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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More Fourier Series

I'm trying to compute the Fourier series of $$f(x)=\frac{1}{2-\cos(x)}$$ on the interval $[0, 2\pi]$. It is an even function, so I need to determine the $a_n$ coefficients. They are given by the ...
3
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1answer
343 views

Computing Fourier Series

$$f(x)=\sin|x|$$ Is it possible to compute the Fourier Series of $f(x)$? It seems that it would admit a Fourier cosine representation because it is even (by looking at the graph), but the periodicity ...
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1answer
156 views

What is this Hilbert space?

The space is $H^s(\mathbb R^d)$. If $f$ is in this space, it means $\int_\mathbb {R^n} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi < \infty$ where $\hat f$ is the fourier transform of $f$: $\hat ...
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1answer
140 views

Fourier transform question

assuming that the integral exists $$ I(u)= \int_{-\infty}^{\infty}dxe^{iux}e^{ax}f(x) $$ using the shift properties of Fourier function is that integral equal to $$ I(u)= \frac{F(u+ia)+F(u-ia)}{2} ...
2
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2answers
292 views

Property of Fejer kernel

Let $$ F_n(x) = \frac{1}{n} \left( \frac{ \sin(\frac{1}{2} n x ) } { \sin(\frac{1}{2} x ) } \right)^2 $$ be the n-th Fejer-Kernel. Then $$ \forall \epsilon > 0, r < \pi : \exists N \in ...
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1answer
4k views

Determine the Fourier Series on a piecewise continuous function

For $x\in[-\pi,\pi]$, $$F(x)=\left\{ \begin{array}{cl} -1 & \text{for}~-\pi\leq x\leq 0\\ 1 & \text{for}~0\leq x\leq \pi \end{array}\right..$$ To what value does the Fourier series converge ...
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1answer
105 views

Fourier-like transform from the time to the phase spectra

I run in to a real problem which must be a classic, only I cannot find the answer. I know Fourier transform shifts a signal from the time spectrum to the frequency spectrum. Is there a similar ...
2
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1answer
171 views

Evaluating a series in closed form

Can I evaluate the series $$ \sum_{k=1}^{\infty} \frac{\ln(k)\sin(2\pi kx)}{k}$$ in closed form, for any specific non integer values of $x$ , without using derivatives of the polylogarithm? It looks ...
2
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3answers
183 views

Series help, fourier series

How do I know if a given function can be represented by a fourier series, that converges to the value of that function at non discontinuities. Also where did Fourier come up with the idea of ...
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1answer
242 views

Sine series of $\pi/2$

I'm studying Fourier series and came across this peculiar problem. I just studied (along with proper reasoning) that if $f(x)$ is an even function, then the fourier series has only Cosine terms and if ...
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1answer
164 views

Fourier series of the fractional part

What is the Fourier series for $\{a\}\{b\}$, i.e. the product of the fractional parts of $a$ and $b$. I know what the Fourier series looks like for a single value of either $a$ or $b$, but I want to ...
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2answers
422 views

Is it possible for cosine functions to have Fourier sine series expressions or sine functions to have Fourier cosine series expressions?

Is it possible for cosine functions to have Fourier sine series expressions or sine functions to have Fourier cosine series expressions? For example, do $\sum\limits_{n=1}^\infty a_n\sin nu=\cos u$ ...
2
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0answers
113 views

Integral of product of two square waves over [0,1]

In Mathematica notation, I am looking for the function f[m,n] for real numbers m and n defined by f[m_,n_]:=Integrate[SquareWave[m x]SquareWave[n x],{x,0,1}]. I'm trying to get a closed form for the ...
2
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1answer
939 views

How can I prove that the Gibbs phenomenon overshoot for a Fourier Series is approximately 9%?

The question is pretty self explanatory, I'm studying Fourier Series with the book Mathematical Methods for Physicists written by Arfken and it does not explain that.
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1answer
202 views

Changing Frequency of a Fourier Transformed signal

How can I change the frequency of a signal after taking its fourier transform? I am taking voice input from user in MATLAB and than I take its fourier transform to convert the signal in frequency ...
3
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0answers
254 views

Questions about the Fourier expansion of $e^{iz\cot(x)}$

By analogy with Jacobi–Anger expansion, one expects that $e^{iz\cot(x)}$ has a Fourier expansion of the form : $$e^{iz\cot(\theta)}=\sum_{n=-\infty}^{\infty}\Lambda_{n}(z)e^{in\theta}$$ ...
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1answer
505 views

Faster than Fast Fourier Transform?

