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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Fourier series for function

Consider the function f(x) = |x| $ - \pi \leq x < \pi $ Compute its Fourier series. $ a_{0} = \frac{1}{\pi} \int_{-\pi}^{\pi}|x| dx = \frac{2}{\pi} \int_{0}^{\pi} x dx $ I ...
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32 views

Signum function and Fourier transform

I'm extracting a portion of my notes which I believe I might have copied wrongly. Given this equation: $$\frac{G(\omega)}{2ic\omega} [e^{ic\omega t}-e^{-icwt}]$$ I want to find the Fouerir ...
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20 views

different formulas for the fourier series.

Quick question, I see both of these $$f(w) = a_0 + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw) \quad f(w) = \frac{a_0}{2} + \sum_{k = 1}^{\infty} (a_kcos(kw) + b_ksin(kw)$$ Why the difference ( ...
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1answer
50 views

Parseval identity for $L^2(a,b)$?

The Parseval Identity states that: $\sum_{n=-\infty}^{\infty}|c_{n}|^2= \frac{1}{2 \pi} \int_{-\pi}^{\pi} |f(x)|^2 dx$ Where $\{c_{n} \}$ are the Fourier coefficients of f. Is there a general ...
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19 views

Fourier series function

$f(x) = x$ , $f(x+2\pi) = f(x) $ on $ [-\pi , \pi] $ How do I know that this function is even or odd? My book says odd, but I don't understand how to work this out? also why does $a_0 = 0$ ...
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24 views

Is $\sin(kx) $ a complete system in $L^{2}(0,b)$ for every k in $\mathbb{R}$?

Is the family $\{ \sin(kx ) \}$ a complete system in $L_{2}(0,1)$ for every $ k \in \mathbb{R}$, $k\geq1$ ? And less specifically, Is it a complete system in $L_{2}(0,b)$ for every b$\in \mathbb{R}$ ...
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2answers
66 views

Fourier series: Why is there a separate formula to determine $a_0$?

$$a_n = \frac 1 \pi \int_{-\pi}^{\pi} f(x) cos(nx) dx \quad\quad n\ge1 $$ Now I am wondering why there is a separate formula for $a_0$: $$a_0 = \frac 1 \pi \int_{-\pi}^{\pi} f(x) dx$$ It looks ...
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1answer
43 views

Fourier coefficients of $\cos(x/2)$

Is there a straightforward way to calculate the fourier coefficients of $\cos(x/2)$ in closed form on the interval $[0,2\pi]$? (I mean in terms of the generic $n$) From a calculation of the first ...
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32 views

Prove that the following function is $C^\infty$ in the point $\xi=0$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any way how to prove this? i think that i must use ...
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41 views

What do real and imaginary parts of phase spectrum represent?

In frequency domain, we can compute phase spectrum of a signal. Usually phase spectrum is complex valued. So my question is what do real and imaginary parts of phases of phase spectrum represent ? ...
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38 views

Approximation of the coefficients of the Fourier Series via the FFT

Is there literature on the approximation of the coefficients of the Fourier Series via the FFT? The approach I'm interested is merely numerical, consisting of computing the integrals with the ...
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27 views

Fourier Series Divergence

Define $b_n = \int_0^{\pi} \cos (nx).\sqrt{x} ~ dx $. Does the following series converge? $$ F(x) = \sum_{n=1}^{\infty} b_n \cos(nx) $$
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45 views

Is there any pde whose solution evolves as a partial Fourier integral?

Is there any partial differential equation such that the its solution evolves as partial Fourier integral (continuous version of partial sum) of a function $f(x)$ which might be an condition or ...
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1answer
50 views

Solving system of equations in polar and Cartesian coordinates, involving a Fourier series

I have an equation of body surface is polar coordinates defined as Fourier series: $$ r=r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] $$ Also I have a line equation in Cartesian coordinates: ...
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1answer
27 views

Evaluate $\sum_{n=-\infty}^\infty \hat{g}(n)$

Let $f:\mathbb{R}\to\mathbb{R}$,$f$ is $2\pi$ periodic such that for $x\in (-\pi,\pi]$: $f(x) = \cos \pi x$. We define $g(x) = f(x + 2010)$. Find $\sum_{n=-\infty}^\infty \hat{g}(n)$. Where ...
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1answer
26 views

Fourier series of piecewise function

Ok, so this is a pretty simple function. I tried to plot the Fourier Series vs the function to see if I did things correctly, but the curves are just not having the same form. Consider the function ...
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1answer
47 views

Definite integral evaluation by means of Fourier series

Does anybody know how to compute the definite integral $\int\limits_0^{2\pi} \dfrac{x\cos{(mx)}}{1-a\cos{x}}dx$, where $|a|<1$, using Fourier series? Help is required. Thanks!
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35 views

