Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials or sines and cosines are used.

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2
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2answers
38 views

Why does $\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$

Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$ This is used in a derivation of the Fourier coefficients. I see why ...
2
votes
1answer
17 views

Unitary Operator on Hilberspace to show that Fourierbasis is a maximal Orthogonal Set

I have looked at the proof Proving that the Fourier Basis is complete for C(R/$2*\pi$ , C) with $L^2$ norm but am having trouble understanding the argumentation about the Hilberspace. I think the ...
0
votes
0answers
51 views

An exercise from stein's fourier analysis

I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows: Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in ...
0
votes
1answer
52 views

Solving a simple trigonometric equation for coefficients

Is it possible to solve the equation $$ a \sin x + b \cos x + c \cos^3 x = d \cos x $$ where $c\neq0$ using some coefficients $a$, $b$, $c$, and $d$? I can't see how to make the frequency of ...
1
vote
0answers
22 views

Determining coefficients to a nonhomogenous differential equation.

Consider the following ODE: $$\frac{d^2 x}{dt^2} + kx = f(t)$$ where $$f(t) = \frac{1}{2} +\sum_{n = 1}^{\infty}{ \frac{-4}{n \pi} \sin\left(\frac{n \pi t}{2}\right)}$$ was derived using fourier ...
1
vote
1answer
43 views

Algebraic way to see why only $n=3$ is a valid coefficient

I'm a bit of a sucker for brute force calculations. Say I want to calculate a coefficient with Fourier theory, in my case \begin{align*} a_n = \int_0^1 \sin (3\pi x) \cos (n\pi x) dx. \end{align*} ...
0
votes
2answers
30 views

Find the Fourier series for the function $f(x) = x^4$

How do I start solving this question, what are the steps? a) Find the Fourier series for the function $f(x) = x^4$ on the interval $[−π, π]$. b) Hence prove that ...
2
votes
1answer
34 views

Particular solution to a non-homogenous differential equation

Consider the following ODE: $$\frac{d^2 x}{dt^2} + kx = f(t)$$ where $$f(t) = \frac{1}{2} -2 \pi \sum_{n = 1}^{\infty}{n\sin\left(\frac{n \pi t}{2}\right)}$$ was derived using fourier series. What ...
0
votes
1answer
57 views

Needs some hints regarding Fourier series

Determine the fourier series for the function defined by: $f(x) = 2x$ for $0 < x < 2\pi$, and $f(x+2\pi) = f(x)$ The Fourier series coefficients are defined as follows: $$A_0 = ...
0
votes
0answers
111 views

Extracting coefficients from a complicated Fourier-Bessel series

I'm solving a problem for my research that involves extracting the coefficients $A_n$ from the following Fourier-Bessel series (with several physical constants omitted for simplicity): $$ ...
0
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0answers
5 views

Independence of two Fourier transfromed random variables

Assume that there is some white noise $Z(\phi)$ on a circle parametrised by an angle $\phi \in [0,2\pi[$. The expectation values of white noise are $$E[Z(\phi)] = 0$$ $$E[Z(\phi)Z(\phi')] = ...
3
votes
1answer
75 views

Hilbert Space multiplication Operator, shift operator

I have this problem and am not sure how to even approach it.. Hilbert space $l^2(\mathbb{Z})$ with orthonormal basis$ $$(e_n)$ and Hamiltonian operator $He_n=i(e_{n+1}-e_{n-1})$ a)I need to use ...
0
votes
1answer
37 views

Fourier transform with second order differentials

How do I start this question: $$\frac{d^2y}{dx^2} = yx$$ for a function $y(x)$ which tends to zero as $x \to \pm \infty$. Show that transform of $\hat{y}(k)$ of $y(x)$ satisfies the first order ...
8
votes
5answers
186 views

How to evaluate $\int_{0}^1 {\cos(tx)\over \sqrt{1+x^2}}dx$?

