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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22 views

In Fourier Series when is it acceptable to just integrate half of period and double the result later to find coefficient?

in finding the coefficient of Fourier Series, $a_0, a_n, b_n$. We integrate the periodic function $f(t)$ over the period $T$. That is $$\frac1T\int^T_0 f(t)\ dt$$ $$\frac2T\int^T_0 f(t)\cos(n\omega ...
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12 views

What is the Fourier series coefficient of $f(\omega t)$?

Sorry for the unclear title. Normally, to find the coefficient of the Fourier series we do the integral $$a_n=\frac2T\int^T_0 f(t) \cos(n\omega t) \, dt$$ $$b_n=\frac2T\int^T_0 f(t) \sin(n\omega t) \, ...
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39 views

Is $\sum_n\exp(ian+ibn^2+icn^3)$ known in terms of anything else?

For arbitrary $a,b,c$, does the series $$F(a,b,c)=\sum_{n=-\infty}^\infty\exp\left(ian+ibn^2+icn^3\right),$$ i.e. an evenly-weighed series of exponentials of cubic polynomials, converge to anything ...
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14 views

Find in terms of f and g, the projection of h upon S, where S = span{f,g}

My attempt so far is: ai) $\operatorname{Proj}_S(x)= Px$, where $P = (w_1w_1^T+w_2w_2^T)$ and $\{w_1,w_2,\ldots,w_m\}$ is an orthogonal basis for $S$. Let $w_1 = f$ and $w_2 = ...
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25 views

fourier transform of a function that is only partially known

I'm interested in periodicity of a function that is not known completely, one way to think about it: our function $x(t)$ is defined at discrete points $t_1, t_2, ...$ but the values at these points ...
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32 views

Prove or disprove the pointwise convergence of Fourier Series

Let $\alpha >1$ and function $f$ defined as \begin{equation*} f(x) = \left\{ \begin{array}{ll} (\text{log}\frac{1}{|\text{sin}(x)|})^{-\alpha} & \textrm{$x\neq k\pi,k\in\mathbb{Z}$}\\ 0 & ...
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1answer
74 views

Expanding Fourier Series of $f(x)=\pi-x$ where $0<x<\pi$ (even and odd)

Please help me solve this Fourier series and correct my solution if it is wrong. it's a non-periodic function which we need to write its Fourier series (even and odd) : $ f(x)=\pi - x $ ; $ ...
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1answer
67 views

Does DFT give exact results on the gridpoints?

Let's suppose we have a periodic function $f(x)$ with period $L$ and we know its Fourier Series coefficients $A_n$. Now I have a set of $N$ equally spaced gridpoints between $[0, L)$ at distances ...
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1answer
23 views

Find a function by its Fourier coefficients

Suppose $x[n]$ ($n$ integers) is periodic with period 8 and its Fourier coefficients are $$ a_k = \cos(k\pi /4) + \sin(3k\pi /4). $$ Prove $x[n] = 4\delta[n-1] + 4\delta[n-7] + 4j\delta[n-3] - ...
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1answer
39 views

Find frequency spectrum of function with unknown period

How can I determine the frequency spectrum and phase of a continuous function, if the period is unknown? If I have a function $f(x)$ with period $T$, the Fourier series ...
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1answer
26 views

How do I calculate the Fourier sine coefficient for this function?

I have a function u(t) which is 0 for $ -\frac{\pi}{\omega} < t < 0 $ and $ Esin(\omega t)$ for $0 < t < \frac{\pi}{\omega}$. I calculated $a_0$ and $a_n$, but when I try to calculate $b_n ...
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1answer
10 views

Moving from general to specific solution in separation of variables PDE

I'm just playing around with separation of variables for a simple 1D diffusion equation for $c(x,t)$, of the form $c_{t} = Dc_{xx}$ where $D$ is a diffusion constant. At some initial time, ...
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72 views

Fourier series convergence and error of approximation

Given $f(t) = e^{-t}$, $|t|\leπ$, Determine the the $3rd$−order Fourier series and then calculate to what values does the Fourier series for the $2π$-periodic function $f(t)$ converge in $|t| ≤ π$ ...
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76 views

Effect of Boundary and Initial conditions for a series solution in a PDE

I'm unsure how to explain and contrast the influence of boundary conditions vs. initial conditions when it comes to the series solution of a PDE. Does it have anything to do with steady-state or ...
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1answer
22 views

Finding fourier coefficients - why do these limits of integration change?

