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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32 views

Test question regarding convergence of Fourier series

I'm preparing for a test and I have no clue how I should solve the following question. Let $f:\Bbb{R}\to\Bbb{R}$ be $2\pi \text{-periodic}$ function such that $f(0)=1$ and $$\forall ...
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2answers
42 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
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0answers
25 views

DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
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1answer
27 views

Fourier series coefficients which do not approach to zero

I want to know whether there are a finite number of coefficients in a Fourier series of a periodic function (with period $P$), whose magnitude are above a certain threshold. Those coefficients can can ...
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1answer
45 views

Fourier Series estimation

I know that the Fourier coefficient of $t\mapsto \frac{1}{\sqrt{\vert t\vert}}$ are given by some Fresnel integral, and behave like $O(n^\frac{-1}{2})$. Reciprocally, if I get a Fourier Series whose ...
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2answers
66 views

Prove that $\int_{-\infty}^\infty P_n(x) \, dx = \pi /n$

Let $P_n(x) = \frac{n}{1+n^2x^2}$. Prove that for every $n\in\mathbb{N}$ $$\int_{-\infty}^\infty P_n(x) \, dx = \frac{\pi}{n}$$ And for every $\delta > 0$: $$\lim_{n\to\infty} ...
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1answer
26 views

Fourier series of function $f(x)=0$ if $0 < x \leq L/2$ and $f(x)=1$ if $L/2 < x \leq L$

I am attempting to work through a very simple problem. Determine the Fourier series expansion for: $$ f(x) = \begin{cases} 0 & 0 \leq x \leq L/2 \\ 1 & L/2 < x \leq L\end{cases}$$ I ...
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1answer
14 views

Show that for every real-valued $L^2$ function $u$ on $S^1$ there is a real-valued $v$ in the same space such that $u + iv\in \widetilde{\mathbf H}^2$

For a homework exercise ($1.8$ in the book An Introduction to Operators on the Hardy-Hilbert Space) I am asked to show Let $u$ be a real-valued function in $L^2(S^1)$. Show that there exists a ...
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1answer
39 views

Summation of 1/n^2 using Fourier series on different intervals

I have been going through my notes on complex Fourier series and came across the following anomaly which I hope someone can help me with. I calculated the complex Fourier series for the function ...
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1answer
43 views

Fourier series: Prove the integral is less than $2\pi$.

Let $f\in R(\mathbb{T})$ which is $C^1$. Also, $\int_0^{2\pi} \left|f'(t)\right|^2 dt < 2\pi$ Prove: $\sum_{n\ne 0} \left|\hat{f}(n)\right|^2 < 1$. $\exists c\in\mathbb{C}: ...
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0answers
37 views

For which algebras Taylor series and the Fourier series can be generalized?

I'm not a professional mathematician. The question is in the title. But most of all I'd like to know about this for quaternions algebra with non commutative multiplication. I'd like to know about ...
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0answers
22 views

Frequency spectrum of signal and is it real?

$x(t) = 2 + 5 cos(-t + \pi/4) - 2sin(3t + 5) + 3(cos(5 t + \pi/2).cos(4t) - e^je^t $ a) To find Fourier series coefficients of the following signal I need to use inverse Euler formula. But I need ...
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0answers
39 views

Fourier series of a parabola

$ x(t) = \left \{ A(t-\frac{T}{4})(t+\frac{T}{4}) , -T/4<t<T/4\right. $ I tried using the derivative rule to get the fourier series from the square wave, but the answer does not match up to the ...
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3answers
68 views

Using Fourier Series to Find Infinite Sum

I'm trying to use the Fourier series for $f(x)=x^3$ on $[-\pi,\pi]$ to show that $$\sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{(2k-1)^3}=\frac{\pi^3}{32} $$ I've found the Fourier series to be $$S(f)(x)= ...
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1answer
26 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
98 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
50 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
55 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
36 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
53 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
47 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
28 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 ...
2
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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
42 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
22 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
28 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 ...
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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
48 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
76 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
20 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
54 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
31 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 ...
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1answer
28 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
20 views

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
21 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
52 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
41 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
199 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
21 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
27 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 ...
0
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2answers
36 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 ...
0
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0answers
47 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 ...
0
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1answer
34 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 ...
0
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3answers
53 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 ...