Is it possible to make an algorithm faster than the fast Fourier transform to calculate the discrete Fourier transform (is there proofs for or against it)? OR, a one that only approximates the ...
2
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2answers
148 views

The difference between m and n in calculating a Fourier series

I am studying for an exam in Differential Equations, and one of the topics I should know about is Fourier series. Now, I am using Boyce 9e, and in there I found the general equation for a Fourier ...
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2answers
293 views

Fourier Transformation

This expression: $x(t)=[e^{-3t+5}] u(t-1)$. I am trying to take the Fourier transformation of the above expression. I know that for $x(t)=[e^{-at}] u(t) \leftrightarrow \frac1{i\omega+a}$. But, ...
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3answers
400 views

How to evaluate this infinite sum?

How to calculate the infinite series $\sum_{n=1}^{\infty }\sin(nx\pi)\sin(ny\pi)$ where $x,y$ are real numbers? I assume this is related to Fourier series, but I can't calculate this.
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0answers
94 views

Partial Differential Equation Eigenvalue of zero question

In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction? That is to ...
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1answer
236 views

How to find the coefficient of this Fourier sine series?

From $$1=\sum_{k\geq 1} a_k \sin((k\pi+\frac{\pi}{2})x),$$ I want to find $a_k.$ My unsuccessful approach is first multiplying both side by $\cos((k\pi+\frac{\pi}{2})x)$. That is, ...
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1answer
168 views

Simple Fourier Series

$$f(x)=\begin{cases}\tfrac{1}{2a},& 0<|x|<a\\ 0,&\text{otherwise} \end{cases}$$ Note that $a<\pi$. $$a_0=\frac{2}{a}\int_0^a \frac{1}{2a}dx=\frac{1}{a}$$ $$a_n=\frac{2}{a}\int_0^a ...
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1answer
85 views

Integral of Scaled Bessel Function With Linear Phase

I am trying to solve a problem part of which includes the following integral ($j=\sqrt{-1}$): $$\int_{k_1}^{k_2} k e^{-jk\sigma} J_n(\rho k) \, \mathrm{d}k$$ The $e^{-jk\sigma}$ term is making my ...
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2answers
538 views

How to integrate $e^{r\cos x} \cos(r\sin x)$

The title says everything. I'm studying fourier series and I've stumbled upon this question: find the fourier series of $f(x) = e^{r\cos x} \cos(r\sin x)$. So that i need to integrate this function ...
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0answers
65 views

discrete fourier transform of $x(m)=e^{\frac{2\cdot \pi\cdot f_0\cdot n}{F_s}}$

Using the power series summation formula, how do I find the discrete Fourier transform of this signal $x(m)=e^{\frac{2\pi *i* f_0 m}{F_s}}$, $m=0,\ldots,(N-1)$, where $f_0$ is the fundamental ...
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1answer
308 views

Complex Fourier Series Coefficient

I am trying to solve the following exercise in my PDEs book: Consider $$ f(x)=\begin{cases}0&x<x_0\\1/\Delta&x_0<x<x_0+\Delta\\0&x>x_0+\Delta\end{cases}. $$ Assume that ...
3
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1answer
92 views

Known facts about a function

In my work I have met the function on the unit circle whose Fourier coefficients are $$ c_n=\frac{1}{|n|}\prod (d_k+1) $$ if $n=\pm\prod p_k^{d_k}$ is the decomposition of the integer $n$ into the ...
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1answer
271 views

Fourier series expansion of a periodic function

I'm trying to sketch several periods of the periodic function $f(x)$ below and expand it in an apporpriate Fourier series. I know there's a formula for expanding fourier series, but when I try I ...
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3answers
905 views

Which Fourier series formula is correct

I'm getting started on Fourier series but I'm confused over the formulae involved. My lecturers notes, including Wikipedia state that, for the interval $(-\pi \le x \le \pi)$ $$a_0 = \frac1\pi \int ...
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1answer
256 views

Fourier transform physical meaning [closed]

What is the physical meaning of the Fourier transform expressed at the spectral density? Also, what is the relationship between the Fourier transform and the total energy of an oscillating system? ...
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0answers
88 views

Sturm-Liouville system boundary conditions

I have a question about the various boundary conditions of a Sturm-Liouville system. I've been told that there are five different possible conditions for the system. However, can't seem to find them. ...
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3answers
182 views

Fourier Series solution giving wrong answer

How this answer came by solving "$a_n$" of Fourier series. $$a_n=\int_{-1}^1 t^2 \cos (n\pi t) dt = 4(-1)^n / (\pi n)^2$$ ? How can I mathematically derive this answer? My answers comes to $$2 ...
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1answer
354 views

How to solve Fourier Series

How can I easily solve fourier series equations given below? I want to solve it quickly and easily in exams cuz I am not good with mathmatics very much.. For Example: $a_n= \int_0^\pi e^{-t/2} ...
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0answers
122 views

Fourier Series and Filter in Function

Let $L=1, A=1$ and $f\in L^2([-L/2, L/2])$, with Fourier series $$f^{t}=\sum_{n=-K}^{K}a_n \exp(2j\pi xn/L),$$ truncated at $K$. Has this function, $f^{t}$, any relation with Fourier Inverse ...
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1answer
106 views

Non-negative function with a non-positive operator.