Find the Fourier coefficient of $f(x)$

Let $$f(x) = \begin{cases} \sum_{k=0}^\infty \frac{e^{inx}}{1+k^2} &\mbox{if } x \ne 2\pi k \\ 0 & \mbox{if } x = 2\pi k \end{cases}$$ Find the Fourier coefficients of $f(x)$ What I ...
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Fourier series of $f(t) = (t-|t|) ^2$

Apparently what I thought was a absolute value was a average down sign, I. E. if the value is 2.9 the sign will make it 2. Just got back from the professor. In my last exam of wave physics that I ...
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221 views

Interesting Integral with Parameters

I would like to compute the following integral: ...
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1answer
38 views

Where do the coefficient equations for Fourier series come from?

I don't see where the equations come from like: $$a_0= \frac{1}{2L} \int_{-L}^L f(x)~dx$$ And like wise for $a_n$ and $b_n$. Also where does the general formula for a Fourier series come from? If ...
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45 views

limits and integrals

Show that $$ \lim_{n \to \infty} \int\limits_{0}^{h} \frac{\sin (n\varepsilon)}{\varepsilon} \;\mathrm{d}\varepsilon = \int\limits_{0}^{\infty} \frac{\sin (t)}{t} ...
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1answer
31 views

Uniform convergence for functions with jumps

We know that Fourier partial sums (integrals) do not converge uniformly for BV functions with jumps due to Gibb's phenomenon. Is there any other types of sums/procedures that use only Fourier ...
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1answer
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Conditions for convergence for a Fourier series

Let $f$ and $f'$ be piecewise continuous on the interval (-p,p); that is, let $f$ and $f'$ be continuous except at a finite number of points in the interval and have only finite discontinuities at ...
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32 views

Fourier Transform Good Articles

I need to use FT in one of my programming projects, but I need to refresh myself on it. Any good books and articles? The last time I studied it was 12 years ago, when I was in college.
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120 views

Why Fourier series has summation and Fourier transform has integration symbol in their respective formulae?

Fourier transform for aperiodic signal is given by $$ X(\omega) = \int\limits_{t=-\infty}^{+\infty} x(t) e^{-j \omega t} dt. \quad (1) $$ Fourier series for periodic signal is given by $$ y(t) = ...
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42 views

If$f \in L_1 (R)$ so that $xf(x), x \hat{f}(x) \in L_1 (\mathbb R)$, show that $\sum|\hat{f}(n)| \le \infty$ and finally Poisson summation formula

PROBLEM (1)$f \in L_1 (\mathbb R)$ so that $xf(x), x \hat{f}(x) \in L_1 (\mathbb R)$, show that $\sum|\hat{f}(n)| \le \infty$. (2)Also, show that $\sum_{n\in \mathbb Z} f(x-2\pi n)$ converges ...
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52 views

Simple Trig Identity?

I have the equation $\sum_1^N \cos\omega_p t\cos\omega_qt$ Where N is an even number representing the number of time steps $\omega_p=\frac{2\pi p}{N}$ p=1,...,$\frac{N}{2}$ I need to prove the ...
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3answers
38 views

Is Fourier transform a generalisation of Fourier series?

Is the Fourier transform a generalisation of a Fourier series or an a different concept? I.e. Can Fourier transforms be used with periodic functions and will it reduce down to the Fourier series ...
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35 views

Fourier series normalize

Let $ \mu \in \mathbb { R} $ and let $$ f ( x ) = e ^{ \mu x } ,\ x \in (- \pi , \pi ] . $$ i) Arguments for that the Fourier series $ \sum_ { k = - \infty } ^ \infty c_k e ^ { ikx } $ for $ f $ ...
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A Simple Fourier Transform [duplicate]

I am studying about the randomprocess thesedays. I am stuck on solving the discrete signal to show the fourier transform the formula is that $$ w_b(k) = {N-|k|\over N} \quad \quad when\ \ |k| ...
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3answers
80 views

Represent $\cos(x)$ as a sum of sines - where is my mistake

Let's look at $f(x)=\cos(x)$ defined on the interval $[0,\pi]$. We know that for any function $g$ defined on $[0,\pi]$ we have: $g(x)=\sum_{k=1}^{\infty}B_k\sin(kx)$ where ...
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23 views

Using fourier expansion to find the constant

I have the expression $$1=\sum^\infty_{n=1} c_n \left(\frac{\pi n}{3}\right) \cosh \left(\frac{2 \pi n y}{3}\right) \sin\left(\frac{\pi n y}{3}\right)$$ I am trying to find the constant $$c_n$$ The ...
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$h_k(x):=\sum_{n \in \mathbb{Z}} F(f_k)(n)e^{-inx} \rightarrow h(x):=\sum_{n \in \mathbb{Z}} F(f)(n)e^{-inx} $?

