2
votes
1answer
164 views

Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
1
vote
1answer
39 views

Evaluate the integral $\int_{-\infty}^\infty e^{-(t²+2t)/2}e^{-i\omega t}dt$

I have this integral: $$\int_{-\infty}^\infty e^{-(t²+2t)/2}e^{-i\omega t}dt$$ I don't know how to solve it, but I have tried, like this: $$ \int_{-\infty}^\infty e^{(-(t²+2t)/2)+(-i\omega t)}dt $$ ...
4
votes
1answer
82 views

How to develop the Fourier Transform in my mind now that I know the Fourier Seires?

I know that we can represent some function $f$ in this way: $$f(t) = \frac{a_0}{2} + \sum_{n=1}^\infty a_n\cos\left(\frac{n\pi t}{L}\right) + \sum_{n=1}^\infty b_n\sin\left(\frac{n\pi t}{L}\right)$$ ...
2
votes
0answers
63 views

An entropy inequality

Let $f:[0,2\pi]\to \mathbb{R}$ be a smooth, positive function such that $f(0)=f(2\pi)$, and $\int_0^{2\pi}fd\theta=2\pi.$ Is it true that $$2\int_0^{2\pi}f\ln fd\theta- 2\int_0^{2\pi}\ln ...
2
votes
2answers
60 views

Find the $L^2[-\pi,\pi]$ projection of $f(x)$

I need to find the $L^2[-\pi,\pi]$ projection of $f(x)=x^2$ onto the space $V_n\subset L^2[-\pi,\pi]$ spanned by ...
1
vote
1answer
71 views

Can integral transforms be viewed as change of basis formulas?

Forgive any lack of rigor. If you have a countable orthonormal basis $B$ for a Hilbert space $H$ , then any function $f \in H$ can be expressed as $$ f(t) = \sum\limits_{g \, \in \, B} \langle f, ...
8
votes
0answers
73 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
1
vote
3answers
39 views

What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...
1
vote
1answer
51 views

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin t}{t}J_m(t)\,dt$

Let $m \geq 1$ be an integer. Evaluate $\int_R \frac{\sin(t)}{t}J_m(t)\,dt$ $$\int_R \frac{\sin(t)}{t}J_m(t)\,dt=\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt =\int_R \hat{\chi_{{(-1,1)}}}(t)J_m(t) \,dt ...
1
vote
1answer
27 views

$|\int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t}| dt \leq G(x,w), G\in L^{1} ? $

Put $\lambda >0,$ and we define, $$F_{\lambda}(x, w)= \int_{\mathbb R} e^{-t^{2}} e^{-(t/\lambda -x)^{2}} e^{-2\pi i w\cdot t} dt;(x,w) \in \mathbb R^{2}$$ we note that, $F_{\lambda} \in ...
4
votes
3answers
66 views

Finding Fourier transform of $\frac{x}{(x^2 + 4)^2}$

So I have this function $$ f(x) = \frac{x}{(x^2 + 4)^2} $$ and I have to find its Fourier transform. This is however much harder than what I have done before so I don't have a clue where to start. I ...
0
votes
0answers
30 views

In general, let $f \in L^2(-1,1)$ and let $g:R \to R$, $g(x)=f( \{ x \} )$. How are the Fourier transforms of f and g related?

Recall that $\{ x \}$ is a decimal part of a real number $x$. For example, if $x=3.41$, then $\{ x \}=-0.41$. Part A: Let $f(x)=2x$, with $x \in (-1,1)$ and let $g:R \to R$, $g(x)=2 \{ x \}$. Sketch ...
1
vote
2answers
36 views

Use the Fourier inversion formula to compute h(x) when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$

Use the Fourier inversion formula to compute $h(x)$ when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(x)=\frac{1}{(1+y^2)^2}=\frac{1}{1+y^2} \times \frac{1}{1+y^2}=\widehat{\frac{1}{2}e^{-|x|}} \times ...
1
vote
0answers
30 views

Check: Find $h(x)$ when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$.

Find h(x) when $\hat{h}(y)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(y)=\frac{1}{(1+y^2)^2} =\frac{1}{1+y^2} \times \frac{1}{1+y^2} =\hat{\frac{1}{2}e^{-|x|}} \times \hat{\frac{1}{2}e^{-|x|}}$$ Let ...
1
vote
1answer
92 views

Is $\hat S$ a function?

Let $S(x)=\sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x)$. Find the Fourier transformation of $S(x)$. Is $\hat{S}$ a function? $$\hat{S}=\int_R \sum_{n=-\infty}^\infty (-1)^n\chi_{(n,n+1)}(x) ...
0
votes
2answers
96 views

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$

Compute $\int_{a-b}^{a+b} \chi_{(-t,t)}(y)dt$. So if I create a number line marking a-b and a+b. If that the integral above has 5 different answers depending on where (-t,t) is located on the number ...
0
votes
0answers
12 views

Use the Fourier inversion formula to compute $\chi_{(-a,a)}*\chi_{(-b,b)}$ where $a,b>0$

Use the Fourier inversion formula to compute $\chi_{(-a,a)}*\chi_{(-b,b)}$ where $a,b>0$. Let $u=\chi_{(-a,a)}(x)$ and $v=\chi_{(-b,b)}(x).$ So, $$u*v=\breve{(\hat{u}\hat{v})}$$ So I calculated ...
0
votes
0answers
26 views

Show that the Fourier transform of f is in $L^p(R)$ for every $2 \leq p \leq \infty$.

