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.

learn more… | top users | synonyms

2
votes
1answer
210 views

Fourier analysis questions

Can anyone give me a hand with the proof of this properties? Prove that: a) The linear span of the set $\left\{T_bh/b\in\mathbb{R}\right\}$ is dense in $L_2(\mathbb{R})$, where $h(x)=e^{-\pi x^2}$. ...
1
vote
1answer
91 views

Fourier analysis question, orthonormal basis.

I need some help with this exercise: Given $A>0$, let $L_{A}^2(\mathbb{R})$ the subspace of $L^2(\mathbb{R})$ of the functions $f$ that satisfy $\hat{f}=\chi_{[\frac{-A}{2},\frac{A}{2}]}\hat{f}$. ...
3
votes
1answer
141 views

Fourier series $\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}$

Does anyone know the sum of Fourier series $$\sum_{m=0}^\infty \frac{\cos (2m+1)x}{2m+1}?$$ I tried WA; it does not return a function.
1
vote
1answer
188 views

Parseval's identity

How to prove the Parseval's identity , I know the formal way but how to justify the interchange between the integral and the sum in a rigorously way , in addition what extra condition does the ...
85
votes
4answers
3k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
6
votes
2answers
451 views

How to expand the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} $?

My Question: My Goal is to determine the Fourier series for $f(x)=\max \{0, \frac{\pi}{2}-\lvert x\rvert \} \quad$ for $x \in [-\pi, \pi ]$ This function is $2\pi$-periodic. My Approach: i found ...
0
votes
1answer
75 views

Determining Fourier series for $\lvert \sin{x}\rvert$ for building sums

My math problem is a bit more tricky than it sounds in the caption. I have the following Task (which i in fact do not understand): "Determine the Fourier series for $f(x)=\lvert \sin{x}\rvert$ in ...
2
votes
1answer
165 views

In my Fourier text book, there are the following exercises to prove. why do some of them have the same left side but have different right sides?

In my Fourier text book, there are the following exercises to prove.why do some of them have the same left side but have different right sides? The demand of these question is to prove these ...
2
votes
1answer
46 views

Need to find a Fourier Series…

I am to find a Fourier Series for the following function: $$ y(x)=\sqrt {R^{2}-x^{2}} $$ about $$ -R \leq x \leq R $$ with the recursion $$ y(x+2R)=y(x) $$ Do I let$\sqrt {R^{2}-x^{2}}$equal $y$ ...
5
votes
2answers
505 views

A Fourier series exercise

Can anyone give me a hand with this exercise about Fourier series? Let $f(x)=-\log|2\sin(\frac{x}{2})|\,\,\,$ $0\lt|x|\leq\pi$ 1) Prove that f is integrable in $[-\pi,\pi]$. 2) Calculate the ...
0
votes
0answers
217 views

Fourier series for $e^x$ over $[0,\pi)$

I am trying to solve the following, Find the Fourier series of $h(x) = \text{e}^x, x \in [0,\pi)$. I'm not sure how to approach it since the question does not specify whether to use an even or ...
0
votes
1answer
22 views

Question about a small part of this Fourier series problem

Calculate the Fourier series expansion for the following function of period 2: $f(t)=2+2t^2$ for $-1<t<1$ I just have a small question for this problem. I've already gotten $A_0$ to be ...
10
votes
4answers
422 views

Singular asymptotics of Gaussian integrals with periodic perturbations

At the bottom of page 5 of this paper by Giedrius Alkauskas it is claimed that, for a $1$-periodic continuous function $f$, $$ \int_{-\infty}^{\infty} f(x) e^{-Ax^2}\,dx = \sqrt{\frac{\pi}{A}} ...
1
vote
0answers
118 views

Intervals where the function is similar to the Fourier series

$$f(x)=\left\{\begin{array}{l l} 0,\quad x \in [-L,0[\\ 1,\quad x \in [0,L] \end{array}\right.$$ I need to know in which intervals the sum of the Fourier series is "equal to the function $f(x)$". ...
-1
votes
1answer
47 views

How to prove this Fourier question?

