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

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1answer
46 views

How to obtain a periodic function from a rapidly decaying function?

Suppose $f(x) = \exp(-x^2)$ with $x \in [0, 3]$. How could I periodise this function to obtain an analytical form of a continuum periodic function $x \in [0, +\infty)$ with period T = 3?
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1answer
35 views

Series convergence in Hilbert space and dual.

I'd like to prove that: $$ \|u_\varepsilon-f\|_*\rightarrow0 \quad\text{in }V^* $$ with $V$ Hilbert and $V^*$ its dual. In particular $u_\varepsilon\in V$. From the precedent points of the proof I ...
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0answers
43 views

Fourier Series and sum help

I have to find the Fourier series expansion of the function $f(x)$=$x^2$ for $-\pi <x< \pi$ and using it I have to show that, i) $1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}...$ = ...
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0answers
16 views

Pointwise convergence of periodic functions

Let ${f_n}$ be a sequence of functions on $\mathbb{R}$ which satisfy $f_n(x+2 \pi) = f_n(x)$ for all $n$ and $x$. Suppose that $f_n \rightarrow f$. Prove that $f(x+2 \pi) = f(x)$ for all $x$. My ...
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0answers
39 views

Want to prove certain sum representation of $\cot(x)$

So here is my problem, I would like to prove an identity I found in a book which was given without a proof. Namely $$-i\sum_{n\in\mathbb Z} \operatorname{sign}(n)\cdot e^{i2\pi nx}=\cot(\pi x)$$ I ...
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1answer
41 views

Fourier series expansion

Is it possible to have a Fourier sine series expansion like $$ \sin\left(\left(\frac{\pi}{2} + n\pi\right)x\right) $$ instead of the normal $$\sin(n \pi x)$$
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29 views

Finding Fourier Series

I know that this is a rather simple problem but i have some confusion here : I have to find the Fourier series representation of $f(x)=x$ for $-\pi<x<\pi$ and for $0<x<2\pi$. My ...
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2answers
25 views

Fourier Transform of $f(t+a)$ if $f(t)$ has tranform $F(k)$?

I know the formula $$f(t) = \int^{+\infty}_{-\infty} F(k)e^{ikt} \, dk$$ and I've seen that for computing $f'(t)$ it's a case of differentiating $e^{ikt}$ inside the integral, so $f'(t)=ikF(k)$ Can ...
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1answer
32 views

Want to prove that the Hilbert transform of a $C^1(\mathbb T)$ function is the principal value of the convolution with $\cot(\pi x)$

So here is my problem, Let $L^2_0:=\{f\in L^2: \hat{f}(0)=0\}$ and consider the Hilbert transform given by the following map $$H:L^2_0([0,1])\rightarrow L^2_0([0,1])$$ $$f\mapsto (\mathcal ...
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1answer
24 views

Plot a fourier series

I'm not a mathematician but I hope my question is easily answered. I'm trying to learn about graphing equations in a computer application (I'm a programmer). From this link a fourier series is ...
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2answers
23 views

How to see this step in deriving an equality in Fourier series?

(Previous steps are omitted.) By convergence of Fourier series, we have $$ \frac{\pi}{4}+\sum_{n=1}^{\infty}\frac{(-1)^n-1}{\pi n^2}(-1)^n=\frac{\pi}{2} $$ Then how come we can get this from the ...
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3answers
39 views

Finding fourier sine series using another cosine series

I have to find the sine series of $x^3$ using the the cosine series of $x^2/2$. $${x^2 \over 2}={l^2 \over 6}+{2 l^2 \over \pi^2}\left[\sum{(-1)^n \over n^2}\cos\left({n\pi x \over l}\right)\right]$$ ...
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0answers
18 views

Fourier Series from product of to functions

I have to calculate the Fourier Series of $x\sin(x)$ beeing $2\pi$ periodic on $[-\pi,\pi]$and i did it the standard way. But then i wanted to solve the problem with multiplication of two fourier ...
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0answers
23 views

Fourier Series Fourier Transform Method

I understand $f$ is even about $\pi$ but i'm struggling conceptually with the part I have underlined.
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1answer
75 views

How to find the Fourier series of $f(x)=x$?