mean to calculate the integral $$\int_{0}^1 {\cos(tx)\over \sqrt{1+x^2}}dx$$ I have tried to consider the integral as fourier coefficient of $f(x)=1/\sqrt{1+x^2}$, however no ideas. Besides, let ...
0
votes
1answer
18 views

Problem about Fourier series and $L^p$ spaces

Need some help with this problems: Is there $f \in C(\mathbb{T})$ such that $\hat{f}(k) = \dfrac{1}{|k|^{1/2}}$, if $k \neq 0$? Suppose the $f_n \in L^1(\mathbb{T})$, $n = 1,2,...$ and $\| ...
0
votes
3answers
36 views

What functions does the Fourier Series work for?

Let us say we have a function $f[a,b]\rightarrow \mathbb{C}$ and we want to find the Fourier series for this function in the interval $[a,b]$ what is the condition on $f$ such that the Fourier series ...
2
votes
0answers
302 views

Intuition behind the DTFT vs Fourier transform of ideally sampled signal

So I am taking a signal processing course in EE and my professor is an Engineer who reallly likes math however his book which we use for the class falls in the dreadfull purgatory of math books in my ...
0
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0answers
19 views

Matrix form of Fourier and Fresnel Transform

I am wondering how to write up the matrix form of, say, Fresnel or Fourier transform. I know that for the case of Fresnel it would be Toeplitz matrix.
-1
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1answer
21 views

Change of scale of periodic function

Could someone show how to prove the following: If $f(x)$ has a period of $p$ show that $f(kx)$ has period of $p/k$.
0
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0answers
19 views

Locating a Reverse “Sawtooth” Wave Function

I am trying to ascertain the best means of determining the function of a non-sinusoidal waveform or "reverse sawtooth wave". I have graph points that produce: function(24) with graph points at ...
0
votes
0answers
19 views

Fourier transform of $\frac{1}{a+b\cos(d x)}$

Is it analytically possible to calculate the Fourier transform of $$\frac{1}{a+b\cos(d x)}$$ where $a, b$ and $d$ are constants and $x$ is the variable. If so, how? Or at least what's the approach? ...
0
votes
0answers
29 views

Find the Fourier Series of f(x)

I am trying to solve the following fourier series: $$f(x) = \begin{cases} \frac{x}{l} &\mbox{if } 0 \leq x \leq l \\ \frac{2l - x}{l} &\mbox{if } l \leq x \leq 2l \\ \end{cases} $$ I know ...
2
votes
1answer
35 views

Solving PDE with Separation of Variables

I have done a few of these now but I'm stuck at this one $$\begin{array}{} u_{xx}=u_{tt}+2u_t \\ u(0,t)= u(\pi,t)= 0 \\ u(x,0)=0, \, u_t(x,0)=\sin ^3x \\ \end{array} $$ for $0<x< \pi$ and ...
0
votes
2answers
30 views

Hyperbolic or exponential solutions to differential equation

I have spent the last couple weeks in my Fourier Analysis course to solve PDEs with the method of separations of variables. However, I have come up with something that annoys me and I can't really ...
2
votes
0answers
23 views

Estimating number of terms for partial fourier sum to minimize error

A function is define as $$ f(t) = \begin{cases} 0 & \text{ if } \pi<x\le -1 \\ x^2 & \text{ if } -1<x<1 \\ 0 & \text{ if } 1\le x < \pi \end{cases} $$ Find the ...
0
votes
0answers
20 views

Getting sum of $2 \pi$ periodic function

I have a $2\pi$ periodic function which in the interval $[0,\pi]$ is $f(t) = \sin{\frac{t}{2}}$. I have to find the sum for $t \in \mathbb{R}$. But do I know anything about $f(t)$ outside of $t \in ...
0
votes
0answers
23 views

Name for a sinc-like function

As far as I understand, with some real number $a$, the function $$f(x) = \frac{\sin(ax)}{ax}$$ is called 'sinc' function. Is there a name for a function like the following one? $$g(x) = ...
2
votes
2answers
66 views

how do I verify that this converges uniformly to $f(x)$?

