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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24 views

Fourier coefficient one-period function

Define a function $f(x) =(2\cos(\pi x))^{10} $$f\in L^{1}$ so it's one-period. I would like to calculate the Fourier coefficient $\hat{f}(2)$. So we get $\displaystyle\hat{f}(n)=\int_{0}^{1}e^{-2\pi ...
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42 views

Fourier Sine Series and Cosine Series

This is the Fourier Series representation for a periodic function with period 2p, given in my lecture note. $\dfrac{a_0}{2} + \sum_{n=1}^{\infty}(a_n cos(\dfrac{n\pi t}{p})+b_nsin(\dfrac{n\pi ...
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1answer
38 views

evaluate arithmetic sum by using fourier series

Hi I've been trying for 40 minutes to evaluate the sum of the following arithmetic series with no luck. $\sum_{n=1}^\infty \frac{sin(2k)}{k}$ I've tried to make this into a fourier series by ...
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35 views

Show that $x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$

Show that $$x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$$ for $ 0<x<\pi$ My idea: I've defined the periodic function $$f(x) = 0 \text{ if } x \in [- \pi, 0) \text{ ...
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61 views

Fourier Cosine Series question

If I have even piecewise periodic function ($T=6$) $$x(t)=\begin{cases} 0 &-3\leq t \leq-2  \\ 2+t &-2\leq t \leq-1 \\ 1 &-1\leq t \leq 1 \\ -t+2 &1\leq t \leq 2 \\ 0 &2 \leq ...
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45 views

Discrete Fourier Transform by hand

I have an assignment where I'm given the DFT of a sequence $x[n]$ as $X[k]=\{4,3,2,1,0,1,2,3\}$ and also $$y[n] = \left\{ \begin{array}[cc] xx[n/2] & \text{if n is even} \\ 0 & ...
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1answer
62 views

Calculating fourier series

I've a fourier series with a period = $2\pi$ that is even. f(t) = \begin{cases} 0 \text{, when: } 0<t<\pi-2 \\ \pi \text{, when: } \pi-2<t<\pi \end{cases} The functions trigonometric ...
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3answers
71 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
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1answer
85 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
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2answers
53 views

Fractional Sobolev space $H^{1/2}(-\pi,\pi)$

Let $H^{1/2}(-\pi,\pi)$ be the space of $L^2$ functions whose Fourier series coefficients $\{c_n\}_n$ satisfy the summability constraint $\sum_n |n| |c_n|^2 < \infty$. Are functions in ...
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1answer
63 views

Finding the Fourier series of a piecewise function

I'm s little confused about Fourier series of functions that are piecewise. Here’s an example of such a function: $$f(x) = \begin{cases} x & -\frac\pi2 < x < \frac\pi2 \\[5pt] \pi - x & ...
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2answers
34 views

Derivative of Fourier series

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ where $a$ and $b$ are lower and upper boundary, ...
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1answer
53 views

Using the Fourier Series of $f(t)=(t-\frac{1}{2})^{2}$ to deduce the sum $\sum_{n=1}^{\infty }\frac{1}{n^{2}}$?

So this is a question in one of the previous tests: My approach (if you want just skip to step 3.):$$$$ 1. Formulation of the problem and calculating the constant term of the series $a_o$ I ...
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27 views

Obtain Fourier series for these functions:

I learned Fourier series early and not expert in that so i want help: for below functions how should we find Fourier series coefficient? I have two problems: in problem 1 if we write this mean ...
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2answers
15 views

Doubt in Fourier Series

When we solve the equation $$\frac 2{\pi}\int_{0}^{\pi}k\sin(nx)dx;$$ after integrating it, we get $\frac {2k}{n\pi}(1-\cos n\pi)$. Why is $\cos n\pi=(-1)^n$?
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35 views

Functions f(x) equal to Taylor series vs Fourier series vs Bessel series

(I had trouble phrasing the question below due partially to the fact that Bessel functions $J_{\alpha}(x)$ and $U_{\alpha}(x)$ are defined for any complex $\alpha$, so below I tried to express an ...
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1answer
61 views

Fourier series for a non-periodic function

My textbook says that: 'If we which to find the Fourier series of a non-periodic function only within a fixed range then we must continue the function outside the range so as to make it periodic.' ...
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1answer
40 views

complex fourier series with odd function

Consider the periodic and hybrid function defined as $$f(t)=x, 0\le x \le 1$$ and $$f(t)=1$$ $$1\le x\le 2$$ Attempt: I need to calculate Cn $$C_n=\frac{1}{2}\int_0^1 xe^{-in\pi ...
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2answers
20 views

Fourier Transform of mix partial derivative

I know FT{$\frac{\partial u}{\partial x}$} = (ik)FT{u}. Give a function $U(x,y)$. Is the following true? FT{ $\frac{\partial^2 U}{\partial y \partial x}$} = FT{$\frac{\partial U}{\partial y}$} ...
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2answers
29 views

Convergence of Fourier Series in $L^1(\mathbb{T})$

Suppose $f \in L^1(\mathbb{T})$ and the sequence of partial sums of its Fourier series converges (in $L^1(\mathbb{T})$) to $g$. How can I prove $f=g$?
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1answer
60 views

Showing that two functions are orthogonal on a rectangle

I was given the following question, and I think I'm nearly there, I just wanted to ask for some clarification in the last step. Derive the eigenvalues and functions of the SL problem $\phi_{xx} ...
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1answer
41 views

Estimating the modulus of continuity of translation in $L^2$ by a Sobolev norm of the function

For any $s\in \mathbb{R}$ define the Hilbert space $H^s(\mathbb{T})$ by means of norm $$\|f\|^2_{H^s}=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ Show that for any $0\leq ...
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43 views

Wiener Algebra, absolute convergence of fourier series

So how do you prove if $f, g\in L^2(\mathbb{T})$, then $f*g\in \mathbb{A}(\mathbb{T})$. $\mathbb{T}$ denote $[0,1)$ and $\mathbb{A}(\mathbb{T})$ denote the Wiener algebra such that if $f\in ...
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1answer
43 views

Prove that $\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth(\frac{\pi}{2})$

I want to prove that: $$\sum_{k=1}^\infty\frac{1}{16k^4 - 1} = \frac{1}{2} - \frac{\pi}{8}\coth\left(\frac{\pi}{2}\right)$$ Using the fourier series: $$\phi(x) = \begin{cases}0 & \text{if ...
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2answers
101 views

Fourier series with half range

Question What are $a_0,a_n,b_n$ equal to with range $-L\leq x \lt0$, rather than the standard $-L\leq x \leq L$? For example: $$f(x)=2x^2,\quad-1\leq x\leq0$$ Instead of $f(x)=2x^2,\quad-1\leq ...
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1answer
29 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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33 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
24 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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1answer
53 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
46 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
37 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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30 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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28 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
46 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
60 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
66 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
31 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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32 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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1answer
35 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 ...
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1answer
241 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
224 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
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 ...
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1answer
42 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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25 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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17 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
64 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
33 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
57 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
67 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
32 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: ...