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
498 views

Fourier series of $f(x)=x^2$ in $x∈[0,2\pi]$ and in $x∈[−\pi,\pi]$? [closed]

Fourier series - what is the difference between the Fourier series of $f(x)=x^2$ in $x∈[0,2\pi]$ and in $x∈[−\pi,\pi]$?
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1answer
132 views

Fourier Series and Solving Differential Equations

I am getting stuck on how to use Fourier Series to solve ODE's. Take the problem where \begin{equation} E(t)=200t(\pi^2-t^2), \end{equation} for $t$ between $-\pi$ and $\pi$ (period of $2\pi$), ...
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1answer
466 views

steady state solution to differential equation - checking my work

EDIT: fixed a stray negative sign. The problem as given: $y'' + 2y' + 5y = 10\cos t$ We want to find the general solution and the steady-state solution. We're using $\mu y'' + c y' + k y = F(t)$ ...
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2answers
345 views

Step function Fourier series

How to do a step function based Fourier series? What I am confused about is how to calculate the time period since the step function doesn't end? And I don't really know the period since the ...
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0answers
122 views

Orthonormal basis for this $L_2$ space of probability measures?

I couldn't find this on math.se or by searching the internet. Thanks for any help! OK, so I have a set $\Omega$. For now, let's think of $\Omega$ as finite, consisting of $n$ elements. Now consider ...
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1answer
282 views

Can we express all doubly periodic functions as one of doubly periodic function?

Singly Periodic Functions $e^{x},\cos(x),\sin(x),\tan(x), .. etc.$ Euler's identity is $$e^{i\alpha}=\cos(\alpha)+i\sin(\alpha)$$ $$e^{-i\alpha}=\cos(\alpha)-i\sin(\alpha)$$ Thus, we can express ...
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1answer
84 views

Computing The Fourier Sine Series.

Compute the Fourier Sine series of the odd function: $f(x) = x^3 - 4x, -2 \leq x \leq 2 $. (Periodically extended with period 4) I know how to compute this of course where: $b_n = ...
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1answer
221 views

Green' s function for harmonic oscillator

Does someone know how to get a solution of differential equation for Green's function $(-d^2/dt^2 + \omega^2) G(t, s) = \delta(t-s) $? There is a periodicity of G, actually $\Delta (t-s) = G(t,s)$ ...
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0answers
94 views

An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
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2answers
158 views

Sum of Fourier Series

I need to find the Fourier Series for $f\in \mathcal{C}_{st}$ that is given by $$f(x)=\begin{cases}0,\quad-\pi<x\le 0\\ \cos(x),\quad0\le x<\pi\end{cases}.$$ in the interval $]-\pi,\pi[$ ...
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1answer
61 views

Periodic Fuctions - Signals -

If, in the periods, the two half's signal periodic have the same form and opposite phases, the periodic signal has symmetry of half wave. If the periodic signal $g(t)$, of period $T_0$, satisfy the ...
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1answer
86 views

These questions are all about Fourier analysis.

Please prove these equalities,these questions appear in the chapter of Fourier series. If you can use other methods,please tell me more about it, and I am glad to know how to solve the questions: ...
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0answers
170 views

Fourier Coefficients : Frequency Shifting

The FS coefficients of a signal $x(t)$ is given by $$x(t) \longrightarrow C_k$$ The frequency shift property says that: if we multiply a signal $x(t)$ by $e^{j m\omega_0 t}$ the fourier series ...
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2answers
116 views

Bessel function to $\sin(kr)$

$J_{\frac{1}{2}}(kr)=\frac{\sqrt{\frac{2}{\pi }} \text{Sin}[\text{kr}]}{\sqrt{\text{kr}}})$ This can be easily obtained by Mathematica, How to do the details?
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0answers
315 views

Is there a particular meaning to the sum of Fourier coefficients $a_{n^2}$?

The formula $$\sum_{n=-\infty}^{+\infty} e^{-in^2x}$$ does not converge in any function space but it is perfectly valid in $\mathcal{D}'(\mathbb{R})$. When applied on a test function $\psi(x) = ...
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5answers
2k views

The Fourier series of $\sin^3 t$ in trigonometric form

I'm trying to calculate the Fourier series of $\sin^3t$ in trigonometric form. In previous excercises I have been able to use trigonometric identities to be able to calculate the coefficents, but here ...
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0answers
69 views

Is harmonicity preserved when taking limits (normal convergence) on the unit disk.

