views

Proving norms inequality for fourier transforms

Let $F\in PW_\pi$. Prove that $\Vert F^\prime\Vert_{L^2(\mathbb R)}\le\pi\Vert F\Vert_{L^2(\mathbb R)}$ I know that the derivative of $F=\hat{f}$ is given by the formula ...
0
votes
1answer
41 views

Calculate $\mathscr{F}((1+t)^{-3})$

Let $$f(t)=\cases{\frac{1}{(1+t)^3}&t>0\\0&t<0}$$ Does: a.$\hat{f}$ is differentiable? b.$\hat{f}\in L^1(\mathbb{R})$? c.$\hat{f}\in L^2(\mathbb{R})$? Seems like we need to calculate ...
3
votes
3answers
457 views

Improper integral - equivalent definition?

Intuitively, it is rather obvious that $$\lim_{l\to\infty}\sum_{n=-\infty}^{\infty}f(n\Delta x)\Delta x = \int_{-\infty}^{\infty}f(x)dx \tag{1}$$ where $\Delta x = \frac{1}{l}$, assuming $f$ is ...
0
votes
1answer
20 views

Convergence of Fourier series for a sum which is not uniform convergent

Given $$\sum_{n=1}^\infty\frac{\cos nt}{n}$$is it a fourier series in a. $L^2(\mathbb T)$? b. $C(\mathbb{T})$? Usually when we get a series we use Weierstrass M test in order to find ...
0
votes
0answers
7 views

Fourier Transformation of $cos(10*\pi*t)*\Pi(t/2)$ where $\Pi(t/2)$ is a rectangular pulse

I am studying for an exam and am trying to find the Fourier Transform of $$cos(10*\pi*t)\Pi(t/2)$$ where $\Pi(t/2)$ is a rectangular function. So far I have: $$cos(10*\pi*t) = ...
0
votes
0answers
8 views

2D Fourier transform of the signal $f(x,y) = 2cos(2\pi(3x - 2y))$ with sampling

Given the signal $f(x,y) = 2cos(2\pi(3x - 2y))$. The signal is sampled with $\Delta x=0.25, \Delta y =0.2$. The questions: 1) What is the spectrum of the original signal? 2) What is the spectrum of ...
3
votes
4answers
105 views

Fourier Transform of Heaviside

I need help with a Fourier Transform. I know Fourier Transform is defined by: $$F(\omega)=\int_{-\infty}^{\infty} f(t).e^{-i\omega t}\, dt$$ where $F(\omega)$ is the transform of $f(t)$. Now, I need ...
0
votes
0answers
16 views

Fourier transform of $|x|^a \times \text{compactly supported function}$ where -1<a<0

THIS IS A HOMEWORK PROBLEM I tried doing what I usually do which is isolate a branch cut for $x^a e^{-ikx}$ and a neighborhood along zero, and hope that the only integrals that survive are the ones ...
2
votes
0answers
23 views

an inequality on convolution

Let $f(x)=A(1+|x|)^{-M}$ and $g(x)=B(1+|x|)^{-N}$ be two functions on $\mathbb{R}$. Here $M\ge N>1$. I'm trying to prove the following inequality about convolution $$ |(f*g)(x)|\le ABC(1+|x|)^N, $$ ...
0
votes
1answer
38 views

Separation of Variables vs Fourier Transform (for PDE)

I would like to know how can I know if I have to solve a PDE (Heat Equation, Laplace Equation, Wave Equation, etc.) using Separation of Variables or Fourier Transform. Which boundary conditions do I ...
0
votes
2answers
33 views

Why is the imaginary part of the logarithm of the gamma function a square wave?