Working through my PDE book, it used the following function as an example to introduce piecewise continuity and periodic extensions, and of which to sketch the fourier series: ...
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21 views

Simple DFT problem s(t)=5^t

I'm seriously confused about my wrong answer in this problem. The correct answer is 1/5, but I get 13020 and I have no idea what is wrong. I need to calculate DFT: $$ s(t)=5^t $$ $$ ...
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1answer
42 views

Trouble plotting Fourier Series in MATLAB

I was wondering if anybody could help me with plotting my Fourier Series in MATLAB. I've had a go at it and I don't believe I have arrived at the correct answer. I've plotted the expanded result fine ...
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1answer
42 views

Expanding Fourier series on the interval $[{-\pi}, \pi]$?

How can I expand $$f(x)=\begin{cases} 0 & -\pi \leq x<0\\ \sin x & 0 < x \leq \pi \end{cases} $$ as a Fourier series on the interval $[{-\pi}, \pi]$. I know that: $$ \sin A \cos B = ...
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3answers
49 views

Fourier sine series for $x^3$

It is asked to find the Fourier Sine Series for $x^3$ given that $$\frac{x^2}{2} = \frac{l^2}{6} + \frac{2l^2}{\pi^2} \sum_{n=1}^\infty (-1)^n \frac{1}{n^2} \cos\left(\frac{n \pi x}{l} \right)$$ ...
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2answers
38 views

Find the half-range Fourier series expansion of $f(x) = cos(x)$

I am stuck on the problem of calculating the half-range Fourier series expansion of $$f(x) = cos(x),$$ $$0 < x < \frac{\pi}{2}$$ I am at the point where I have calculated the definite integral ...
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1answer
109 views

Upper bound for the error on the Fourier series for $e^{x}$

I have been given the following problem: Find the Fourier series for $e^{x}$ over the interval $-\pi \le t \le \pi$. Hence find the upper bound of its error. To spare me typing a huge expression and ...
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1answer
29 views

Sign flip for every TWO terms in a sum? (Instead of one)

It's easy to see that $(-1)^n$ in a summation with index n will yield alternating signs on the terms, but what if I want to alternate the sign every 2 terms? I haven't found a way to do this. ...
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1answer
44 views

Fourier transform invariant functions other than the bell curve? [closed]

Are there any functions that are their own Fourier transforms other than $e^{-\pi x^2} $?
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1answer
61 views

Calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$

I have to calculate $\sum\limits_{n=1}^{\infty} \frac{a\cos(nx)}{a^2+n^2}$ for $x\in(0,2\pi)$. I have used the function $f(x)=e^{ax}$ and I have calculated the Fourier coefficients which are: ...
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1answer
25 views

Proving the infinite series solution for the Dirichlet heat equation on $0\leq x\leq\ell$ converges to 0 as $t\rightarrow\infty$

I have a last minute homework problem that I'm struggling with, and ANY help is greatly needed (and extremely appreciated)! Problem (exactly how it is given): Recall that the series solution to the ...
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0answers
28 views

Convergence of the sum of fourier coefficients

I have a function $f$ defined on $[-\pi, \pi]$, $f(\pi) = f(-\pi)$, and has continuous first derivative. How can I prove that the sum of the absolute value of the Fourier coefficients converges? The ...
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1answer
69 views