I got a question which is too simple: Find the Fourier series of $f(x)=x$ in the interval $[-\pi,\pi]$ and show that this function doesn't converge to its Fourier series. I found the series as ...
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3answers
199 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 ...
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1answer
41 views

Finding Fourier of $x^3$ by Fourier of $x^2$

I found the cosine series of $x^2/2$ to be (by first finding Fourier sine series for $x$ on $(0,l)$ and then integrating that term by term) $${x^2 \over 2}={l^2 \over 6}+{2 l^2 \over ...
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0answers
83 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
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1answer
41 views

Series expansion Fourier-Legendre

Can anyone explain me how can I expand this function using the Fourier-Legendre expansion? f(x) = x ; -1<=x<=1 Thanks.
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2answers
32 views

Parallel between Fourier Series and orthogonal projections

My professor made an analogy between Fourier series and orthogonal projections and I was hoping someone could explain that someone more. Basically, as I understand it: $$ a_n = \frac1L \int_L^L ...
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0answers
44 views

Question on Fejér Theorem

I need to solve a problem, which tells me to find the Fourier coefficients of the function $f(x) = |x|, \quad \text{for} \quad x \in [-\frac12, \frac12]$ and show that $$f(x) = \frac14 + \lim_{n ...
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1answer
122 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
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0answers
37 views

Fourier Series to Laurent Series

Given a periodic function $f(\sigma)$ with period $T$, one can compute its Fourier series, $$f(\sigma)=\sum_{n\in\mathbb{Z}} c_n e^{i \omega n\sigma}$$ where $\omega=2\pi/T$ and the coefficients of ...
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1answer
32 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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0answers
27 views

What is the fourier transform of rect( ( x - a) / L)

I thought of it as follows: $$ F \left[ \Pi \left( \frac{x - a}{L} \right) \right] ( k_x ) \\ = F \left[ \Pi \left( \frac{x}{L} - \frac{a}{L} \right) \right]( k_x ) \\ = \exp \left( -2 \pi i k_x ...
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1answer
43 views

determine the Fourier series for the following function

determine the Fourier series for the function up to n= 3 given that $$f(t) = \begin{cases}-2 & \text{ if }\quad-\pi < t < -\frac{\pi}{2}\\ 0 & \text{ if }\quad -\frac{\pi}{2} < t ...
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0answers
20 views

$f\mapsto \sum_{n\in \mathbb Z} |\widehat{F(f)}(n)|$ lower semi continuous?

Let $T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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1answer
41 views

Degree of smoothness of real functions and Fourier series

I was wondering, why is the degree of smoothness $S$ of functions an integer? Why can't there be functions with $S=2/3$ ? The way we determine how smooth a function is by how many continuous ...
2
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1answer
51 views

Fourier Series trouble

"For $f(x) = x^2$ on the interval $[-1,1]$ with period $2$, determine the Fourier series. Show that $\pi^2 / 6 = \sum_{n=1}^{\infty}(1/n^2)$". How is the first part of this exercise related to the ...
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1answer
45 views

Separation of variable of PDE

For any $u_0,u_1\in L^2(0,\pi)$ and $f\in L^2((0,\pi)\times(0,+\infty))$ find using separation of variables and Fourier series a formal explicit expression of the solution of the problem ...
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2answers
207 views

How to sketch a graph for fourier series

I had to find fourier series for $f(x) = x$, $-\pi < x < \pi$. I found that the Fourier series for $f$ is $$\sum_{n=1}^{\infty}(-1)^{n+1}\cdot\frac{2}{n}\cdot\sin(nx).$$ Now I have to sketch ...
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2answers
21 views

Fourier coefficients assume a maximum and minimum?

Let $f:\mathbb R\to\mathbb R$ be continuously differentiable and periodic with period $2\pi$. The Fourier coefficients are defined by $$\hat f_n=\int_{-\pi}^\pi f(x)\exp(-inx)dx$$ My questions: Is ...
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0answers
71 views

The coefficients of the Fourier series of the product of two real valued functions

Consider two piecewise continuous, twice integrable functions $f, g: [-\pi, \pi] \rightarrow \mathbb{R}$, and suppose they have the following convergent Fourier series expansions: $$ \begin{aligned} ...
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1answer
40 views

Dirichlet vs Neumann Boundary Conditions of a PDE?