I'm reading Koosis's book on $H^p$ spaces and have a question. He is proving a $L^p$ version of the Dirichlet problem which states that if $F(t)$ is in $L^p$ on the unit circle then $$ ...
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2answers
1k views

Differentiating a triangular wave

I was really stuck and tried many times to differentiate the following series, and tried to convince myself that the differential form of a triangular wave is the square wave. But I couldn't work it ...
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1answer
59 views

Convergence of eigenmodes of a Sturm Liouville operator.

Is there any "eassy to see" proof for: "The eigenmodes of a Sturm Liouville ODE in a closed interval [a,b], with given boundary conditions, form a complete, orthogonal basis for continuous functions ...
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1answer
96 views

Exponential Fourier Series.

Determine the exponential Fourier series(which invovle exp(jkwt) terms) of the following. x(t)=cos(t)+cos(2t)+0.5 I calculated C0 and got the following. C0=0.5 however, I calculated Cm to be 0 ...
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1answer
328 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
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1answer
173 views

Inverse Fourier transform to find out $\hat c_1$

If we have an integration which is need to solve inversely $$a_0 e^{-r^2/R^2} = \int_0^\infty \hat{c}_1(k) \frac{\sin(k r)}{r} dk,$$ If I transform the $\sin(kr)$, then we get imaginary part. Please ...
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1answer
128 views

Fourier integral representations using only cosine functions.

Hi I have a question about Fourier integrals. Can Fourier cosine integrals represent any function, or just even functions?
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3answers
2k views

Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ [duplicate]

Find Fourier Series of the function $f(x)= \sin x \cos(2x) $ in the range $ -\pi \leq x \leq \pi $ any help much appreciated I need find out $a_0$ and $a_1$ and $b_1$ I can find $a_0$ which is ...
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1answer
149 views

Fourier Transform and amplitude of waves

Given this definition of the fourier transform: $$f(t) \rightarrow \hat{f}(\omega)=\int\limits_{-\infty}^{+\infty}f(t)\,e^{-i\omega t}\,dt$$ and now ...
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1answer
46 views

Positive functions with negative Fourier tail

As the title indicates, my question is: Question: Does there exist a nonnegative function $f\in L^1(\mathbb R)$ such that the Fourier transform of $f$ satisfies $$\hat f(\xi)<0$$ for all ...
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1answer
103 views

Uniform boundedness of Fourier series of indicator functions

Suppose $f\in L^1[0,2\pi]$, denote by $S_n f(x)$ the partial sum of the Fourier series of $f$. I am interested in whether $S_nf(x)$ is uniformly bounded independent of $x$ and $n$, i.e. $$(*)\ \ \ \ ...
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1answer
383 views

Fourier Series of $f$ on the given interval

my goal is to find the Fourier series of f on the given interval: $$f(x) = \begin{cases} 0, & \text{if } -\pi < x < 0 \\ \sin(x), & \text{if } 0 \le x < \pi \end{cases}$$ I know ...
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2answers
217 views

Conceptual question about Discrete Fourier Transform

On the wikipedia page for the discrete Fourier transform, the first sentence says: In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a ...
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3answers
351 views

Further studies on Fourier Series and Integrals.

If you had to choose two books from the following list, which pair would you chose, and why? If you haven't read any, would you pick any pair among the list based on the author of the book? I am ...
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1answer
212 views

generating a random periodic function with bounded amplitude and bounded fourier coefficients

I would like to generate (i.e. repeatedly compute via a computer) a random periodic function $f(x)$ with period $T$ such that $|f(x)| \leq M$ and the kth Fourier coefficient $|A_k| \leq g(k)$ for a ...
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1answer
97 views

A PDE problem about $(\partial_x^2 + \partial_t^2)u = 0$ using Fourier series.

