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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39 views

Theoretical question about Fourier Series, I'm confused!

If I have a function f(x) defined on $[0,L)$, said to be periodic of period $L$ and such that $f(0)\neq0$, how should I get the Fourier coefficients? I'm hesitating between taking the even extension ...
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152 views

Parseval Identity and Fourier Series Question on function $f(x)=|x|$

Trying to compute the fourier series for $f(x)=|x|$ for $f$ on $[- \pi, \pi]$ using the trig method. I have a question as to the absolute value function. I'm using the definition of absolute value ...
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31 views

Hilbert space (nonseparable): ONB

Every Hilbert space admits an ONB by axiom of choice. For separable Hilbert spaces this can in fact be constructed by Gram-Schmidt. For nonseparable Hilbert spaces there can be no general construction ...
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35 views

Using a family of functions to find fourier series

I'm given a family of functions $$T= \left \{\frac{1}{\sqrt{2\pi}},\frac{1}{\sqrt \pi} \cos n\pi, \frac{1}{\sqrt \pi} \sin n \pi: n=1,2,3,\ldots \right \} , $$ on the interval $[-\pi, \pi]$ ...
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92 views

Fourier Series of $f(x)=e^x$ on $[0,\pi)$ as a function of period $\pi$

Can you tell me what you get? I've tried computing it, I've got some result but I don't think it's right since I need to use it for something else and it doesn't work at all... What exactly I'm trying ...
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25 views

Conditions for Uniform Convergence of Fourier Series

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a 2$\pi$ periodic function such that $\exists$ $C>0$ and $\epsilon>0$ with $|f(x)-f(y)|\leq C|x-y|^{.5+\epsilon}$. Show that the the Fourier series ...
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54 views

Let f be a continuous real valued function on R, and prove that $f(x)$ is constant using the fourier series.

I missed my class where we went over the fourier series and am having extreme issues with this homework question. $f(x) = f(x + 1) = f(x + \sqrt{2})$ Is there anyone who could be kind enough as to ...
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24 views

Inverse Fourier Transform

I need help solving the following Fourier transform question. Given, $$ X_s(f) = \frac{1}{\Delta T} \sum_{n = -\infty}^{\infty} X\left(f - \frac{n}{\Delta T} \right) $$ $$ H(f) = \begin{cases} 1 ...
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45 views

What function does the Fourier series $\pi^2 / 6 + \sum^{\infty}_{k=1} \frac {-1} {k^2} \cos(kx) $ converge against?

What function does the Fourier series $$\pi^2 / 6 + \sum^{\infty}_{k=1} \frac {-1} {k^2} \cos(kx) = \pi^2 / 6 + \sum^{\infty}_{k=1} \frac {-1} {2k^2} (e^{ikx} + e^{-ikx})$$ converge against ? I've ...
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44 views

trigonometric interpolation of a sampled signal

Given N sampled points, using the FFT we can get the Fourier transform of those N points $X_k$. With N/2 the Nyquist frequency and $X_0$ the DC value. Using the inverse we can then get back the ...
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32 views

fourier series representation

Find the Fourier series with period $2$ of $$f(x) = -x,\qquad-1<x<1$$ so I find that $a_0$ and $a_n$ both are $0$ since odd functions so the Fourier series is on the form: ...
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45 views

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$.

Find the Fourier series for $f(x) := |\sin(x)|$ and the sum of $\sum_{n=1}^{\infty} \frac {(-1)^{n+1}} {4n^2-1}$. I have computed $$c_n = \frac 1 {2\pi} \int^{\pi}_{-\pi} |\sin(x)|e^{-inx} dx = ...
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28 views

Redundant assumption in an exercise concerning fourier series?

So here is my problem, I have to solve the following exercise, Let $\phi\in L^1[0,1)$ and $\psi\in L^{\infty}[0,1)$, both of period 1 and $\int_0^1\psi(t)dt=0$. Show that $$\lim\limits_{n\rightarrow ...
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23 views

A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
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23 views

Prove that $||f||_2 \le \sqrt{2 \pi} ||f || _{\infty}$

Let $||f||_2=\sqrt{\int_{-\pi}^{\pi} f^2(x) dx}$ $||f||_{\infty}=\sup \{ |f(x)| \mid x \in [-\pi,\pi]\}$. Suppose $f: \mathbb{R} \to \mathbb{R}$ an in the space of piecewise continuous functions ...
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3answers
116 views

Calculating own dft via matlab?

We are asked to code our own dft function from the formula : If everything is done correctly it should give the same result with matlab's own dft function, in the end I'm comparing them but they ...
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1answer
47 views

Uniform convergence of the Fourier Series using Bessel's inequality

Consider the Fourier series of $f$, $$ \frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx) $$ Let $$f_n(x)= a_n \cos(nx) + b_n \sin(nx)$$ Then to show that $f_n(x)$ is uniformly ...
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19 views

Complex form fourier series of a sum of e

The heart of the problem is finding a fourier series in its complex form for: $\displaystyle\sum _{k=-\infty }^{\infty } e^{-4|t-k|}$ The form I know of is $\displaystyle\sum_{k=-\infty}^{\infty} ...
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43 views

Find Fourier series coefficients of $f(x)$.