Say I have the PDE $$u_{tt}=4u_{xx}, u(x,0)=f(x),u_t(x,0)=g(x), 0<x<L$$ How does the solution change if I am given the boundary conditions $$u(0,t)=u(1,t)=0$$ versus $$u_x(0,t)=u_x(1,t)=0$$? ...
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1answer
134 views

proof following Bessel's inequality and Fourier series

Letting $a_n$ denote the coefficients in the Fourier cosine series of any function $f(x)$ on $(0,\pi)$ , how can I show that: $\sum_{n=1}^N c_n^2 \le ||f∥^2$, where $c_n$ are the Fourier constants ...
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1answer
68 views

Plotting partial sums of the fourier series

I need to find and plot the fourier series of $\sin^{2}(x)$. I know that the Fourier Series for this function is clearly $\frac{1}{2} - \frac{1}{2} \cos(2x)$ which is the reduction formula for ...
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0answers
12 views

fourier series of $f \circ R_\alpha$

I have a problem with the following: If $f \in L^2 (T)$ and $\displaystyle f(x)= \sum_{n=-\infty}^{\infty} a_n e^{2 \pi i nx } $ in $L^2(T)$ the Fourier expansion, then why the Fourier expand of $ f ...
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1answer
25 views

Not understanding one step in derivation of Dirichlet kernel

I was reading some notes on the Dirichlet Kernel and they have a proof of how it reduces to $\sin(2\pi(N+ 1/2)t)/\sin(\pi t)$. I could follow the steps except for one early step which is the ...
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0answers
82 views

What is the Fourier series of $e^{\mu\cos\theta}$?

Motivation: I want to solve this convolution problem on the circle: find $f$, given $g$ and $$ g(\theta) = \int_{S^1} e^{\mu\cos(\theta-\phi)}g(\phi)\ d\phi. $$ To do this, I want to find the Fourier ...
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0answers
48 views

Full series expansion of the floor function

We know if $x$ is not an integer we have $$\left \lfloor x \right \rfloor=x-\frac{1}{2}+\frac{1}{\pi }\sum_{k=1}^{\infty}\frac{\sin(2\pi kx)}{k}$$ Is there an series expansion of floor function ...
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1answer
36 views

Question about the frequency domain and the fourier transform

if you have a signal say x(t) in continuous time and you transform it using the Fourier transform for continuous time you get X(w) which is the frequency domain representation of this signal x(t). ...
0
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1answer
30 views

Complex Fourier Series of $|x|$

How would I write the Fourier series for $|x|$ in complex form over the interval $[-2,2]$? I have already tried writing $$|x|=\sum c_ne^{i\pi nx/2}$$ where ...
4
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1answer
53 views

Leibniz series for $\pi$ using an integral of the Dirichlet kernel

I'm trying to create a proof of the Leibniz series $\sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} = \frac{\pi}{4}$ using the Dirichlet Kernel. What I did is start with the kernel $$1+2 \left ( 1+\cos\theta ...
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2answers
128 views

Are these statements of my professor about periodicity of harmonic processes in time series analysis correct?

Assume $X_t$ is a harmonic stochastic process, i.e., $$X_t = \sum_{j=-k}^k A_j \exp(i \lambda_j t)$$ where the frequencies $\lambda_j$ are given and $A_j$ are uncorrelated random variables with zero ...
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2answers
41 views

fourier series, parsevel's identity

i need to solve the following question using parsevel's identity. $\displaystyle\int_{-\pi}^{\pi} \cos^{4} x\, dx = \frac{3\pi}{4}$. $\displaystyle\int_{-\pi}^{\pi} \cos^{6} x\, dx = ...
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1answer
72 views

Is it true that the Fourier coefficient of convolution is the product of the coefficients?

what I mean by the title is the following: if we define the convolution between two $2\pi$-periodic, $C^1$ functions as $f*g(x) = (2\pi)^{-1}\int_{-\pi}^\pi f(x-y)g(y)dy$, is it true that the Fourier ...
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1answer
137 views

How to find the Fourier series of a periodic function

Find the Fourier series of the function $f(t)=3t^2$, $-1\le t\le 1$. How do I solve this problem? What is the general formula and the way to solve this?
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2answers
49 views

fourier series analysis, show that for every integer n, using euler's formulas relating trigonometric and exponential functions

Show that for every integer $n$, $$\int_0^{\pi} \cos nt~\sin t~\mathrm{d}t = \begin{cases} \dfrac{2}{1-n^2} & \text{if } n \text{ is even} \\[10pt] 0 &\text{if } n \text{ is odd} ...
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0answers
22 views

When do Fourier series and Fourier transform coincide

The other day I proved that if $f \in \ell^1 (\mathbb Z)$ then its Gelfand transform $\widehat{f}$ is a map $S^1 \to S^1$ such that $$ \widehat{f}(z) = \sum_{k \in \mathbb Z}f(k) z^k$$ and that ...
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0answers
22 views

What is the formula for single frequency generation function obtained from FFT?

What is the correct formula of a function that generates specific tone from fourier transform? I thought that having: transformata - an array with FFT of a source sample. v = transformata[freq] - ...