How to prove this Fourier question? I hope for a procedure in detail.
1
vote
0answers
196 views

Fourier Analysis of Prime Counting Function

I was thinking about the following: Denote $\pi(x)$ as the prime counting function such that: $$ \pi(x) = \#\text{ of prime numbers}\leq x $$ It is well known from the prime number theorem that $$ ...
1
vote
1answer
262 views

What is the odd Fourier extension of $\sin x \cos(2x)$?

odd half range extension of $$f(x) = \sin x \cos(2x)\text{ with limits $0$ to $\pi$}$$
2
votes
1answer
55 views

Convergence of Fourier series $\frac 1 {2i} \sum_{n \neq 0} \frac { \exp (inx)} n$

Let $$ f(x) := \begin{cases} -\frac \pi 2 - \frac x 2 && x \in (-\pi,0) \\ \frac \pi 2 - \frac x 2 && x \in (0, \pi) \\ 0 && x = 0 \end{cases} $$ I have to show that $\frac 1 ...
5
votes
1answer
8k views

What is the Fourier transform for $f(x)=e^{-x^2}$

I remember their being a special rule for this kind of function but I cant remember what it was. Anyone know how ? thanks
1
vote
0answers
41 views

Understanding the indices in a Fourier series

Sometimes the truncated Fourier series of a function with Fourier coefficients $\hat{u}_k$ is written $$\sum_{k=-N}^N\hat{u}_ke^{ikx}$$ which is a linear combination of $\cos(nx) +i\sin(nx)$ for ...
0
votes
1answer
52 views

Prove that the given sum is not Fourier series

Book browsing Banach spaces of Analytic function of the author Kenneth Hoffman on page 74 is one example. This example is compiled in this way: Prove that $$ \sum_{n=1}^{\infty}\frac{1}{\log ...
19
votes
1answer
673 views

Seeking Fourier series solution on Laplace equation…still looking, am I on track?

Okay, I've been working at this a couple of days now, I will try to give relevant details but will omit some intermediate steps. The problem as given says: Consider the BVP for $u=u(x,y)$: ...
2
votes
1answer
75 views

Computation of standard series

I am stuck on the computation of the following sum: $\sum_{k=1}^\infty e^{-n^2}\cos(n)$. Simple tricks fail and also i have no idea how to fit it for Fourier series. Are there any other ways?
2
votes
2answers
148 views

fourier series by lebesgue integral

hw: anyone knows how to find fourier series over the function $$ f(x)= \begin{cases} 1 & \text{if $x$ is irrational}\\ 0 & \text{if $x$ is rational} \end{cases} $$ by lebesgue integral? ...
0
votes
1answer
55 views

Show that Fourier coefficients approach zero uniformly

Let $f(t)$, $g(t)$ be piecewise continuous functons on $[-\pi,\pi]$, periodically continued on $\mathbb R$. I want to show that $$ a_n(x) = \frac{1}{\pi} \int\limits_{-\pi}^{\pi} f(x+t)g(t) ...
1
vote
2answers
5k views

Calculate the Fourier transform of ${\rm b}\left(x\right) = 1/\left(x^{2} +a^{2}\right)$

I need help to calculate the Fourier transform of this funcion $${\rm b}\left(x\right)=\frac{1}{x^{2} + a^{2}}\,,\qquad a > 0$$ Thanks.
0
votes
2answers
561 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
0
votes
1answer
92 views

Wave-Function Series?

So I was basically exploring the function: $\displaystyle {\text{frac}(x)}$ which is the fractional part function and I noticed that it has a nice fourier series definition which is: ...
1
vote
1answer
408 views

Fourier series of square wave with DC component (mean component) - amplitude question

Do I subtract the DC component (mean value) from the amplitude in my sine terms? $f(t)=\left\{ \begin{array}{l l} 0 & \quad -5\le t\leq 0\\ 1 & \quad 0< t\leq 5 \end{array} ...
0
votes
1answer
73 views

Fourier transforms - don't understand this concept!!! Please help me on this

I have two Fourier transforms to solve, but the problem is that a I have a characteristic bijection or some etching that I don't know what it is and I don't know how to solve this... Please help ...
2
votes
2answers
1k views

Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...
1
vote
0answers
28 views

Complex Fourier series of a function [duplicate]

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
3
votes
1answer
437 views

Complex Fourier series

I need to find the complex Fourier series of this function, and I'm having problems calculating these integers: $$|a|<1$$ $$x\in [-\pi,\pi]$$ $$f(x)=\frac{1-a\cos(x)}{1-2a\cos(x)+a^2}$$ ...
3
votes
2answers
5k views

