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
14 views

Uniform convergence of the series $\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $

I am trying to find if this series is uniformly convergent: $$\sum_{n=1}^{\infty} \frac{\cos(2nt)}{4 n^2 - 1} $$ So far I have (using the Weierstrass M-Test): $$| \frac{\cos(2nt)}{4 n^2 - 1}| \le ...
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0answers
11 views

Expand function using Maclaurin's series(infinite form)

Expand the function f(x)=log(1+x) in powers of x in an infinite series stating the validity of such expansion for x belonging to (-1,1]. The question actually asks to show that cauchy's remainder or ...
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0answers
14 views

Fourier coefficients of a (finite, regular, positive) measure are absolutely summable => the measure has a density

Let $\mu$ be a finite, regular, positive measure on $[0,1)$ such that $\sum_{n\in\mathbb{Z}} |\hat{\mu}(n)| < \infty$. How can I prove that there exists $f(x)$ such that $\mu(dx) = f(x)dx$? ...
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0answers
25 views

Fourier series of $\phi(t) =0$ if $ -\pi \lt t < 0$ and $\phi(t)=\sin(t)$ if $0 \le t \lt \pi$ [duplicate]

I am having trouble finding the Fourier series of: $$\phi(t) = \begin{cases}0 & -\pi \lt t < 0 \\ \sin(t) & 0 \le t \lt \pi \\ \end{cases}$$ I am trying to use: $$\phi(t) = ...
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0answers
6 views

What's the best way to recognize a shape o a function with N-points

I've many shapes with points in theirs countours, how is the best way to recognize a shape? I think the DTF is available but i don't know whether this is the optimal way. P.S. I think if i will ...
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0answers
18 views

Why does the point spread function not violate the linearity of the Fourier transform?

In radio astronomy the point spread function is the Fourier inverse of the $uv$-sampling function of a telescope. The $uv$-sampling function is a sum of sampling functions (one for each baseline). So ...
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0answers
28 views

Fourier Series in different forms

I am trying to write the Fourier series of $(1-x)$ in $[0,1]$ in two different ways: $$f(x)=\frac{a_0}{2}+\sum_{n=1}^\infty (a_n\cos(2\pi nx/L)+b_n\sin(2\pi n x/L)),$$ $$f(x) = ...
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2answers
26 views

How to orient the integration contour when applying the residue theorem?

For fixed $s$ and $k$ real positive numbers, I consider the $2\pi$-periodic function $f:\mathbf R\to\mathbf C$ defined by $$f(x)=\frac1{s+\mathrm i k\cos x}$$ and want to compute its Fourier series ...
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0answers
20 views

How Many Negative Eigenvalues of $-\frac{d^{2}}{dx^{2}}$ on $[0,L]$?

What is the maximum number of eigenvalues $\lambda < 0$ for the trigonometric problems?: $$ \begin{array}{c} -\frac{d^{2}f}{dx^{2}}=\lambda f,\\ ...
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2answers
41 views

A simple Fourier Transformation

I am a bit stuck with this small basic signal. I have this $$y(t)=\frac{\sin(200\pi\,t)}{\pi\,t}$$ and I want to take its Fourier Transformation. Obviously it looks like the sinc function. But that ...
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1answer
13 views

Solving the wave equation bounded by one free end and one fixed end

Given that $\{\sin\left[\frac{(2n-1)\pi}{2L}x\right] : n\in\mathbb N\}$ is the complete set of eigenfunctions of a regular Sturm-Liouville with boundary points $0$ and $L$ and weight function $1$, and ...
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1answer
43 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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1answer
16 views

Fourier transform and Z transform question?

Lets suppose we have an exercise where I have to find the Z transform and its region of convergence.I find the Z transform and the region.How do I determine if the Fourier transform exists from this ? ...
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1answer
27 views

How to prove that this equality is the development of a fourier series?

how can I show that this identity is a development of a fourier series? $$f(x)=\sin^3 x=\frac{3}4 \sin x-\frac{1}4 \sin 3x$$ I tried this: obtain the Fourier coefficients whih $$b_n=\frac{2}\pi ...
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0answers
24 views

Write the Fourier series to $f(t)=|\sin t|$

I have this function which I should write the Fourier series for: $f(t)=|\sin t|$ I now that the period is $\pi$ and that it is an even function. Because it is even, I only need to calculate the cos ...
2
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1answer
96 views

Fourier serie of $\sqrt{1 - k^2 \sin^2{t}}$

I'm struggling with a Fourier serie. I need to find the Fourier series of the following function. That's the function under study: $f(t)=\left[\sqrt{1-k^2\sin^2t}\,\right]$. The function ...
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1answer
75 views

How to get fourier series of 8-bit character to be transmitted?