$T=2$ $$f(x) = \begin{cases} 1, & \text{$-\frac12\le x \le\frac12$} \\[2ex] |2x|, & \text{$\frac12 < x \le1\frac12$} \\ \end{cases}$$ The image: I found that $a_0=\frac12$. Since ...
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19 views

Explicit formulas for Fourier coefficients from its Tayor expansion

In my research, I need to determine unique coefficients $a_k$ in terms $b_k$: $$\sum_{k=0}^n a_k \cos(\frac{k}{n+1}t)+O(t^{2n+1})=\sum_{k=0}^n b_k t^{2k}$$ This problem showed up in my search of ...
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59 views

Square-summable sequence and Fourier series

Every square-summable sequence $(a_{n})_{n}$ is represented by $a_{n}=\widehat{f}(4^n)$, where $\widehat{f}(i)$ is Fourier coefficient of continuous function $f$. Where can I find proof of this ...
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54 views

What is this waveform?

Consider the following infinite series: $\text{f} \left( x \right) =\displaystyle \sum \limits_{n=1}^{\infty} \frac {\sin \left( n x\right)}{n^2}$ We know that $\text{f} \left( x \right)$ is ...
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110 views

Let $f(x) = |\cos(x)|$. Prove the corresponding Fourier series converge point-wise or uniform and show identity.

Consider $f(x) = |\cos(x)|$ for $x \in \mathbb R$. I've proved the n'th fourier coefficient $c_n = \int^{\pi}_{-\pi} f(y)e^{-iny} \ dy = \frac 1 {2\pi} \frac {(-1)^{n-1}} {n^2-\frac 1 4}$. However, ...
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30 views

Fourier series, estimate the values of $a_0$, $a_n$ and $b_n$.

Using the following periodic function (period of $2\pi$) $$F (x) =\begin {cases} 4.&-\pi \lt x \lt -\pi/2\\ -2.& -\pi/2 \lt x \lt \pi/2\\ 4.&\pi/2 \lt x \lt \pi ...
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25 views

Half range expansion

This is a exercise of sine half range expansion I do the bn expansion and is not the final result
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47 views

How to obtain a periodic function from a rapidly decaying function?

Suppose $f(x) = \exp(-x^2)$ with $x \in [0, 3]$. How could I periodise this function to obtain an analytical form of a continuum periodic function $x \in [0, +\infty)$ with period T = 3?
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35 views

Series convergence in Hilbert space and dual.

I'd like to prove that: $$ \|u_\varepsilon-f\|_*\rightarrow0 \quad\text{in }V^* $$ with $V$ Hilbert and $V^*$ its dual. In particular $u_\varepsilon\in V$. From the precedent points of the proof I ...
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43 views

Fourier Series and sum help

I have to find the Fourier series expansion of the function $f(x)$=$x^2$ for $-\pi <x< \pi$ and using it I have to show that, i) $1+\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+\frac{1}{25}...$ = ...
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16 views

Pointwise convergence of periodic functions

Let ${f_n}$ be a sequence of functions on $\mathbb{R}$ which satisfy $f_n(x+2 \pi) = f_n(x)$ for all $n$ and $x$. Suppose that $f_n \rightarrow f$. Prove that $f(x+2 \pi) = f(x)$ for all $x$. My ...
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40 views

Want to prove certain sum representation of $\cot(x)$

So here is my problem, I would like to prove an identity I found in a book which was given without a proof. Namely $$-i\sum_{n\in\mathbb Z} \operatorname{sign}(n)\cdot e^{i2\pi nx}=\cot(\pi x)$$ I ...
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1answer
42 views

Fourier series expansion

Is it possible to have a Fourier sine series expansion like $$ \sin\left(\left(\frac{\pi}{2} + n\pi\right)x\right) $$ instead of the normal $$\sin(n \pi x)$$
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29 views

Finding Fourier Series

I know that this is a rather simple problem but i have some confusion here : I have to find the Fourier series representation of $f(x)=x$ for $-\pi<x<\pi$ and for $0<x<2\pi$. My ...
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25 views

Fourier Transform of $f(t+a)$ if $f(t)$ has tranform $F(k)$?

