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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Sketching a function, finding fourier series.

I have a question relating to fourier series which is as follows; $$f(x)=\frac{1}{2h},\quad |x|<h$$ $$f(x)=0, \quad h<|x|<\pi $$ where $h$ is a constant such that $0<h<\pi$ This ...
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1answer
31 views

Can the sum $\sum_{k=1}^\infty (1/k)^{3/2}\sin(kx)$ be evaluated using Fourier series or otherwise?

I have to compute this sum, and I was wondering if it can be evaluated using Fourier series. It seems familiar to me but have forgotten the Fourier tricks I used in the past, so time for revision. ...
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1answer
89 views

Relationship between the Fourier transform the Fourier series?

What is the relationship between the Fourier transform of a periodic function and the coefficients of its Fourier series? I have a following piecewise function: $y(t) = \begin{cases} 0, & -\pi ...
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46 views

Calculation of a Fourier Coefficient.

I need some help calculating this Fourier coefficient. Periodic signal, six-steps. Odd periodic signal. I've made the calculations myself, not using any software, and the results are these: ...
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25 views

How to solve this fourier transform

Function is: $v(t)=4$ for $0< t< \frac\pi2$ $v(t)=-4$ for $-\frac{\pi}{2}< t< 0$ $v(t)=0$ for $-\pi< t< -\frac \pi2$ and $\frac\pi2< t< \pi$ I solved this and got : ...
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1answer
50 views

Why is the complex fourier series defined this way?

The definition of complex fourier series of a function is always given as the limit of symmetric partial sums $S_N(x)=\sum_{-N}^N c_n\exp(2\pi i nx)$, provided that series is convergent. Why do we ...
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1answer
48 views

Suppose $f$ is a continuous real-valued function on $\mathbb{R}$ such that $f(x+1)=f(x)$ for every $x$, Let $\gamma$ be an irrational number. [closed]

Suppose $f$ is a continuous real-valued function on $\mathbb{R}$ such that $f(x+1)=f(x)$ for every $x$, Let $\gamma$ be an irrational number. Prove that $$ \lim_{n\to ...
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76 views

A closed form for a series

This maybe duplicated, please let me know if it is so. I would like to find a closed form for the following sum $$ \sum_k \frac {1}{k^2}\cos (kx)$$ Any suggestion would be helpful.
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14 views

Does inverse of all Fouriers transforms have a corresponding function in time domain?

I am trying to cancel out the following transfer function of a system: $$\frac{( 1 - e^{(i*k*T)} ) }{ (i*k)}$$ I thought it would work if I find the inverse Fourier transform of $$\frac{ (i*k)}{( 1 ...
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18 views

Cannot take inverse fourier transform of a function

I am trying to get the inverse fourier transform of the following function: $$\frac{ (i*k)}{( 1 - e^{(i*k*T)} ); }$$ Where t is constant. Is there a way to approach this? I have tried the ...
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17 views

Can this function be expaned as a fourier series?

Let $\phi,\tilde{\phi}$ be functions in $L^{2}(\Bbb{R}^{2})$. Let $G(\omega)$ be defined as$$G(\omega)=\sum_{k\in\Bbb{Z}^{2}}\hat{\phi}(\omega+2k\pi)\overline{\hat{\tilde{\phi}}(\omega+2k\pi)}$$ My ...
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1answer
12 views

Sum into closed form

I'm working with spectral approximations and I ran into this problem. Hope someone knows how to solve it! $(D_N)_{lj} = \frac{1}{N} \sum_{k=-N/2}^{N/2-1} i k e^{2 i k (l-j) \pi /N} $ ...
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16 views

how to go from reel to complex fourier series

I have this fourier series on reel form where $a_n = 0$ og $b_n = \frac{1}{n^3}$ I am then trying to find the complex form. I've gotten that $c_0 = 0$, $c_n = \frac{1}{2}(a_n - ib_n) = -i ...
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1answer
30 views

Using Fourier series of $x^2$ to evaluate Fourier series of $x^3+\pi^2 x$

Let $f,g$ be functions such that $f(x)=x^2 \ , \ g(x)=x^3+\pi^2 x$. Use Fourier series of $f$ to evaluate Fourier series of $g$. Domain is $[-\pi,\pi]$. My try: The Fourier series of ...
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2answers
21 views

Calculating cosine coefficient of a Fourier series

I have highlighted steps for indefinite integration of the following function: $$f\left( x \right) =\int { { e }^{ a x }\cos(n x)dx } \\$$ After integration by parts twice the following results is ...
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1answer
26 views

Eigenfunction representation of the L2 derivative

I think the main idea of the definitions that follow is to define some sort of generalized double derivative on a subset of $L^2[0,1]$ Define $D(K)$ to be the subset of $C^1[0,1]$ made up of ...
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21 views