I have been reading this in a book, but can't understand how he used the 8-bit in fourier series equation to get the result below. The transmission of the ASCII character ‘‘b’’ encoded in an 8-bit ...
2
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1answer
29 views

2 similar question about how to find the $a_n$s and $b_n$ of a Fourier series

Find the terms $b_n,\ n\geq 1$ so that $$x-\frac{\pi}{2}=\sum_{n=1}^{\infty}b_n \sin nx$$ for all $x\in (0,\pi)$. A similar one: Find the term $a_n, \ n \geq 0$ so that ...
2
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1answer
37 views

Calc $\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$

The answer to this integral:$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty -\frac{t}{1+t^2}(\delta (\omega-t-\pi)-\delta(\omega-t+\pi))dt$$ is ...
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0answers
19 views

Need a closed form for fourier coefficients (if it exists)

i have a set of 53 fourier coefficients. the dc term is 0. the 26 positive frequency amplitudes (coefficients) are given below. the 26 negative frequency amplitudes are the same. {0.014451, ...
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0answers
16 views

Iterating a correct Sigma sign to odd zeros.

Trying to compute this integral for my $b_n$'s i a Fourier series exercise I came out with this, eventually: $$b_n=\frac{8}{\pi n^3}$$ for odd integers and $$b_n=0$$ for even integers [Which is ...
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1answer
43 views

Proving $\sum_{k=1}^{\infty}\frac{\sin kx}{x}=\frac{\pi-x}{2}$ for $0\le x\le 2\pi$

Refer to this OP: Sign of a series, we have the following equation \begin{equation} \sum_{k=1}^{\infty}\frac{\sin kx}{k}=\frac{\pi-x}{2} \end{equation} defined for $0\le x\le 2\pi$. Here is ...
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1answer
26 views

Fourier Transform and $f*g$ convolution

Given the 3 following: $$\mathfrak{F}(e^{-|t|})=\sqrt\frac{2}{\pi}\frac{1}{1+\omega^2}$$ $$\mathfrak{F}(r(t))=\sqrt\frac{2}{\pi}\frac{\sin \omega}{\omega}$$ where $$r(t)=\left\{\begin{matrix} 1, ...
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1answer
42 views

What do I do with $f(x+1)=f(x)$, seems to be a fourier question

It would seem having $f(x+1)=f(x)$ should just give me a straight line, since say $f(1)=2$,$f(2)=f(1)=2$ etc. So all $x$ are assigned to the one $y$ value, hence (here) I would have the line $y=2$. ...
-1
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1answer
51 views

Fourier series of rescaled cosine function [closed]

How would I find the Fourier series of $\cos\left(\, 5x/2\,\right) $ on $\left[-\pi,\pi\right]$? Progress $$A_0={1\over 2\pi}\int_{-\pi}^\pi \cos(5x/2)dx={2\over 5\pi}$$ $$A_n = {1\over \pi} ...
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2answers
30 views

What is the $L$ in the Fourier series term?

I am a bit confused about this: I want to calculate the Fourier series $S^f$ of $f(x)$, where $f$ is periodic with period $k\in \mathbb{R}$. I know that the equations for my terms are: ...
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1answer
54 views

Calculate $\int _0^\infty \frac{\sin^2 (\pi \omega)}{(\omega ^2 -1)^2}d \omega$

We define $f: \mathbb{R}\to \mathbb{R} $ by: $$f(x)=\left\{\begin{matrix} \sin x, & |x|\leq \pi\\ 0, & |x|>\pi \end{matrix}\right.$$ A. Find the Fourier transform of $f$. Answer: ...
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2answers
21 views

Plotting a $\cos$ function within a specific domain

Lef $f$ be an odd function with period $\pi$ defined by $f(x)=\cos(x)$ where $0<x\leq \pi/2$. Plot the graph of $f$ on $[-\pi, \pi]$. The answer in my book is this: But I don't understand why it ...
4
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1answer
47 views

Fourier series of $\sin(x)$

I know that this series has been calculated here for more then one time but I need help with a specific thing. We define $f$ as an even function with period $2 \pi$ by $f(x)=\sin (x) $ where $0 \leq ...
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0answers
10 views

Generalized Fourier transform for any periodic function

What is the general transform that represents any kind of wave as a sum of infinitely many waves of another kind? For example, a sine wave as a sum of triangle waves.
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0answers
12 views

Problems filtering a signal

The signals I'm inspecting are taken from an accelerometer. Up until now I've been filtering noise by decomposing the signals using a Fourier series and getting rid of all modes greater than a certain ...
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1answer
31 views

Degrees of freedom in each domain in Discrete, Continuous and Mixed Fourier Transforms

I'm having trouble with the different infinities involved in the Discrete and Continuous Fourier Transforms. In the DFT, we have a finite number $N$ time domain samples $x(i), 0\leq i<N$, which ...
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2answers
16 views

Heat Equation Existence of Fourier Series

I'm currently doing a bit of digging with the Heat Equation and the Fourier Series. It seems that the boundary condition $u(x,0)=f(x)$ can be arbitrary. At some point, we get something like (in a ...
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1answer
27 views

simplifying an expression with even and odd integers

I got this expression for my $b_n$ to a Fourier series: $$b_n=\frac{(2- \pi^2 n^2)\cos(\pi n) -2}{4( \pi n)^3}$$ Now I want to write it in a closed form without the use of $\text{when } n \text{ is ...
3
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1answer
13 views

Fourier series to a function

Assume that $f(x)$ is periodically extended outside the original interval. Find the Fourier series of the extended function. $f(x)=2(1-x^2), -1\leq x<1$ So I find that $a_0 =\frac{4}{3}$ and to ...
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0answers
19 views

Does $\mu\coth(\mu)=A\mu^{2}+B$ have at most two positive solutions $\mu$?