Question: I would like to know if there is any simple function that is $\geq 0$ but with its partial sums $S_{m} \leq 0$? Note: After much discussion, it would seem this question is not possible to ...
3
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1answer
85 views

Summation over a Vector

I am trying to find the Fourier series of a 3D function, $e^{-\alpha(x^2 + y^2 + z^2)}$ with bounds $-\ell_1 < x < \ell_1$, $-\ell_2 < y < \ell_2$, $-\ell_3 < z < \ell_3$. I have ...
3
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1answer
472 views

Is the Taylor series comparable to Fourier series and spherical harmonics?

I am currently trying to grasp spherical harmonics and try to digest that we proved that the sine and cosine functions are a basis for the $L^2$ space of the squared-integrable functions. So as far ...
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4answers
30k views

Difference between Fourier series and Fourier transformation

Whats the difference between Fourier transformations and Fourier Series? As I've been working with Fourier Series in my maths lectures yet a friend of mine also doing engineering has been working with ...
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3answers
1k views

Can any continous,bounded function have a fourier series?

In particular,can an oscillatory function with some decay term ( i.e e^(-t)*cos(kt) have a fourier series representation? All the articles I read said that the function has to be periodic,but this one ...
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1answer
680 views

Fourier Sine Series of a Piecewise Smooth Odd Function

I am trying to find the Fourier sine series of the following function: $$ f\left(x\right)= \begin{cases} 1&x<L/2,\\ 0&x>L/2.\tag{1} \end{cases} $$ Let $L=1$. Then, this is what ...
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2answers
4k views

Fourier Series of The Sine Function

I am computing the Fourier series of $$f(x)=\sin\frac{\pi x}{L}.$$ The Fourier series of a piecewise smooth function $f(x)$ defined on the interval $-L\leq x\leq L$ is given by $$f(x)\sim ...
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0answers
3k views

Fourier Series Coefficients of a Full-wave Rectified Cosine

I need to calculate the Fourier Coefficients $a_k$ of $\left | \cos(2 \pi f_c) \right |$ (a full-wave rectified cosine) so that $\left | \cos(2 \pi f_c) \right | = a_0 + ...
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1answer
142 views

Fourier Series with Signals

So the question is: Determine the fourier series representations for the following signal: Here the formula for the fourier series $$C_k=\frac{1}{T}\int_T \! x(t)e^\frac{-j2\pi kt}{T} \, \mathrm{d} ...
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1answer
3k views

Fourier series for $\sin x$ is zero?

I have no practical reason for wanting to do this, but I was wondering why the Fourier series for $\sin x$ is the identical zero function. I'm probably doing something wrong or missing some important ...
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368 views

Effect on magnitude of coefficients when time-shifting a fourier series

Suppose a periodic function, $f(t)$ with period $2\pi$ has a Fourier series of $\sum_{k=-\infty}^{\infty} c_ke^{ikt}$ Now suppose we time shift the function to obtain $g(t) = f(t-t_0)$ My question ...
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55 views

Proving that two representations of a Fourier series are the same

I have to show that $$\sum_{n=0}^\infty A_n\cos\left({xn\frac{2\pi}{T}-\theta_n}\right) \equiv \sum_{n=-\infty}^\infty c_n \mathrm{e}^{\left({ixn\frac{2\pi}{T}}\right)}$$ I have tried two ...
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1answer
111 views

Help finding inverse Fourier transform of a function.

I have the function $f(x) = 1 -|x|$ for $|x|\leq 1$. And zero everywhere else. I'm supposed to find the inverse Fourier transform of the function but I only have a formula for the inverse Fourier ...
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96 views

Show $\left\lvert \frac{1}{\pi+2} \sum_{0 \le k \le 2n,k\ne n} \frac{i}{k-n} e^{ikt} \right\rvert > c \ln n, c>0$

The partial sum of the Fourier series of $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ can be written as \begin{align} S_n(f,t) &= \sum_{0<|k| \le n} \frac{i}{k} e^{ikt} \; (1) \\ & = ...
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1answer
144 views

Convergence of the Fourier series of $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$

The partial sum of the Fourier series for the function $f(t)=(t-\pi)\chi_{\left(0,2\pi\right)}$ is $$ S_n(t)=-2 \sum_{k=1}^{n} \frac{\sin kt}{k} $$ We saw a theorem which states that the Fourier ...