Compute Fourier series using Dirac Delta

Define $f(t) = t^2$ for $|t| \leqslant 2\pi$ and let it have period $2 \pi$. Compute the Fourier series of $f(t)$ indirectly by computing the Fourier coefficients of $f''(t)$ (using the Dirac delta ...
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1answer
34 views

Solution to $y′(t) + y(t − 1) = \cos^{2}\pi t$

Determine a solution with period 2 of the differential-difference equation $y′(t) + y(t − 1) = \cos^{2}\pi t$. Solution: period: $T = 2 \Rightarrow \Omega = \frac{2 \pi}{2} = \pi$ Fourier series: ...
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1answer
15 views

Sum of Fourier Series for $f(t) = (t+1) \cos t$ at $t = 3\pi$

So I have this question here (I write out the whole question): If $f(t)=(t+1)\cos (t)$ for $-\pi < t < \pi$, what is the sum for the Fourier series for $f(t)$ at $t=3 \pi$? My solution is ...
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38 views

Compute the Fourier Series for $f(x)=\cos(\frac{x}{2})$ for the domain $(-\pi<x≤\pi)$ and $f(x)=f(x+2\pi)$

I know that $b_{n}$ is an even function, due to the function being an even function, however I am struggling to compute $a_{n}$.
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21 views

Approach on finding the Fourier series solution

Given the expression: $u(x,t) =e^{-Ct} y(x,t) - v(t)$ How would one approach this when looking for the Fourier series solution, or where to start? Is it possible to see from this which Fourier ...
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1answer
35 views

finding Fourier series of $|x\sin(x)|$

I have this function: $f(x)=|x\sin(x)|$ Now, since this is an even function I know the $b_k \equiv 0$. I tried calculating the $a_k$ coefficients and got $$a_k = \frac{2(-1)^{k+1}}{k^2 - 1}$$ so ...
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47 views

fourier series, complex coefficients.

I've been told that a function $f$ can be written by the Fourier series: $$f(x) = 1 + \frac{1}{2}\sin(x)+ \sum_{n=1}^\infty \frac{1}{2^n}\cos(nx)$$ The way I get it is that a Fourier series is ...
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2answers
38 views

Fourier sine series in solution to 1D Heat Equation

I have reduced my solution of a 1D heat equation boundary value problem to the following: $$W(z, t) = \sum_{n=1}^\infty b_n \sin(\lambda_n z) e^{-\lambda_n^2 \alpha t}$$ To get the coefficients ...
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34 views

What is the Fourier transform of $1/|x|$?

I looked it up in several tables and calculated it in Mathematica and Matlab. Some tables say that the answer is simply $$\frac{1}{|\omega|}$$ and in other table it is ...
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36 views

How to show the convergence of the Fourier Series of a 2pi-periodic real-analytic function,

This is an old complex analysis exam question. EDIT: I would like to use complex analysis methods to solve this problem, so the link offered by Normal Human seems a bit out of scope, but I will ...
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47 views

Proof that $\int f(x)\sin(Nx)\ dx \to 0$ as $N \to \infty$

I'm studying Fourier series out of Rudin's "Principals of Mathematical Analysis". In the proof that the Fourier series $s_N(f;x)$ converges pointwise to $f$, it assumes that at a point $x$, there is ...
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1answer
63 views

Inner product from Fourier-like kernel

A kernel $K\colon [0,1]^s\times [0,1]^s \rightarrow\mathbb{R}$ is a symmetric and positive semi-definite function (meaning that for any $v_1,\ldots,v_m\in [0,1]^s$ and any $m\geq 1$, the matrix ...
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34 views

Uniform error of approximating the Heaviside function by a partial sum of its Fourier series

Suppose $f(x)=H(x-.5)$ where H = Heaviside function on $0<x<1$ is approximated by the first five nonzero terms of its Fourier sine series. Compute the uniform error (i.e maximum error, max p(x) ...
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1answer
65 views

Convergence of a sine series

Using Mathematica, we claim that the following series is convergent: $$\sum_{n=1}^{\infty}\frac{\sin(n^2 t)}{n}$$ Any idea how we prove this?
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23 views

Fourier series - explaining step in proof

I have seen a proof of the claim that the Fourier coefficients of $f\ast g$ equals $\displaystyle \hat{f(n)}\cdot{\hat{g(n)}}$. I can't understand a step of it. The proof goes like this: Let $f,g$ ...
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1answer
23 views

Complex Fourier series and its represntation

I'm trying to tackle the following question, but I'm not sure that my solution is correct. Let $f$ be real-valued $2\pi$ periodic function which is continuous almost everywhere, such that its ...
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2answers
41 views

Compute infinite sum using fourier series

I'm trying to tackle the following question: Use Fourier series of the function $f(x)=x(\pi+|x|)$ in $[-\pi,\pi]$ to compute the infinite sum $$ \sum_{n=1}^{\infty}\frac{(-1)^n}{(2n-1)^3}$$ So, ...
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30 views