Is it true that $$ \frac{\mu\cosh(\mu)}{\sinh(\mu)} = A\mu^{2} + B $$ has at most two solutions $\mu > 0$ for any choice of $A$, $B$? I believe this is true; it looks true when I ...
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1answer
26 views

Solve this heat equation using separation of variables and Fourier Series

I'm working on a practice question and just a little confused at some parts, would greatly appreciate some help. Here is the question: $ \frac{\partial u}{\partial t} = K \frac{\partial^2 ...
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2answers
54 views

what value this equal $\sin\left((2n-1)\frac{\pi}2\right)=$?

The answer is $(-1)^n$ And why is this wrong? $$\begin{align}\sin\left[(2n-1)\frac\pi2\right]&= sin(nπ-π/2)\\ &= \sin(n\pi) - \sin \left(\frac\pi2\right)\\ &= 0 - 1 = -1 ...
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1answer
48 views

To what value will the series converge?

I have done a Fourier series expansion and get $$\frac{12}{\pi(2n-1)}\sin((2n-1)x)$$ How to find the value it converges at $x=\frac{\pi}{2}$? isn't it divergent? Please show me the correct way step by ...
2
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1answer
72 views

Convergence of a Series involving Cos and Log

Does the following series converge? $$\sum_{k=1}^{\infty}\int_0^{\pi}\int_0^{\pi}\cos(2k(x-y))\log\big(\sin|\frac{x-y}{2}|\big)\,dx\,dy$$
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1answer
43 views

proving Orthonormal basis

I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions: $$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$ where $f_n(x)= ...
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0answers
42 views

Fourier transformation example

I have been studying Fourier transform and to make things completely clear I wanted to make a simple example for myself and I wanted to present it here, in order to verify that I have a correct ...
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4answers
64 views

Why $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$?

I am working out a Fourier Series problem and I saw that the suggested solution used $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$ to simply the expressions while finding the Fourier Coefficients ...
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1answer
18 views

Completeness condition for periodic function

I know that for a real-valued function set $\{f_n(x)\}$, its completeness condition is $\Sigma_n f_n(x)=\delta(x-x')$. That is, this condition guarantees that a well-behaved function can be write as a ...
2
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1answer
43 views

Why does the Fourier sine series of $x^2$ on $[0,l]$ converge to 0?

When expanding, for example, $x^2$ on $[0,l]$ as a sine series, we get $f(x) = \sum_1^{\infty}b_n sin(\frac{n\pi x}{l})$ If we plug in $x=l$ to this expansion, we get $f(x)=0$. Why aren't we getting ...
4
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1answer
80 views

Short form of few series

Is there a short form for summation of following series? $$\sum\limits_{n=0}^\infty\sum\limits_{k=0}^n\dfrac{(-1)^kn!\alpha^{2n}((2y-1)^{2k+1}+1)}{2^{2n+1}(2n)!k!(n-k)!(2k+1)}$$ ...
4
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2answers
139 views

Nontrivial solutions of $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$

Let $a=(a_n)$ with $a_n\in\mathbb{C}$ be a vector indexed over all $n\in\mathbb{Z}$, and consider the system of equations $\sum\limits_{-\infty}^\infty\overline{a_n}a_{n+k}=\delta_{k0}$ for all ...
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0answers
29 views

Vanishing Fourier terms

Which Fourier coefficients vanish for a periodic function $ f(\theta) $ of period $ 2\pi $ satisfying $ f(\theta) = f(\pi − \theta) $? What about $ f(\theta) = - f(\pi − \theta) $ 􏰖Hint: ...
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0answers
29 views

Fourier Transform, Laplace Transform, but what about…

I have a question regarding the fourier and laplace transform. First, the Fourier transform essentially takes a function, divides it by a frequency (imaginary exponential), and then sees how much of ...
4
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0answers
42 views

Find a relationship between $\hat f$ and $\hat g$, given $g(x)=\int^{x+1}_x f(t)\,dt$, $f\in L^2(\mathbb R)$,

Given $f\in L^2(\mathbb R)$ and $g(x)=\int^{x+1}_x f(t)\,dt$ (a) Relate $\hat f$ and $\hat g$ (b) Prove $g\in H^1(\mathbb R)$ Part A: $$\hat g(k)= \int^{\infty}_{-\infty} \int^{x+1}_{x} ...