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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25 views

What is meant by “what value does a Fourier (co)sine series converge to over some interval”?

Let us say there is a function $f(x)$. Let us say that that it has a Fourier (co)sine series representation $$g(x) = \sum_{n=1}^{\infty} a_n\sin(kx) = f(x)$$ I am having difficulty understanding a ...
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1answer
97 views

Fourier series: Understanding a proof

Let $f:[0,2\pi]\to\mathbb{R}$, continuous, such that for all $n\in\mathbb{Z}$:$$\int_0^{2\pi} f(x)e^{i(n+\frac{1}{2})x} dx = 0$$ Prove that $f(x)=0$. The solution: We can rewrite the integral ...
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1answer
35 views

$L_1 \cap L_2$ is dense in $L_2$?

We were talking about Fourier series the other day and my professor said that the requirement that a function be in $L_1 \cap L_2$ wasn't a huge obstacle, because this is dense in $L_2$. Why is this ...
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1answer
45 views

Question about Parseval's theorem

Parseval's theorem claims: $$\sum_{n=-\infty}^\infty \left| \hat{f}(n) \right|^2 = \|f\|^2$$ Isn't the absolute value redundant, because of the square?
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0answers
40 views

Wave Equation with outgoing wave boundary conditions

I need some help with this problem: I have a to solve the wave equation with two initial conditions and with outgoing wave boundary conditions; i.e., $$\begin{cases} u_{tt}-u_{xx} & =0\\ u(x,0) ...
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0answers
35 views

What is meaning of $FFT(\vec{E}(x,y ))$

What is the meaning and how one takes fourier transformation of vector that has spatial distrubution. Let say electric field (with transfer x, y distibution) with direction $$FFT(\vec{E}(x,y ))$$ ...
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1answer
52 views

Fourier series: Show that $\sum_{n\ne 0} \bigl| \hat{f}(n) \bigr|^2 \le 1/(4\pi^2)$

Let $f:\mathbb{R}\to\mathbb{C}$ which is $1$-periodic and $f\in C^1$. Also, $\int_0^1 \left| f' \right| \le 1$. Show that $\sum_{n\ne 0} \left| \hat{f}(n) \right|^2 \le \frac{1}{4\pi^2}$. (*) ...
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1answer
22 views

Showing decay of Fourier coefficients $C_n = 1/2\pi \int_{-\pi}^\pi e^{-inx} \phi(x) dx$

I'm looking at the Fourier coefficients of $\phi \in L^1([-\pi, \pi])$ defined as $$ C_n = \frac{1}{2\pi} \int_{-\pi}^\pi e^{-inx} \phi(x) dx$$I want to show that $\lim_{|n| \to \infty} C_n = 0$ I ...
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1answer
34 views

Variants of Dirichlet's theorem on Fourier series

The following is Dirichlet's theorem on Fourier series: Theorem: If $f(t)$ is a bounded periodic function which in any one period has at most a finite number of local maxima and minima and a finite ...
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0answers
21 views

Convergence of Fourier Series (Dirichlet Conditions)

So in one of the books that I'm reading, there is something called the Dirichlet Conditions (4 conditions), which if the function satisfies these conditions, then its Fourier Series (FS) will converge ...
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2answers
28 views

Removing $e^{-in\pi x/\ell}$ from an integral

I'm considering a proof of the convergence of the Fourier series. It begins by considering the full Fourier series of the periodic extension of $\phi$ defined on $[-\ell, \ell]$. The full Fourier ...
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0answers
41 views

Bernoulli monosplines

Please help me with Bernoulli monosplines. Let's consider $2\pi$-periodic cubic spline, which is consist from $N$ ranges $0<x_1<x_2<\cdots<x_N<2\pi$. We can introduce a periodic ...
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0answers
20 views

Complex serier Fourier

Im Having some problems by calculating some Complex Form of Fourier Series. I did it for $x$ and for $x^2$ with real numbers but now I´m trying to calculate de Fourier Series of $f(x)=x$ in $[- \pi , ...
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0answers
26 views

coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$

I want to prove that $\forall n \in \mathbb{N}$ at least one of the Fourier coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ is greater than 1( The ...
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2answers
32 views

Evaluate $\int_{-\pi}^{\pi} te^{-int} dt $

Evaluate $\int_{-\pi}^{\pi} te^{-int} dt $ Using integration by parts: $$\int_{-\pi}^{\pi} te^{-int} dt = t\frac{e^{-int}}{-in}|_{-\pi}^\pi - \int_{-\pi}^\pi \frac{e^{-int}}{-in}dt $$ Let's ...
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1answer
29 views

Convergence of $\frac{4}{\pi}\sum_{m=1}^{\infty}\frac{2m-1}{4m^2-4m-3}\sin[ (2m-1)x]$

I was reading this question, and made a wrong contribution which I deleted. Now I would like to understand things. Here is the problem: Consider $f(x)=\cos 2x$ on $[0,\pi]$: $f(x)$ is not even on ...
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1answer
51 views

Computing the Fourier series of $f = \cos{2x}$?