Fourier series, infinite series

I need help in computing this infinite series. What value of $t$ do I use? The fourier series I've been given is: $$ \sum_{k=1}^\infty \frac{4(2k-1)}{\pi \left(4(2k-1)^2-1\right)} (-1)^k $$
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1answer
21 views

Some properties of the homogeneous spaces on $\mathbb{T}$

I am reading the first chapter of Y. Katznelson "An introduction to harmonic analysis". There is a definition of homogeneous space $B$ on $\mathbb{T}$, and then it is proved that the trigonometric ...
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71 views

Solving Laplace's equation on an infinite strip

I have come across (and indeed, solved) variations of this problem on a finite strip, but not for an infinite strip as in my version of the problem. It is the standard two-dimensional Laplace's ...
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17 views

Derivative of series

I have to use Parseval's theorem for calculating: $$\sum_{n=1}^{\infty}\dfrac{1}{(a^2+n^2)^2}$$ After using the function $e^{ax}$ in period $(o,2\pi)$ I ended up with: ...
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31 views

Proof of Properties of Fourier series in CT

I feel problem in understanding the proof of Fourier series properties 1) Time scaling $$b_k = \frac{1}{T}\int_{T}x(t)e^{jk\omega_0t}dt$$ a= scaling factor $$ = ...
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1answer
38 views

How can I calculate $\int_{-\pi}^{\pi} \cos^6(x)dx$ by using Parseval's theorem [closed]

How can I calculate by using Parseval's theorem: $$\int_{-\pi}^{\pi} \cos^6(x)dx$$
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19 views

Fourier transform of a complex function of constant magnitude

For $f(t) \in \mathbb{Z}$, suppose we have $|f(t)| = c$ where $c$ is some constant. What does that imply about the Fourier transform/series of $f$? Since $|f(t)|^2 = c$, Then by Parseval's identity, ...
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1answer
27 views

Effect of sampling frequency on Discrete Fourier Transform?

I don't get it. I have the following form of the DFT: $$ Y_N(e^{j\omega_n})=\sum_{k=1}^{N-1}y(k)e^{j\omega_n k}\quad\omega_n=\frac{2\pi n}{N}\quad n=0,1,...,N-1 $$ But this assumes that the sampling ...
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65 views

Asymptotic expansion for the solution of linear KDV eq.

Hi, The question arises from the book Solitons by P. G. Drazin about the linearized KDV eq. $$ u_t+u_{xxx}=0 $$ My first step was to take a Fourier transform of the equation, find that the ...
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1answer
27 views

Interpreting function notation?

We seek to compute $\int_0^{2\pi}g(x)^2dx$ with the following given: $$f(x) = \frac{\pi-x}{2}, x \in \left[0,2\pi\right] $$ $$g(x) = f(x+1)-f(x-1)$$ $$f(x) = \sum_{1}^\infty \frac{\sin(nx)}{n}$$ ...
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1answer
27 views

Why is this true for this Fourier series $exp(ax)$ $x\in(0,\pi)$

I have found for $a_0, a_n, b_n$ but as you can see in picture that in first equation the coefficient $b_n$ is removed and is written $2$ in front of coefficient $a_n$. Almost the same for equation ...
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0answers
29 views

Sine representation of cosine [duplicate]

I am trying to establish the following partial sum equivalence: $$\sum_{n=1}^{N}\cos (n\pi u) = \frac{1}{2}\left[\frac{\sin(N+\frac{1}{2})\pi u}{\sin(\pi u)/2} - 1\right]$$ with a hint that the ...
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0answers
22 views

Fourier series and $f(x)=\sum_{h=-\infty}^{+\infty} y(h) e^{-i h x}$.