I just stumpled upon it and it made me curious. Why is the imaginary part of $\ln(\Gamma(x))$ a square wave for $x < 0$ ? The square wave has a period of 2 and a amplitude of $\pi/2$. How can one ...
1
vote
2answers
90 views

Doubt in Rudin's Proof:

Once I go through the proof of the below theorem, I could encounter that he used dominated convergence theorem to prove $(f)$, in that how they claim that $$\frac{e^{-ix(s-t)}-1}{s-t}\leq |x|$$ Kindly ...
1
vote
0answers
7 views

Existence of compactly supported Fourier transforms on LCA groups

I'm trying to prove the following theorem: The following are equivalent for a locally compact abelian group: $G$ has an open compact subgroup. There exists a nonzero $f\in C_c(G)$ such that ...
0
votes
1answer
10 views

Does $(1+|\cdot|^2)^{k/2} f\in L^2(\mathbb R^n)$ imply $f^\vee\in C^k(\mathbb R^n)$ and $\partial^\alpha f^\vee\in L^2(\mathbb R^n)$?

Let $f$ be a measurable function such that $(1+|\cdot|^2)^{k/2}f\in L^2(\mathbb R^n)$ where $k\in\mathbb N_0=\mathbb N\cup\{0\}$. This implies $f\in L^2(\mathbb R^n)$ for \begin{align*} ...
2
votes
2answers
49 views

Coefficients in Fourier series

This is my first post so please go easy, I don't know all the rules yet. I was reviewing Fourier series for the 5th time and I realized that every explanation I read goes into the orthogonality of ...
3
votes
2answers
52 views

What is the meaning of $1^\lor=\delta$

I tried to look up the definition of a $\lor$ and it does not seem to explain this particular usage $$1^\lor=\delta$$ This is used in a proof that inverse fourier transform of $1$ is $\delta$, but ...
3
votes
1answer
74 views

Upper bounds for the dimension of a binary cyclic code

Let $\mathbb{F}_2 = \{0,1\}$ denote the field with two elements. Consider a binary $N$-tuple $a = a_0 a_1 \ldots a_{N-1}$, of elements $a_i \in \mathbb{F}_2$. The cyclic code $C_a$ corresponding to ...
0
votes
1answer
66 views

From fourier series to continuous fourier transform

In derivation of fourier transform, we start with the fourier series coefficients. If we let $T \to \infty$, it's common to say the spacing between consecutive fourier coefficient will approach $0$, ...
0
votes
0answers
23 views

Harmonic analogue of the Weierstrass approximation theorem

The Weierstrass approximation theorem says that, given any continuous function $f(x)$ on a closed interval, there is a polynomial which approximates it arbitrarily closely. I'm looking for a theorem ...
2
votes
1answer
45 views

Fourier COSINE Transform (solving PDE - Laplace Equation)

I'm trying to solve Laplace equation using Fourier Cosine Transform (I have to use that), but I don't know if I'm doing everything OK (if I'm doing everything OK, the exercise is wrong and I don't ...
0
votes
0answers
13 views

3D Fourier transform in cylindrical coordinate

I know this question has asked already, but I do not understand exactly how it works, since I am still a Matlab novice. Previous post: Fourier transform in cylindrical coordinates What I got from ...
3
votes
1answer
43 views

Complex Measure Agreeing on Certain Balls

I came across this problem and am lost as to how to solve it. Let $r>0$ be fixed. Suppose $\mu, \nu$ are complex Borel measures on $\mathbb{R}^d$ such that for each open ball B of radius $r$, ...
0
votes
1answer
18 views

Can inverse fourier transform be formulated in terms of residue?

Today I ran into a peculiar problem when trying to perform the inverse fourier transform of $\frac{1}{a+jw}$ where a is some number $$ \mathcal{F^{-1}}(\frac{1}{a+jw}) = ...
-1
votes
1answer
28 views

PDE, separation of variables

I need some help here. It's regarding question 5a. I am pretty lost as i've got no clue regarding how I should use the boundary conditions(i would've known if they weren't derivatives)
0
votes
1answer
33 views

How do you verify this identity of fourier transform involving $\delta(x-x')$

How do you prove that $$ \delta(x-x') = \frac{1}{2\pi} \int\limits_{-\infty}^{\infty} e^{-ik_x(x-x')} \, \mathrm{d} k_x $$ Attempt, take fourier transform of delt function and using the sifting ...