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.

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1answer
102 views

Fourier series: term-by-term Laplace transform.

Quick question: If a Fourier series is uniformly convergent should the term-by-term Laplace transform of the series equal the result of the periodic function theorem for the Laplace transform?
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1answer
46 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 ...
3
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1answer
71 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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2answers
51 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
56 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
36 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
40 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
55 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
206 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
426 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
48 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
39 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 ...
2
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1answer
68 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 ...
4
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1answer
276 views

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

I'm struggling with a Fourier series. 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
510 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 ...
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1answer
32 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
45 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
35 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
18 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
132 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
43 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
60 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 $y=2$. ...
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1answer
84 views

Fourier series of rescaled cosine function

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
40 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
69 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
24 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 ...
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1answer
108 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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1answer
134 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
37 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
39 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
29 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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1answer
80 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
126 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
54 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 ...
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1answer
95 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
79 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
114 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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7answers
11k 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 $a_0$, ...
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1answer
133 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 ...
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1answer
68 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 ...
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2answers
108 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
161 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
55 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 ...
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0answers
66 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} ...
3
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1answer
193 views

Fourier series of f(x)

I want to find the Fourier series of $f(x)$ defined by $f(x)=\begin{cases} 1 , -L\le x<0\\ 0, 0\le x<L. \end{cases} $ Well, to find $a_0$ I do this integral: $$a_0=1/L \int _{-L}^0 dx +1/L ...
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1answer
49 views

Discrete Fourier Series

I have a series of discrete values that are periodic and I am looking to calculate the Fourier series of it. I learnt all of this in college but I can't for the life of me remember now. The discrete ...
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1answer
43 views

characterization of unital Fourier multipliers on $L^\infty(\mathbb{R})$?

Does there exist a characterization of Fourier multipliers $T \colon L^\infty(\mathbb{R}) \to L^\infty(\mathbb{R})$ which are unital, i.e. $T(1)=1$? In the case of the torus $\mathbb{T}$, it is easy ...
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0answers
132 views

Fourier series and Riemann integral

On the heuristic level, one often says that given a periodic function with period L, its Fourier series converges when $L \rightarrow \infty$ towards a Riemann integral. In other words, the ...
22
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1answer
420 views

Using Fourier Series to compute sums

I have just started learning the basics of Fourier series and have some doubts about it. I am aware that Fourier series can be used to compute infinite sums. For example, $\zeta(2)$ and $\eta(2)$ can ...
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3answers
69 views

What are the concepts that I need to understand before studying Fourier Analysis?

Background ( Long Story Short ) : For some reasons, I am taking a class in my university that focus on Fourier Analysis Laplace Transform, and Partial Diffiential Equations Problem : I have done ...