After some calculation on real function $f(x)$, I obtain the following $$f(x)=\sum_{h=-\infty}^{+\infty} y(h) e^{-i h x}, \ \ x\in\mathbb R $$ where $y(h)$ are equally spaced samples of a real ...
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33 views

Fourier analysis on groups, and the isomorphism of characters in the “classical” setting

I am reading these lecture notes By Daniel Bump about Character Theory on Abelian Groups. If $G$ is a group, then $G^*$ denotes its characters, the set of homomorphisms $\pi : G \to \mathbb ...
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1answer
48 views

1D diffusion equation with Robin boundary conditions

Solving $u_t = \alpha^2 u_{xx}$ with boundary and initial conditions $u(0,t)=0$, $u_x(1,t)+h u(1,t)=0$, $u(x,0)=x$. (Following the book by Farlow, "PDEs for scientists and engineers", page 54) ...
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1answer
20 views

Exponential conversion

While solving Fourier series coefficients if found this problem. Anyone help me to tell that how they converted $\frac{1}{2}e^{j\pi/4}=\frac{\sqrt{2}}{4}(1+j)$
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13 views

Can discrete-time Fourier transform formula be extended to non-unifor sampling cases?

Discrete-time Fourier transform of $x(t)$ is defined as $X_{1/T}(f) \stackrel{\mathrm{def}}{=} \sum_{n=-\infty}^{\infty} \underbrace{T\cdot x(nT)}_{x[n]}\ e^{-i 2\pi f T n}$. Can we extend this ...
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26 views

Selection rules for integral of triple product of sines and cosines

Let $\Phi_m(\varphi), m = -l, \dots, l$ be the truncated Fourier basis, that is $$ \Phi_m(\varphi) = \begin{cases} \sin |m|\varphi, &m < 0\\ \frac{1}{\sqrt{2}}, &m = 0\\ \cos m \varphi, ...
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33 views

$\Sigma^{\infty}_{n=1}(-1)^n[ \frac{\pi sin(n\theta)}{n}-\frac{2cos(n\theta)}{n^2}]=0$

Is $\Sigma^{\infty}_{n=1}(-1)^n[ \frac{\pi sin(n\theta)}{n}-\frac{2cos(n\theta)}{n^2}]=0$ true? This is came out from my Fourier series computation, according to the answer, that sum should be zero ...
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1answer
30 views

Proof for determining Fourier coefficients

While determining Fourier coefficients we have this equation $$\int^{T}_{0} x(t) e^{-jn\omega_0t} dt = \sum^{+\infty}_{k\ =\ -\infty} a_k [\int^{T}_{0} e^{j(k-n)\omega_0t}dt]$$ I want to ask that how ...
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0answers
43 views

Numerically solving a non-linear PDE by an ODE on the Fourier coefficients

I need to solve numerically a PDE of the form $$ u_t(x,t)=u_{xx}(x,t)+u_x(x,t)^2-a(x)u_x(x,t)-a_x(x) $$ with initial condition $u(x,0)=u_0(x)$. I can assume that both $u(\cdot,t)$ and $a(\cdot)$ are ...
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1answer
28 views

Fourier series of arbitrary polynomial

I am working through the following problem: Let $P(x)$ be some arbitrary polynomial over the interval $[-1, +1]$. Then define $$A_n(P) = \int_{-1}^{+1} P(x)\cos{(n\pi x)}\,\mathrm{d}x$$ I am require ...
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1answer
30 views

proof of the convergence of a series of Fourier coefficients

Let $a \in (0,1/2]$ and define $ f:\mathbb T\rightarrow \mathbb R $ by $$ f(x) = \begin{cases} 1, &\text{if $x$ is between $-a$ and $a$} \\ 0, &\text{otherwise} \end{cases} $$ I figured out ...
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1answer
35 views

Find Discrete Time Fourier coefficients of $(-1)^n x[n]$

Given that $x[n]$ is an N-periodic sequence with Fourier coefficients $a_k$, I want to find the Fourier coefficients of $$(-1)^n x[n]$$ for the situation in which $N$ is odd. I'm also interested in ...