Let $$f(x)=\sum_{n=1}^\infty \sqrt{n} \chi_{(\frac{1}{n+1},\frac{1}{n})}(x)$$. The Fourier transformation of f is $$\hat{f}(y)=\sum_{n=1}^\infty ...
1
vote
1answer
35 views

missing $j*\omega$ in integral

let us consider following integral according to property of delta function,we can write this intgeral as $\int^{t=\infty}_{t=t_0} e^{-j*\omega*t}$ or we can write as ...
0
votes
0answers
33 views

Solution of definite integral of product of bessel function and exponential

I have an integral $I=\int_{\theta} \int_r J_m(k_1r)e^{-j[P_x r \cos(\theta)+P_y r \sin(\theta)]} r dr d\theta$ $0\leq\theta\leq2\pi; r<\infty$ is there any method to solve this?
0
votes
2answers
25 views

Fourier Transform-1

I am trying to solve a Fourier transform problem and I am stuck. The problem is: $$f(t)= \frac{\sin(2t)}{e^{|t|}}.$$ I have used integration, but the answer that I come up with is different than ...
4
votes
1answer
198 views

For every $a>0$, show that $\langle f_a, \psi\rangle= \int_{|x|>a} \frac{\psi(x)}{|x|}dx+\int_{|x|>a} \frac{\psi(x)- \psi(0)}{|x|}dx$

To prove that the inner product is a distribution it must satisfy the following property" $$|T(\phi)|=|\langle T,\psi\rangle| \leq C_N \sum_{|\alpha| \leq N} \|\partial^\alpha \psi\|_\infty$$ Part ...
1
vote
1answer
44 views

$\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{(-1/2k,1/2k)}$

How do I calculate the following limit: $$\lim_{k \rightarrow \infty} g_k(x) =\lim_{k \rightarrow \infty} k(1+\cos(2k \pi x)) \chi_{\left(\frac{-1}{2k},\frac{1}{2k}\right)}$$
2
votes
0answers
53 views

Fourier transform of a sinusoidal function

Let us consider following table which I want to calculate myself $$ x(t)=\frac{\sin(\omega_bt)}{\pi t}\quad\iff\quad X(j\omega)= \begin{cases} 1 & \text{if $|\omega|<\omega_b$}, ...
3
votes
1answer
184 views

Can this be simplified?

$$ e^{-i\frac43\pi n} - e^{-i\frac23\pi n}, n\in \mathbb{N} $$ I am trying to simplify this but cant. Any ideas appreciated.
2
votes
1answer
44 views

Levy processes, vanilla option and Fourier Transform

The context to this problem is mathematical finance, although the answer does not need specific knowledge of the area. I am trying to work out the expression for the price of a call option using Levy ...
0
votes
2answers
28 views

$\int\exp(-jnw_0t)\,dt$ integral calculus.

I seem to forgot these parts of integral calculus. I am trying to determine the Fourier coefficient in complex exponential form. Here, $t$ is the variable being integrated and $n$ is for all ...
2
votes
1answer
62 views

the series $\sum_{k=1}^\infty a_k$ converges implies the series $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational

Let $\sum_{k=1}^\infty a_k$ be a convergent series. Then can we obtain $\sum_{k=1}^\infty a_k\sin (k\pi x)$ converges for $x$ irrational? If $\sum_{k=1}^\infty a_k$ converges absolutely, then I can ...
3
votes
2answers
101 views

Computing the inverse Fourier transform of $\frac{1}{1+|\xi|^2}$ for $\xi \in \mathbb{R}^n$.

I'm trying to compute the integral $$ \int_{\large\mathbb{R}^n} \frac{ e^{\large ix \cdot \xi}}{1 + |\xi|^2} ~d^n\xi. $$ I know that for an integral like $$\int_{\large\mathbb{R}^n} \frac{ 1}{1 + ...
1
vote
1answer
42 views

A naive example of discrete Fourier transformation

We know a discrete Fourier transformation with discrete $n$ and continuous $x_1,x_2$: $$ \sum_{n\in\mathbb{Z}} e^{-in(x_1-x_2)\frac{2\pi}{L}}=L\delta(x_1-x_2) $$ with Dirac delta function $\delta$. ...
0
votes
1answer
14 views

Showing that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set

I have the following problem: Show that the set $\displaystyle \{e^{2\pi i (mx+ny)}\}_{n,m=-\infty}^\infty$ is an orthonormal set in $L^2(D)$, where $D$ is any square whose sides have length ...
0
votes
0answers
43 views

Symbol of self-adjoint pseudodifferential operator

It seems that the following result should hold, but I can't find it explicitly anywhere. If $A=A^*$ is a properly supported pseudodifferential operator, does this imply that ...
1
vote
1answer
31 views

Can one express $f'(x)$ with the same basis as one uses for $f$?