I had to find the fourier series for $f(x)=|x|, -\pi \le x \le \pi$ I got the fourier series as $$f(x)=\frac{\pi}{2}-\sum_{n=0}^{\infty}\frac{4}{\pi(2n+1)^2}cos((2n+1)x)$$ but now I have to verify ...
0
votes
0answers
14 views

Solve for a Fourier series problem

I am learning Fourier series and I have a problem which has me confused and would like to here others take on it. $$F(t)=\begin{cases}v_0 & \ \ 0 \le t\le T\\ 0 & \ \ T\le t \le \ ...
1
vote
1answer
70 views

Show that $\frac{1}{2} + \sum_{j=1}^\infty \cos(jx) = 0$ in the Cesàro sense

So, we have $s_n = \frac{1}{2} + \sum_{j=1}^n \cos(jx)$ and we wish to show that $s_n = 0$ in the Cesàro sense. My understanding of the Cesàro method is to take $\frac{\sum_{n=1}^N s_n}{N}$ which ...
0
votes
1answer
45 views

Calculation of Fourier series

Let us define function $$V(t) = \begin{cases}3,& \text{for $0\leq t < 6$}\\ 4,& \text{for $6\leq t < 12$} \\ 3,& \text{for $12\leq t < 18$} \\ 0,& \text{otherwise}. ...
7
votes
2answers
87 views

Absolute convergence of fourier series

Let's say I have a piecewise continuous function which has the fourier series $\sum_n\ c_{n}e^{inx}$ and I assume that $ \sum_n\ n|c_{n}|$ converges, then I know the following holds: The fourier ...
1
vote
1answer
37 views

Prove that $\frac{1}{2\pi}\int_0^{2\pi}|p(e^{i\theta})|^2\,d\theta=\sum_{n=0}^N|a_n|^2.$

Let , $\displaystyle p(z)=\sum_{n=0}^Na_nz^n$ be a complex polynomial. Then show that , $$\frac{1}{2\pi}\int_0^{2\pi}|p(e^{i\theta})|^2\,d\theta=\sum_{n=0}^N|a_n|^2.$$ ...
0
votes
1answer
28 views

Fourier series of a truncated Gaussian

I came across the following discussion concerning the computation of the Fourier series of a truncated Gaussian: Fourier transform of a truncated Gaussian function Numerical simulations suggest that ...
1
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1answer
32 views

Fourier cosine series expansion of $f(x)=1$

Fourier cosine series expansion of $$f(x)=1,~~~ x\in (0,\pi)$$ Hint is "thought is better than calculation".
1
vote
1answer
15 views

Short-Time Fourier Transform - why does the index range from negative to positive infinity?

I'm new to Fourier Transform. Could anyone explain to me in the Short-time Fourier Transform Equation (wikipedia): $$STFT\{x[n]\}(m,\omega) \equiv X(m,\omega) = \sum_{n = -\infty}^\infty ...
0
votes
0answers
21 views

Detailed study of the continuity of $f(x)=\sum_{n=1}^\infty (-1)^n \frac{f_1(nx)}{n}$

Let $f_1(x) = \vert x \vert -\frac{1}{4}$ for $\vert x \vert \le \frac{1}{2}$ and let $f_1$ be defined for other values of $x$ by periodic continuation with period $1$. Then $f$ is defined on ...
0
votes
2answers
16 views

Is it possible to write a sinusoid of one frequency $f_1$ as a linear combination of sinusoids of different frequency than $f_1$?

If this is impossible (which I suspect it is), how can we formally show this using the notions of Fourier series and/or Fourier transform?
0
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0answers
25 views

Show that the series converges and is equal to the following.