Let $f_n \rightarrow f$ be a sequence of functions in the $L^1$ sense. Then the Fourier transform implies $||F(f_n) -F(f)|| \rightarrow 0 $ uniformly. Now, I was wondering. Does this imply that the ...
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What is the range on a fourier transform?

In particular, I want to know the range of the coefficients on the type-IV discrete cosine transform. Assuming no normalization factor or window is applied, what interval can I expect the coefficients ...
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1answer
26 views

Complex version of Fourier series

Let a be a positive real number and f $2\pi$ periodic function defined by: $f(x)= \begin{cases}0 & \text{if }-\pi\lt x\lt 0 \\ 1 &\text{if }0\le x\le a \\ 0 &\text{if }a\lt x\le ...
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1answer
29 views

Convergence of a subsequence in $(C(\mathbb{T}), \|\cdot\|_2)$

Problem: Define, $ \mathbb{T} := \mathbb{R}/{2\pi\mathbb{Z}} $. Consider a sequence of functions $(g_n)_{n\in \mathbb{N}} \in C^4(\mathbb{T})$ such that, $ \sup_{n \in \mathbb{N}}(\| g_n \|_2 + \| ...
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1answer
88 views

A problem similar to $L^2$ Fourier transform, but in the setting of Borel measure.

Problem: Let $\mu$ be a finite Borel measure on the real axis, supported on a countable set $\mathbb{Q} \subset \mathbb{R}$ (I'm not sure whether here $\mathbb{Q}$ is all rational numbers ). And let ...
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1answer
44 views

Series in hyperbolic sines.

I was looking into a problem and I arrived to something in which I want to expand some function $\varphi(x)$ in series of hyperbolic sines, something like: ...
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37 views

How to find coefficients in a power of a trig series?

Suppose $m$ is a positive integer, and we know that the following identity holds for all $0<x<2\pi$: ...
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140 views

Is Fourier series used always for periodic signals and Fourier transform for aperiodic signals only?

I want to ask basic question. In our mathematics classes ,while teaching the Fourier series and transform topic,the professor says that when the signal is periodic ,we should use Fourier series and ...
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Obtaining $\sum_{n=1}^{\infty} a^n \cos{(n\theta)} = \frac{a \cos{\theta}-a^2}{1-2a\cos{\theta}+a^2}$

This is a homework problem. From Fourier Series and Boundary Value problems, Brown/Churchill 8th ed. I should begin with $2\cos{A}\cos{B}=\cos{(A+B)}+\cos{(A-B)}$, substitute with $A=n\theta$ and ...
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On converting from real to complex Fourier series

Let a real-valued function $f$ be defined as following: $$f\left ( x \right )=\left\{\begin{matrix} 2k-x, x\in\left [ 2k-1,2k \right ) & \\ x-2x , x \in \left[ 2k,2k+1\right )& ...
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2answers
42 views

Finding the particular solution of a pde.

I have solved a PDE up to the point of finding the particular solution. I am trying to find the constant $$C_n$$ I have the expression $$3x-x^2=\sum_{n=1}^{\infty} C_{n} \, \sin\left(\frac{\pi n ...
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Fourier synthesis of periodic signals

I was reading the Fourier synthesis of periodic signals But I didn't understand the sentence i.e. "Although the calculation of $a_0, a_1, b_1, a_2, b_2$, is a mathematically straightforward ...
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Proving Inverse DFT

I have trouble understanding the proof I was provided of the IDFT, here is what I have: $$ \nu_n = \frac{n}{\Delta N} \\ x(t) = \int_{-\infty}^{\infty}X(\nu)e^{i2\pi\nu_n t}d\nu \\ $$ the next step I ...
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43 views

If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...
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108 views

Difficult Fourier transform exercise

I am currently dealing with a problem in functional analysis where I want to show the following. Let $\phi_{k+1}(x):=\sqrt{2} \sum_{n \in \mathbb{Z}} h(n) \phi_k(2x-n)$ and $\phi_0= \chi_{[0,1]},$ if ...
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22 views

Reference on Gibbs phenomenon

I need a reference that explains the following result (also known as the Gibbs phenomenon) Let $g$ be a $2\pi$-periodic function, $C^1$ per pieces (i.e., there exists a partition $x_1 < \cdots ...
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25 views

Is there any mistake? Proof related to the Poisson summation formula.

I need to prove the following statement: Let $\varphi \in C(\Bbb R)$ with compact support. Then, $$ \Big \Vert \sum_{k\in \Bbb Z} \varphi(k) e^{ikx} \Big \Vert_{L_1(0,2\pi)} \leq C \Vert ...