If I have an orthonormal basis $\{\phi_n\}_1^\infty$ in space $L^2(a,b)$ and the generalized Fourier series expansion for $f$ would be: $$f= \sum \langle f, \phi_n\rangle\phi_n,$$ then can one use ...
0
votes
1answer
34 views

Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in ...
2
votes
1answer
26 views

Proving that $\langle f, g\rangle = \sum_n \langle f, \phi_n \rangle \overline{\langle g, \phi_n \rangle}$

I have the following problem to solve: If the set of functions $\{\phi_n \}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ and the functions $f, g \in L^2(a,b)$, then show that: ...
1
vote
3answers
52 views

Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval ...
0
votes
0answers
22 views

compute the derivative of a function defined by an integration over the whole real line

Let $h(t)=\int _{\mathbb{R}}e^{-\frac{(x+it)^2}{2}}d\lambda(x)$, where $\lambda$ is the Lebesgue measure. I want to prove that $h$ is differentiable and compute the derivative of $h$: ...
2
votes
2answers
112 views

Greens function of 1-d forced wave equation

[ORIGINAL PROBLEM] You are given hat the Green's function $g(x,t,\xi, \phi)$ is $\frac{\partial^2g}{\partial t^2} - \frac{\partial^2g}{\partial x^2}=\delta(t-\tau)\delta(x-\xi)$ with ...
2
votes
1answer
114 views

$\sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90}$ [duplicate]

How we can do this sum? $$ \sum_{n=1}^\infty \frac{1}{n^4}=\frac{\pi^4}{90} $$ I know that we could possibly use a Fourier series decomposition however I don't know what function to start with. I ...
1
vote
1answer
30 views

Proving that orthonormal set is an orthonormal basis

If I know that the set of functions $\{\phi_n\}_1^\infty$ forms an orthonormal basis on $L^2(a,b)$ and the set $\{\psi_n\}_1^\infty$ is an orthonormal set on $L^2(\frac{a-d}{c}, \frac{b-d}{c})$, with ...
1
vote
1answer
22 views

Proving that a set $\{\psi_n(x)\}_1^\infty = \{\sqrt{c}\;\phi_n(cx+d)\}_1^\infty$ is an orthonormal basis

I have the following problem I need to solve: Suppose $\{\phi_n\}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ (set of square-integrable functions on $[a,b]$). Suppose $c>0$ and $d\in ...
1
vote
1answer
24 views

Why does $\overline{\langle f, \phi_n\rangle }\langle f, \phi_n\rangle = |\langle f, \phi_n\rangle|^2$

If $f(x)$, $\phi_n(x) \in \mathbb{C}$, $\;\;-\infty<a<x<b<\infty$ and $$\langle f, \phi_n\rangle = \int_a^b f(x)\overline{\phi_n(x)}\;dx,$$ then why does: $$\overline{\langle f, ...
1
vote
1answer
37 views

Enlarging the space $PC(a,b)$ to include functions with one or more infinite singularities

I'm reading a Fourier analysis book and on the chapter about convergence and completeness of orthogonal sets of functions I have one part which I don't understand. I have uploaded the part as an image ...
2
votes
3answers
207 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
2
votes
2answers
78 views

Dirac's delta integration

What about the following integral? $$\int_0^a x^3 \delta(x-1) dx$$ If $a$ is more or less than 1 it's all clear, but what if $a=1$. Is the integral is equal to $1/2$ ? Edit: this is my motivation, ...
3
votes
1answer
223 views

Integrate $\int_{-\infty}^{\infty}\exp\left(-\frac{\pi^2t(2x+1)^2}{2c^2}\right)\cos\left(\frac{(2x+1)\pi y}{c}\right)\exp(-2\pi i kx)dx$

By the poisson summation formula we have: $$\frac{1}{c}\sum\limits_{k=-\infty}^{\infty} \exp\left(-\frac{\pi^2t(2k+1)^2}{2c^2}\right)\cos\left(\frac{(2k+1)\pi ...
0
votes
1answer
61 views

Proving that the function set $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set

I have the the following problem from my Fourier analysis book: Show that $\{ (2/l)^{1/2}\sin(n-\frac{1}{2})(\pi x/l) \}_1^{\infty}$ is an orthonormal set in $PC(0,l)$, i.e. class of piecewise ...
0
votes
1answer
19 views

Close fourier transforms implies close time domain functions?

What conditions do we need so that $A(f)\approx B(f) \Rightarrow \mathcal{F}^{-1}\left\{A\right\}(t)\approx\mathcal{F}^{-1}\left\{B\right\}(t)$ The Fourier transforms of my two things look alike. In ...
0
votes
2answers
56 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
0
votes
0answers
24 views

conflictions of analytic functions to the boundary and Schwarz reflection principle

Let $\Omega$ be an open subset of $\mathbb{C}$ and $f:\Omega\longrightarrow \mathbb{C}$ be a holomorphic function. Then for any $z\in \Omega$ and any $r>0$ such that $D(z,r)\subseteq \Omega$, $f$ ...