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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Q: Bases and Frames using Fourier Series

Define $w: \Bbb R \rightarrow \Bbb C$ by \begin{equation} w(t) =\begin{cases} 1/\sqrt{2\pi} & t \in [0, 2\pi)\\ 0 & \text{otherwise}. \end{cases} \end{equation} and for $n \in \Bbb ...
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189 views

Origin of coefficients of fourier series?

I was wondering how we derive these formulas, and why we have a separate formula for $a_0$? All I know from advanced engineering mathematics text book are following formulas but where do they come ...
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124 views

Prove that the Fourier series of $\dfrac{1}{f}$ is absolutely convergent.

I have a problem: Let $f$ be a continuous function on the unit circle $(\Gamma)$: $$\Gamma=\{e^{i\theta}: \theta\in [0, 2 \pi]\}$$ Assume that $f \ne 0$ on $\Gamma$, and the Fourier ...
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267 views

Fourier Sine Series extension

If $\phi(x)$ is any function on $(0, l)$, derive the expansion $\displaystyle\phi(x) = \sum c_n \sin\left(\left(n + \frac{1}{2}\right) \frac{\pi x}{l}\right)$ for $0 < x < l$ by the following ...
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38 views

Find the completeness radius of the prime numbers

As the title says, I'm trying to find the completeness radius of $\{2,3,5,7,11,\ldots\}$. The completeness radius of a sequence $\Lambda=\{\lambda_n\}$ is $R(\Lambda)=\sup\{A~|~\{e^{i\lambda_n ...
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63 views

(Real Discrete) Fourier Series: Normalisation Factor

If you have the equation: $$f(t) = \sum_{k=0}^N \left( A_k \cos \omega_k t + B \sin \omega_k t\right)$$ To compute the $A_k$ Fourier coefficient you have two cases: $$A_k = \color{\red}{{2 \over ...
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55 views

Is $f$ is non-prime, Can we say $|f|$ is also non-prime ; in convolution algebra?

By Schwartz-inequality and Riesz–Fischer theorem, one can deduced that, $$L^{2}(\mathbb T) \ast L^{2}(\mathbb T) = A(\mathbb T)(:= \{f\in L^{1}(\mathbb T): \sum_{n\in \mathbb Z} |\hat{f}(n)| < ...
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42 views

Trigonometric functions are dense in Sobolev Spaces

So I am trying to prove that the following functions $\{f_n=\frac{1}{\sqrt{2 \pi}}e^{inx}\}_{n=-\infty}^{n=\infty}$ are dense in the space $H^{n}[0,2\pi]$. For the proof it would be safe to assume ...
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235 views

Condition on Fourier coefficients for real-valued function

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is real-valued if and only if ...
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127 views

Evenness of Fourier coefficients implies even function

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$Assume that the Fourier coefficients determine an $L^1$ ...
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53 views

Evenness of Fourier coefficients

Let $f\in L^1(\mathbb{R}/2\pi\mathbb{Z})$ and let $F(n)$ denote its Fourier coefficients $$F(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx$$ I want to prove that $f$ is even if and only if ...
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625 views

Is it possible to do a half-range sine expansion on the sine function?

Suppose $f(x)=sinx,\quad 0<x<π$. Can you do a half-range sine expansion on f(x)? I tried, but I got $a_0=a_n=b_n=0$. If you requrie me to show my steps (i.e. I should have not gotten ...
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70 views

Does $\int_{0}^{\pi/2n}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t\leq \pi$ hold for $n \geq 2$?

Today I am trying to prove an integral inequality: $$\frac{1}{\pi}\int_{0}^{\pi/2}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t<\frac{2+\ln n }{2}$$ where $n\geq 2$ and $n \in \Bbb{N}$. First, ...
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80 views

Find the Fourier coefficient for the function $f(t)=t^3-2t$

I'm trying to find the $a_5$ Fourier coefficient for $f(t)=t^3-2t$ from $-π<t<π$ I think I'm supposed to get $a_5=0.32$. However, I don't know how to get there. If that is the right answer, ...
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76 views

Prove that $\sum\limits _{n=-\infty}^{n=\infty}\cos\left(2\pi nt\right)=\sum\limits _{n=-\infty}^{n=\infty}\delta\left(t-n\right)$

I've tried using Fourier transforms on both but didn't quite get anything useful. I'd really appreciate some help.
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973 views

fourier series of $|\sin x|$

I need to find the fourier series of $$|\sin x|$$. Im not sure my way is right, would be happy if someone fix me. I found $$a_0=4/\pi$$, the function is even, so $$b_n=0$$ but how do I calculate: ...
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113 views

Why is there an exponential in Fourier's defining integral?