I'm currently attempting to solve the following problem: Given the function $f$ defined on the interval $(0, \pi)$ by $f(x) = \cos{2x}$, find the $2\pi$-periodic, even extension of $f$ and compute ...
2
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1answer
30 views

Possible existence of weight function $\rho (t)$

Consider $L^2[-\pi,\pi]$. We define an inner product on this space by $$\langle f,g\rangle=\int_{-\pi}^{\pi} f(t)\overline {g(t)} \, dt \quad\to(1)$$ Suppose if we introduce a weight function ...
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1answer
46 views

Why does a fourier series have a 1/2 in front of the a_0 coefficient

I am reading up on the fourier series, and I keep seeing it as being defined as: $$ f(\theta)= \frac{1}{2}a_0 + \sum_{n=1}^{\infty}(a_n \cos(n\theta) + b_n \sin(n\theta)) $$ where $$ a_n = ...
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2answers
72 views

Fourier Series of $\frac{\sin(x)}{x}$

Good afternoon! My teacher of signals and systems put in my test that calculate the Fourier coefficients for the function $f(x) = \frac{\sin x}{x}$. But ... How I can do? I know that the function is ...
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1answer
17 views

how to find out how many Fourier coefficients there are (which are not zeros)

given a real periodic (with period $T_0$) signal $x(t)$ with fourier transform in which $$X(jw)=0\ \ \forall |w|\ge {6\pi \over T_0}$$ I know that the fourier series will have finite coefficients (5 ...
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2answers
52 views

What's amiss in this Fourier convergence analysis?

I worked out this solution to this basic PDE Fourier series convergence problem, but I suspect the result is "too easy to be correct," because all the answers point to no restriction on either $m$ and ...
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0answers
24 views

Finding Fourier series coefficients numerically

Given a known function $f$, I am wondering how fast (depending on $n$) we can numerically approximate the Fourier coefficients $\int_0^1 f(x) e^{2\pi i n x} \, \mathrm{d}x$, either for fixed $n$ or ...
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1answer
21 views

Taylor series of $r:x \mapsto \begin{cases} e^{-{1\over x}}, & \text{if $x>0$} \\ 0, & \text{if $x \le 0$}\end{cases}$ at $0$

Prove the following lemma: The function $$r:x \mapsto \begin{cases} e^{-{1\over x}}, & \text{if $x>0$} \\ 0, & \text{if $x \le 0$} \end{cases}$$ is $C^{\infty}$ (and x=0),that has ...
0
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1answer
27 views

The product of trig polynomials is a a trig polynomial

Given trig polynomials $$T(x)= \sum_{k=0}^{n} a_k\cos(kx)+b_k\sin(kx) $$ $$V(x)= \sum_{k=0}^{l} \alpha_k\cos(kx)+\beta_k\sin(kx)$$ I want to show that the product $T(x)V(x)$ is also a trig ...
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1answer
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sum of geometric series $\sum_{j=-N}^N e^{i \cdot j \cdot ξ \cdot λ}=\sin(N+{1\over 2}) \cdot (ξ \cdotλ)\over \sin({ξ \cdot λ \over 2})$

Prove that $$\sum_{j=-N}^N e^{i \cdot j \cdot ξ \cdot λ}={\sin(N+{1\over 2}) \cdot (ξ \cdot λ)\over \sin({ξ \cdot λ \over 2})}$$ i think about sum of geometric series but if my thinking is correct how ...
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1answer
18 views

Fundamental Fourier Series Question about a0 and am

Question: Calculate the Fourier series of f (x) = e^x on the interval −π ≤ x ≤ π. I am new to Fourier Series. I managed to find a0 and am. However, I have no idea where does the second am comes ...
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0answers
45 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 ...
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0answers
39 views