Define $F(x) = i(-\pi - x)$ if $-\pi \leq x < 0$ and $F(x) = i(\pi - x)$ if $0<x\leq \pi$, with $F(0) = 0$. Show that if $x \neq 0$ mod $2\pi$, then the series $E(x) = \sum_{n=1}^{\infty} ...
1
vote
1answer
16 views

Finding Coefficients of a Double Fourier Series Related to Bessel Function

This is from the lecture notes of MIT. I am confused about the sentence "where the $a_{nm}$ and $b_{nm}$ are found from the ICs". It is from problem 3 in this pdf. I tried to use ...
0
votes
2answers
10 views

Why is $2Re\{c_n e^{\frac{j2n\pi t}{T}} \}$ equal to $2Re\{c_n \} cos{\frac{2n\pi t}{T}} - 2Im \{ c_n \} sin{ \frac{2n\pi t}{T}} $?

Why is $2Re\{c_n e^{\frac{j2n\pi t}{T}} \}$ equal to $2Re\{c_n \} cos{\frac{2n\pi t}{T}} - 2Im \{ c_n \} sin{ \frac{2n\pi t}{T}} $ in the Fourier series? See images below: $C_n$ Derivation
1
vote
0answers
30 views

Fourier series concerning Gibbs constant and the divisor function.

It is quite a remarkable function I found. It seems, though, that I may be staring at something trivial, which is hopefully not the case. I would like some opinions. The function is ...
0
votes
1answer
26 views

Writing Sine and Cosine functions as Fourier Series

What are the fourier series for: $\sin(\pi*x)+\cos(3\pi x)$ and $\sin(3x)$ I was taught that the purpose of the Fourier series was to describe periodic functions in the form of an infinite sum of ...
0
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0answers
27 views

Fourier Series and epicycles - How to extract the radii and angular velocities from the Fourier Series expansion of a function.

NOTE: I am attaching Mathematica code for those who may want to check it out and understand what I'm asking for. The rest of the question is pretty mathematical in nature, I'll also try the ...
0
votes
1answer
16 views

Calculate the Fourier Transform of $A(x)$

I want to calculate the fourier transform $\hat{A}$ of the following function: $$A(x) = \begin{cases} 1, & \text{if $\lvert x \rvert \le \frac{b}{2}$ } \\[2ex] 0, & \text{else} \end{cases}$$ ...
1
vote
1answer
31 views

Nonhomogeneous pde - problem with Fourier series with cosine

I've been trying to solve the following equation $$u_t - a^2 u_{xx} = tx, \ \ \ x \in (0, \pi), \ \ t>0$$ $$u_x(0,t)=u_x(\pi, t) = 0, \ \ \ t \ge 0$$ $$u(x, 0) = 1 \ \ \ x \in (0, \pi)$$ So I ...
0
votes
0answers
13 views

Power Spectrum and Line Spectrum (Fourier Series)

I want to know more about Fourier Series and about some applications. Which is the difference between Power Spectrum and Line Spectrum? Which is the computing method? Thanks!
1
vote
1answer
21 views

Equivalence of two definitions of Fourier Series

I want to know why the following two definitions of Fourier series are equvalent: 1. $\displaystyle f(t)=\frac{a_0}{2}+\sum^{\infty}_{n=1}{(a_n\cos n\omega t+b_n\sin{n\omega t}})$ 2. $\displaystyle ...
0
votes
0answers
20 views

Show $|\frac{e^{-ixs}-1}{s}|\leq |x|$ , $x,s$ real. [duplicate]

I am writing some code on a minor project for the fourier transform, and I've noticed that this inequality is true, I've also ran across this in textbooks, like Big Rudin, when describing the ...
0
votes
0answers
20 views

Fourier series - showing that an expression is true

I have this function: $f(t) = \frac{1}{12} (\pi^2 t-t^3)$ and I have a Fourier series defined by: $$f(t) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^3} \sin (nt)$$ I am then suppose to show that ...