I am having a hard time relating integration with Fourier series. Basically, I just get lost where there is an exponential in the integration to convert into the frequency domain. If someone can ...
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306 views

Show a Fourier series converges uniformly

I need to show that the Fourier Series of |x| in the interval $(-\pi, \pi)$ converges uniformly to |x| in $[-\pi, \pi]$. I know that |x| = $\frac{\pi}{2}$ + ...
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234 views

Is the matrix Wn from the DFT a Hermitian operator?

A homework question asks me whether or not the matrix $W_N$ from the matrix representation of the Direct Fourier Transform is a Hermitian operator. From what I understand an Hermitian operator does ...
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154 views

Multi layer perceptron activation function

How can you show that the Fourier series approximation of a function (so $f(x)=\sum\limits_{n=0}^{\infty} (a_n cos(nx) + b_n sin(nx))$ can be approximated to arbitrary precision by a feedforward ...
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1answer
135 views

What does it mean for a function to converge at a point of discontinuity?

In Fourier analysis, if $x$ is a point of discontinuity of $f(x)$, then $f(x)=\frac{f(x^+)+f(x^-)}{2}$. How is this Convergence? Uniform convergence? Pointwise convergence?
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178 views

Fourier Series: Shifting in time domain

I am reading "Fourier Transformation for Pedestrians" from T. Butz. He speaks about what happens to the Fourier coefficients when the function is shift in time. I have copied the equation I have a ...
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42 views

inverse transform of $Z(\omega) =\frac{a}{\alpha-i\omega}$

I am stuck at calculating the inverse transorm of $Z(\omega) =\frac{a}{\alpha-i\omega}$. Can someone help me please? thanks
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Fourier's Method Question

I've been asked to use Fourier's method to obtain the following solution; $$u(x,t) = \sum_{n=1}^{\infty} B_n e^{-(n \pi C / L)^2 t} \sin(\frac{n \pi x}{L})$$ $$B_n = \frac{2}{L} \int_0^L \sin(\frac{n ...
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355 views

Why output of FFT is same as input data size ?

What I understand from DFT formula below I can decide the N my self. I can try to use just 16 bins to describe a function or I may even use 4 , it won't be very accurate but I can do it right? The ...
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1answer
56 views

Fourier sine series for function F(t) = t for 0<t<L

How to get to the part circled with red? I tried to compute it on Wolfram alpha.. (http://www.wolframalpha.com/input/?i=2%2FL+*Integrate+x+sin%28%28n+pi+x%29%2FL%29+dx+from+0+to+L) Still confused. ...
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353 views

Discrete Fourier Transform: Understand Negative Frequencies

I am trying to learn DFT on my own. I have been struggling for a while now around understanding the concept of negative frequencies and notably what happens when $k$ is greater than $N/2$ in the ...
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370 views

3D fourier series

I wonder how I can write a function $f(\textbf{r})$ as a fourier series, when $f$ is periodic, in the sense that there exists a $ \textbf{T}_i \neq \textbf{0} $ so that $f(\textbf{r} + \textbf{T}_i) = ...
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Is it possible to calculate a single frequency bin in $O(\log N)$ time (considering the $N\log N$ performance of FFT algorithms)?

Fast Fourier transform (FFT) algorithms are able to calculate the discrete Fourier transform (DFT) in only $O(N\log N)$ asymptotical time. Since there is roughly $N\log N$ operations for computing $N$ ...
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63 views

Fourier series of a square wave signal with a bias

Given a $f(t)$ of the kind: $$f(t)=1, \{kt_0\le t\le kt_0+\tau\}$$ $$f(t)=a,\{kt_0+\tau\le t\le (k+1)t_0\}$$ with $a\lt 1$ what is the Fouries series development of f(t)? Thanks
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355 views

Find the Fourier series S(t) of the period 2$\pi$

Find the Fourier series S(t) of the period 2 function $f(t)=\begin{cases} -1& \text{if −$\pi$ < t < 0;}\\ \;\;\;1& \text{if $\:$0 < t < $\pi$;}\\ 0&\text{if $t = −\pi, 0, ...
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1answer
105 views

Discrete Fourier Series: What Happens After N/2?

I am really confused! I started to study Fourier series. I think I understand the theory approximately (I am still new to it). I was curious so started to read about DFT which I thought would be ...
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35 views

Does this transformation have an inverse?