Fourier Series of $f(x) = x$

I am having trouble finding the complex Fourier series of $f(x) = x$ and using that complex series to find 1)the real Fourier series of $f(x)$ and 2) the complex and real Fourier series of $h(x) = ...
0
votes
1answer
539 views

Parseval's Theorem Q

I have this question: I know Parseval's theorem is given by $2a_0^2 + \sum_1^{\infty} (a_n^2 + b_n^2) = \frac {2}{T} \int_{-T/2}^{T/2} f(x)^2 dx$, where T is the period. $f(x)$ is even, so I ...
0
votes
1answer
51 views

A function whose derivatives always have a convergent fourier series

I am looking for a solid example that such a function that its derivatives can always be found by taking derivatives component-wisely in its Fourier series. A function with finitely many Fourier terms ...
1
vote
1answer
170 views

Steady Temperature Distribution Pipe

I was wondering if anyone can show me what approach to finding the steady state temperature distribution in this problem. The image is in the link below. ...
2
votes
2answers
132 views

How to solve this equation by Fourier series?

$$ y''+3y=\sin ^4 x ,\quad y=\frac{1}{8} +\frac{\cos2x}{2}-\frac{\cos4x}{104}.$$ Now the text book states the solution, but I don't know the process of solving this equation. I need your help!
2
votes
0answers
295 views

Upper bound on truncation error of a fourier series approximation of a pdf?

Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation: $$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$ ...
2
votes
1answer
192 views

Pointwise convergence of double Fourier series

I'm looking for theorems that deal with the pointwise convergence of double Fourier series expansions for a special class of functions. Let $D \subset [-\pi, +\pi]^2$ be an arbitrary set of finite ...
0
votes
1answer
235 views

Exponential Form of Fourier Series

Problem Suppose $f$ is a continuous function on interval $[-\pi,\pi]$ such that $\sum_{n\in\mathbb{Z}} |c_n| < \infty$ where $c_n = \dfrac {1}{2\pi} \int_{-\pi}^\pi f(x)\cdot \exp(-inx)~dx$, the ...
1
vote
1answer
86 views

Sine Fourier Series? How do I get to this answer?

Calculate the Sine Fourier series expansion for $\displaystyle f(t) = t^2 $ in $\displaystyle 0 < t < 2.$ I know I need to use $\displaystyle ∑ B_n \sin\left(\frac{nπt}{2}\right)$ I know the ...
2
votes
1answer
484 views

Roots of a finite Fourier series?

In general, are there any clever tricks to help find the roots of a finite Fourier series? Presumably there aren't analytic methods, but can we use the fact that our function is a finite Fourier ...
0
votes
1answer
42 views

Pointwise convergence of sine series of $x^{-2}$

I was wondering if the sine series of $x^{-2}$ converges pointwise on the open interval $(0,1)$. What is the most general criterion to ensure pointwise convergence of a Fourier series?
2
votes
1answer
76 views

Computation of $\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy$

Find the value $$\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy,\qquad k\in N^{+}$$ My idea: \begin{align} ...
1
vote
1answer
62 views

Fourier Series Convergence

Going over some revision. Not really sure what to do for the last bit of aii) I know at $x = 0$, it will converge to $0$ and at $x = \frac{M}{2}$ it will converge to $1$, I'm not seeing how this ...
0
votes
1answer
59 views

Calculating value of the fourier series

In the following fourier series, how the red marked numbers are calculated?
0
votes
1answer
80 views

How to calculate integral of fourier series?

I know how to integrate, but I can't understand how the integral of this fourier serie is calculated. my problem is with integral of the sigma. fourier: integral: Can anyone say me how this ...
1
vote
0answers
335 views

How can I find the compact trigonometric Fourier series from these signals?

I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
3
votes
1answer
60 views

How to calculate $\sum_{k=1}^{\infty}\frac{1}{(a^2+k^2)^2}$ after calculating $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Parseval identity?

The task is to calculate sum $\sum_{k=1}^{\infty}\frac{1}{a^2+k^2}$ using Fourier coefficients of $f(x)=e^{ax}$. First of all I calculated Fourier coefficients of the sum: ...