I'm trying to solve the following initial value problem (from Folland, pg. 277, exercise $48$a) using Fourier series: Let $x \mapsto f(x)$ and $x \mapsto u(x, t)$ be periodic functions on ...
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0answers
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calculation wave function

I have a bunch of points from a segment (~1.5 periods) of a wave. The wave looks like a cosinus wave, but it isn't. The length between the left maximum and the minimum is shorter than the length ...
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0answers
340 views

Fourier-Bessel series coefficients

When finding the coefficients of a Fourier-Bessel series, the Bessel functions satisfies, for $k_1$and $k_2$ both zeroes of $J_n(t)$, the orthogonality relation given by: $$\int_0^1 ...
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0answers
187 views

Prove: $\int_0^{\infty}\left(\frac{\sin x}{x}\right)^2dx=\pi/2$

I am dealing exercise 12 in Chapter 8 of Rudin's Principles of Mathematical Analysis. Given the function $f$: $$f(x) = \begin{cases} 1, & \text{if $|x|\le\delta$} \\ 0, & \text{if ...
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1answer
331 views

Even and odd functions using integrals

If $f: [-r, r] \to\mathbb{R}$ is an even function, show that $g(x) = \cos(nx)$ is an even function and $h(x) = \sin(nx)$ is an odd function. Consider: $\int_{-r}^{r} f(x)\cos(nx)dx = 2\int_{0}^{r} ...
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2answers
698 views

Programming discrete fourier coefficients in matlab

Alright so I am having the following issue: I want to figure out how to find the fourier coefficients of the following function: $$D(X)=\frac {a'(x)} {1+a'(x)^2}$$ Where $a(x)$ is an arbitrary ...
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1answer
122 views

Is this infinite series a Fourier series?

I have what looks like a Fourier series but I don't quite understand how (or if) it is possible to recover a function from this. ...
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1answer
101 views

Finding the fourier series representation for a piecewise function

Expand the given function in the appropriate Fourier series: $$\begin{align} f(x) = \begin{cases} x+1 &\mbox{if } -1 \leq x \leq 0 \\ x-1 &\mbox{if } 0 \leq x \lt 1 \end{cases} \end{align}$$ ...
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2answers
92 views

What is the notation of 'a single term in the DFT'

I have a notation/terminology question: I am writing a paper in not-quite-my-area and can't figure out the right way to phrase/notate the following: I have a discrete function $p[x]$, of which I can ...
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1answer
156 views

Why is it so difficult to prove that the discrete Fourier transform (DFT) cannot be calculated in faster time than $N \log N$?

As the title says, why is it so difficult to prove that the discrete Fourier transform (DFT) cannot be calculated in faster time than $O(N \log N)$? This is a famous open problem in ...
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1answer
69 views

Fourier series convergence in $L^2$

Consider a function $g \in L^2(-\pi,\pi)$ such that it is continuous at $x \in (-\pi,\pi)$. Prove that if the Fourier series of g converges at x then that implies g(x) is its limit. I was thinking ...
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1answer
65 views

What is the formula for complex fourier series?

I am watching this video on complex Fourier Series where the instructor states the formula as: $$ f(x) = C_0 + \sum_{-\infty}^{\infty}C_ne^{inx} $$ where as the notes on the same topic by ...
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Showing that $\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} \mathrm{G}_t(x-2\pi n)\in\mathbb{C}^\infty((0,\infty)\times\mathbb{R})$

Welcome everybody :) I need your help in answering the following question: Let $t > 0$ and $\mathrm{G}_t(x) = (2\pi t)^{-1/2}e^{-x^2/2t}$ Show that $$\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} ...
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0answers
73 views

how a discontinuous function converges to Hermite- Fourier series?

I have the proof using a text that if a function $f (x)$ is square integrable with weight function $e^{-x^2}$ and also is piecewise continuous, then $f (x)$ converges to ...
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2answers
60 views

Confused between multiple representations of Fourier Series' formula

I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$. Now, suppose ...
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1answer
455 views

Function not satisfying pointwise convergence and Fourier series

Can you show an example of a function that does not satisfy pointwise convergence theorem hypotheses for Fourier series but that is still expressible as Fourier series? [Added after comment] In ...
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1answer
617 views

Absolute convergence of Fourier series of a Hölder continuous function

Suppose that $f$ is $2 \pi$ periodic and Hölder continuous of order $\alpha > 1/2$. Show that the Fourier series of $f$ converges absolutely. So we know that $f(x+2 \pi t) = f(x)$ for all $t \in ...
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1answer
180 views

Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
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1answer
33 views

Am I understanding this integration right?

This is the snippet of a problem from this PDF here. What I dont understand is why they retain the $Sin$ part for evaluation after integration when all that it is going to evaluate to is 0. If I ...