Intuition behind alternate expression of impulse train

So it's known that $\sum_n \delta(x-nT) = \frac{1}{T}\sum_m e^{2\pi imx/T}$. This can be proven by expressing the left hand as a Fourier series and finding $c_m$. But it's just mindboggling that this ...
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13 views

How can I calculate the inverse fourier transform of $jw$

I am trying to solve $h_I(t)$, which is satisfying $h(t)*h_I(t)=\delta(t)$. e.g. If $h(t) = \delta(t+c)$, then $h_I(t)=\delta(t-c)$. If $h(t) = u(t)$, then $h_I(t)=\delta'(t)$. Q) What is the ...
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30 views

Littlewood-Paley Decompositions and Periodic Besov Spaces

I'm currently working on some problems in $2$-dimensional periodic space, and it seems that the framework of Besov spaces will be useful to me. Since we're working in periodic space, we can consider ...
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1answer
23 views

The $n^{th}$ partial sum of the Fourier series of a continuous function is a $o(\ln(n))$

I stumbled upon te following result: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ be a continuous $2\pi$-periodic function and $S_n$ denote the $n^{th}$ partial sum of its Fourier series. Then ...
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1answer
41 views

using fourier series to find the sum of series

I have a question about this. Using only the fact that $$ 1 + \frac{1}{3^2} + {1 \over 5^2} + .... = {\pi^2 \over 8} ,$$ can we show $$ 1 + \frac{1}{2^2} + {1 \over 3^2} + {1 \over 4^2} + .... = { ...
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8 views

Finding corresponding signal from Fourier coefficients

If I have Fourier coefficients that are i at k = -1, -i at k = 1, and 0 for all other k how do I find out what the signal is? In my class we only barely covered ...
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1answer
32 views

Determining Fourier series coefficients

I'm just beginning to learn about Fourier series and I'm trying to figure out how to find the Fourier series coefficients for $x(t) = e^{j100\pi t}$ I know also that $$x(t) = \sum_{-\infty}^{\infty} ...
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Fourier sums convergence

I am given the Sin series expansion: (1)$$x=2\sum_{1}^{\infty}\frac{(-1)^{n+1}}{n}Sin(nx)$$ But how do I deal with a solution given by: ...
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59 views

Fourier integration of $f(x)=\pi e^{-x}$

I've tried below Fourier integration and reached some answer.I would appreciate if anyone takes a look at this and enlighten me if something is wrong (or if it is right): $ f(x)=\begin{cases}\pi ...
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2answers
51 views

To show that the limit of the sequence $\sum\limits_{k=1}^n \frac{n}{n^2+k^2}$ is $\frac{\pi}{4}$

Show that $$\lim_{n \to \infty} \sum\limits_{k=1}^n \frac{n}{n^2+k^2} = \frac{\pi}{4}.$$ I am familiar with Taylor series and Fourier series of the standard functions. I tried to compare with those ...
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1answer
26 views

Is the double fourier series just the product of the single series?

Let $f(x)=\begin{cases}x & 0 \le x \le 1 \\ 2-x & 1<x\le 2 \end{cases}$. Find the double Fourier series of $f(x)f(y)$ on $R_{2,2}$ To find the double Fourier series can I just multiply ...
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18 views

Fourier sine and cosine series of these functions?

What are the Fourier sine and cosine series of these functions: for $0 < x < \pi$: $x$ $x(\pi − x)$ $e^{-\alpha x}$ a function which is 1 when $0 < x < \frac{\pi}{2} $, -1 when ...
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34 views

Fourier series, Gibbs Phenomenon, etc.

Is there a periodic function whose nth partial sum is bounded has at most finitely many discontinuities of the first kind and is bounded below by $C\log(n+1)$ where $C>0$?
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31 views

How is this Fourier Transform animation realized from the given equations

I've completed my graduation in Computer Engineering and I've used Fourier Transform many times, yet when I see this animation, I fail to understand how they're realized. Can anyone please explain me ...
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43 views

How do you explain the sine function of a basic triangle wave?

I am working on an investigation focusing on mathematics in music. Modelling various different chords and their mathematical functions, I now need to understand (in relative detail) how the most basic ...
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1answer
25 views

Need help proving the identity $2^{2n} \cos ^ {2n} (\pi x) = \sum _0 ^{2n} {2n \choose k} e_{n - k}$

Let $e_j = \cos(2 \pi j x) + i \sin( 2 \pi j x)$ (the $j^{th}$ fourier basis term on the unit circle $[0,1])$ How can I show $2^{2n} \cos ^ {2n} (\pi x) = \sum _{k=0} ^{2n} {2n \choose k } e_{n - ...
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13 views

Apodization and convolution theorem

Suppose that we exponentially suppress high frequencies by multiplying the Fourier amplitude $\tilde{f(k)}$ by $e^{-\epsilon |k|}$$. Show that the original signal f(x) is smoothed by convolution with ...