Find the Fourier Series of e^x

Hello I am having some issues the following fourier series $$f(x)=e^{x}, -\pi<x<\pi $$ I have no issues with the immediate steps, solving for $a_n$ and $ b_n $, i believe that I am having some ...
5
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1answer
195 views

Coefficient of Fourier cosine & value of full series

I am working on a simple Fourier question from an introductory PDE text by John Davis. The question begins with a graph that can be reduced into piecewise: $$f(x) = \begin{cases} 1, &0 \leq x ...
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0answers
19 views

calculating fourier coefficients for piecewise function

Take the piecewise function: F(x) = 1, x < L/2 and 2, x > L/2 Now a fourier series is defined over a full period of -L < x < L Just using the fourier sine coefficiencts as an example, ...
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0answers
25 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 ...
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2answers
33 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 ...
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0answers
42 views

A discrete fourier-bessel series?

A function $f$ on an interval $[0,b]$ can be expanded as a sum of Bessel functions, using the inner product $$\int_0^b f(x) g(x) x\mathrm dx$$ under which these functions are orthogonal, for example ...
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1answer
30 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 ...
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3answers
47 views

Fourier Series - Periodicity

I don't know if I'm doing something wrong in this exercise. $f(t)=\pi-t$, if $0<t<\pi$ $f(t)=0$, if $\pi<t<2\pi$ I have to find the Fourier Series of $f(t)$ I define the Fourier ...
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2answers
94 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 ...
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1answer
94 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 ...
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1answer
8 views

Am I using sums appropriately in simplifying this Fourier series?

I was reading about Fourier series here when I can across the following sum: \begin{equation} \frac{a_0}{2} + \sum_{n=1}^{\infty} \frac{a_n-ib_n}{2} e^{inx} + \sum_{n=1}^{\infty} \frac{a_n + ib_n}{2} ...
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1answer
31 views

Check if a vector-space is closed or non-closed

Let $V$ denote the subset of $C[-\pi,\pi]$ consisting of all finite linear combinations of functions $1, \cos x, \cos 2x, ... \cos nx, ..., \sin x, \sin 2x, ... \sin n2, ... $ I want to examine if ...
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1answer
81 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 $0$, ...
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1answer
31 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 ...
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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)
3
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1answer
24 views

Legendre-Fourier series for $x^n$

What is the full Legendre-Fourier series for $x^n$? I realize that this depends upon if $n$ is odd or even. Progress I wrote out the first three coefficients for the series, which are dependent ...
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1answer
62 views

Finding Fourier coefficients of $x\mapsto e^{2\cos x}\cos(2 \sin x)$

Consider the function $$g:[0,2\pi]\to \mathbb{R}, \quad x\mapsto2e^{2\cos x}\cos(2 \sin x) -1$$ I would like to find its Fourier coefficients. Since $g$ is an even function, the Fourier coefficients ...
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0answers
55 views

Applying DFT to signal twice

I am doing a excercise on discrete fourier transforms. The excercise asks to find the resultant signal after applying DFT twice. I was able to figure it out by thinking it of in this way, multiplying ...
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1answer
44 views

Using Bessel's inequality to prove the Riemann-Lebesgue lemna

Let $f$ and $f'$ be piecewise continuous function on $[-L,L]$. Use Bessel's inequality to show that $$\lim_{ n\to \infty} \int_{-L}^L f(x)\cos \bigg(\frac{n \pi x}{L}\bigg) dx=\lim_{n\to \infty} ...
2
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0answers
58 views

Basic Fourier analysis questions [closed]

Could someone point me in the right direction to prove the following: Let $f \in L^{1}(\mathbb{T})$ and suppose $S_{N}f \to g$ in $L^{p}(\mathbb{T})$ then $\|f-g\|_{L^{1}(\mathbb{T})}=0$, where $1 ...
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0answers
32 views

Why are complex exponentials orthogonal over sums?

Consider the following sum: $$ \sum_{j=0}^{J-1} \omega_j^k \bar{\omega_j^l} $$ Where $\omega_j^k=\exp \left( \frac{2\pi i j}{J} \right)$. It is easy to show that if $k \equiv l$ modulo $J$, then ...
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1answer
26 views

An example of a function which is not piecewise continuous, but has Fourier series

Would you Please give an example of a function which is not piecewise continuous, but has Fourier series? It means that the coefficient in the Euler-Fourier formulas can be computed. In fact, the ...