I know the formula $$f(t) = \int^{+\infty}_{-\infty} F(k)e^{ikt} \, dk$$ and I've seen that for computing $f'(t)$ it's a case of differentiating $e^{ikt}$ inside the integral, so $f'(t)=ikF(k)$ Can ...
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33 views

Want to prove that the Hilbert transform of a $C^1(\mathbb T)$ function is the principal value of the convolution with $\cot(\pi x)$

So here is my problem, Let $L^2_0:=\{f\in L^2: \hat{f}(0)=0\}$ and consider the Hilbert transform given by the following map $$H:L^2_0([0,1])\rightarrow L^2_0([0,1])$$ $$f\mapsto (\mathcal ...
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Plot a fourier series

I'm not a mathematician but I hope my question is easily answered. I'm trying to learn about graphing equations in a computer application (I'm a programmer). From this link a fourier series is ...
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23 views

How to see this step in deriving an equality in Fourier series?

(Previous steps are omitted.) By convergence of Fourier series, we have $$ \frac{\pi}{4}+\sum_{n=1}^{\infty}\frac{(-1)^n-1}{\pi n^2}(-1)^n=\frac{\pi}{2} $$ Then how come we can get this from the ...
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41 views

Finding fourier sine series using another cosine series

I have to find the sine series of $x^3$ using the the cosine series of $x^2/2$. $${x^2 \over 2}={l^2 \over 6}+{2 l^2 \over \pi^2}\left[\sum{(-1)^n \over n^2}\cos\left({n\pi x \over l}\right)\right]$$ ...
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Fourier Series from product of to functions

I have to calculate the Fourier Series of $x\sin(x)$ beeing $2\pi$ periodic on $[-\pi,\pi]$and i did it the standard way. But then i wanted to solve the problem with multiplication of two fourier ...
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23 views

Fourier Series Fourier Transform Method

I understand $f$ is even about $\pi$ but i'm struggling conceptually with the part I have underlined.
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78 views

How to find the Fourier series of $f(x)=x$?

I got a question which is too simple: Find the Fourier series of $f(x)=x$ in the interval $[-\pi,\pi]$ and show that this function doesn't converge to its Fourier series. I found the series as ...
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207 views

Any good introductory book/tutorial on Fourier Transform (up to FFT) with plenty of exercises and solutions?

I wonder what could be a good book to start learning in depth all aspects of the Fourier transform up to the FFT algorithm, and beyond. I am going to dedicate quite some time on the subject, so I ...
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1answer
41 views

Finding Fourier of $x^3$ by Fourier of $x^2$

I found the cosine series of $x^2/2$ to be (by first finding Fourier sine series for $x$ on $(0,l)$ and then integrating that term by term) $${x^2 \over 2}={l^2 \over 6}+{2 l^2 \over ...
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89 views

Fourier Series; odd and even half-range expansion

I have some standard Fourier series questions which I cannot solve. My fourier series is defined like this: $$s(x)=\frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos (nx) + b_n \sin (nx))$$ For $f(t) = ...
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45 views

Series expansion Fourier-Legendre

Can anyone explain me how can I expand this function using the Fourier-Legendre expansion? f(x) = x ; -1<=x<=1 Thanks.
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32 views

Parallel between Fourier Series and orthogonal projections

My professor made an analogy between Fourier series and orthogonal projections and I was hoping someone could explain that someone more. Basically, as I understand it: $$ a_n = \frac1L \int_L^L ...
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44 views

Question on Fejér Theorem

I need to solve a problem, which tells me to find the Fourier coefficients of the function $f(x) = |x|, \quad \text{for} \quad x \in [-\frac12, \frac12]$ and show that $$f(x) = \frac14 + \lim_{n ...
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1answer
123 views

Numerical approximation of trigonometric polynomial

I have the following problem: Let $g$ be a trigonometric polynomial of degree n (there are complex coefficients $c_k$ with $k = -n, ..., n$ such that $g(t) =\sum\limits_{k = -n}^n c_{k}\exp(ikt). $ ...
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40 views

Fourier Series to Laurent Series

Given a periodic function $f(\sigma)$ with period $T$, one can compute its Fourier series, $$f(\sigma)=\sum_{n\in\mathbb{Z}} c_n e^{i \omega n\sigma}$$ where $\omega=2\pi/T$ and the coefficients of ...
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33 views

pointwise limit of uniformly bounded sequence in $A(\mathbb T)$ is again in $A(\mathbb T)$?

Let $\mathbb T$ be a circle group, and $\hat{f}(n)= \frac{1}{2\pi}\int_{0}^{2\pi} f(t) e^{-int} dt;$ $(n\in \mathbb Z, f\in L^{1} (\mathbb T)).$ Put $A(\mathbb T)= \{f\in C(\mathbb T): \hat{f}\in ...
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29 views

What is the fourier transform of rect( ( x - a) / L)

I thought of it as follows: $$ F \left[ \Pi \left( \frac{x - a}{L} \right) \right] ( k_x ) \\ = F \left[ \Pi \left( \frac{x}{L} - \frac{a}{L} \right) \right]( k_x ) \\ = \exp \left( -2 \pi i k_x ...