Let $f(n)$ be a complex sequence. Then for prime $p$ define $\hat{f}(p) = \sum_{n = 1}^{\infty} a_n e^{-i 2 \pi n / p}$. Then let the transformation of sequences be $T$, i.e. $Tf = \hat{f}$. Is ...
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1answer
101 views

How can we prove that a Fourier Series exists?

How does one show that an arbitrary periodic function, so long as it is reasonably well behaved, can always be represented as a sum of sine and cosine functions? It sounds like the first thing you ...
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107 views

Sinusoid sum of cosine and sine

I am studying Fourier series right now. I asked a question before of math.statckexchange regarding Fourier series. This question is related and hopefully quite simple: Generally Fourier series works ...
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78 views

Harmonic Oscillator and Fourier Series

I am currently studying Fourier Series (on my own). I am using a few different references/sources. Some are more trying to give an intuition about Fourier Series and others are more rigorous. ...
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119 views

Interpreting the sign of fourier coefficients

I am studying the Fourier series right now. Hopefully it's going okay. Now I have been playing a little bit with taking the product of a wave function (a sine or cosine with some phase) with a sine ...
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42 views

Working with infinite sequences in $\ell^2(\mathbb{Z})$

Let $\ell^2(\mathbb{Z})$ be the set of all two-sided sequences $(a_i)$ in $\mathbb{C}$, such that $\sum_{n\in \mathbb{Z}} |a_n|^2 \lt \infty$. What considerations do I have to take into account when ...
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91 views

Fourier Series: going from $a_n$ and $b_n$ to $c_n$

I sort of understand the principle of the Fourier series, but when I watch the wiki page I don't understand how to get from: ${a_0 \over 2} + \sum_{n=1}^N[a_n cos({2\pi n x \over P}) + b_n sin({2\pi ...
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138 views

The Fejer kernel has this $\sin$ closed form.

Let $D_N$ be the $N$th Dirichlet kernel, $D_N = \sum_{k = -N}^N w^k$, where $w = e^{ix}$. Define the Fejer kernel to be $F_N = \frac{1}{N}\sum_{k = 0}^{N-1}D_k$. Then $$F_N = ...
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312 views

If $f_n \rightarrow f$ and $f$ is uniformly continuous, then does $f_n \rightarrow f$ uniformly?

Let $f_n$ be a sequence of functions, $f_n : S\rightarrow T$, not necessarily continuous and suppose that $f_n \rightarrow f$ as $n \rightarrow \infty$. Let $f$ be uniformly continuous. I.e. for ...
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196 views

The Fourier series converges absolutely $\implies$ it converges uniformly.

Let $S_N(f)$ be the $N$th partial sum of the Fourier series for $f$. I.e. $$S_N(f) = \sum_{n = -N}^{N} \hat{f}(n) e^{2\pi i n x / L}$$ Suppose that the Fourier series converges absolutely, i.e. ...
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57 views

How do they do this w.l.o.g. so freely (Fourier series).

Theorem 2.1. Suppose that $f$ is an integrable function on the circle with $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$. Then $f(\theta_0) = 0$ whenever $f$ is continuous at the point ...
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1answer
830 views

Error Term for Fourier Series?

Suppose I have a piecewise smooth $2 \pi$-periodic function $f$ on $\mathbf{R}$ with a Fourier series $\sum_{n \in \mathbf{Z}}a_n e^{inx}$, a number $x_0 \in \mathbf R$, and $N>0$. I would like an ...
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111 views

Can I calculate approximately a definite integral of a function by integrating its Fourier Sine Series term-by-term?

I'm not sure how to put fancy formulae here because I'm a fairly new user. So bear with me for a moment as we go through a formulae-less reasoning. 1) I have a function $f(x)$. 2) I want to ...
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1answer
71 views

Find the first four terms in the Fourier series for a solution of the wave equation

The question is to find the first four terms in the Fourier series for $u(x,t), t>0$. It is for a plucked string of length L, has zero initial displacement (I.e. $u(x,0)=0, 0<x<L$) and ...
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532 views

fourier expansion of $\coth$ and justifying an identity

The problem: Justify the following equalities: $$\cot x = i\coth (ix) = i \sum^\infty_{n=-\infty} \frac{ix}{(ix)^2+(n\pi)^2}=\sum^\infty_{n=-\infty}\frac{x}{x^2+(n\pi)^2}$$ I am trying to figure ...
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123 views

Checking work on Fourier series for $10 \cos t$

Well, I am checking this out because even though I know the problem (a 2nd order differential) can be solved more easily, I want to try this out. So we have $10 \cos t$ and want the Fourier ...
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1answer
505 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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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 $2\pi$), ...