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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1answer
47 views

CT Fourier Transform

I need to find the Fourier Transform of the given signal below; $$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$ I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = ...
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8answers
26k views

Real world application of Fourier series

What are some real world applications of Fourier series? Particularly the complex Fourier integrals?
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3answers
703 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
2
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0answers
50 views

Seeking better understanding of Fourier transform?

I'm quite confused on the one part of the Fourier transform. I don't understand what is the term $\left(u*x + v*y \right)$ mean. I mean $u$ and $v$ are the axis for frequency domain and $x$, $y$ are ...
3
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0answers
27 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
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1answer
90 views

How can I know if my Fourier Series coefficients are correct?

I want to find Fourier Series coefficients ($a_n$ and $b_n$) for this signal: $$f(t) = \frac{A}{t_s}t[u(t) - u(t-t_s)] + A[u(t-t_s) - u(t-(t_s + t_{on}))] + ...
2
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1answer
341 views

How to properly prepare for a graduate level PDE course using the books by Evans and Strauss

For my undergrad background, I have Calculus 1-3, Linear Algebra, one semester of ODE, one semester of real analysis. Never had any PDE before. Thus I know this background is hardly enough to do well ...
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1answer
90 views

Calculation of coefficients of a Fourier series

Calculating the Fourier series of a periodic function I need to evaluate these integrals: $$1) \int_{-\pi}^{+\pi}dt\left(\cos^{-1}(\alpha t-1)+2(1-\alpha t)\sqrt{\frac{1}{2}\alpha ...
2
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1answer
65 views

Fourier Series and Summation

$\sum_{n=1}^\infty \frac{1}{n^2}$ can be computed in straight-forward way by computing the Fourier co-efficients of $f(x)=x$ and applying Parseval's identity. Likewise, $\sum_{n=1}^\infty ...
3
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2answers
73 views

Relationship between Fourier coefficients of $f\left(x\right)$ and $f^{-1}\left(x\right)$

Say I have a function $f\left(x\right)$, which can be expressed as a Fourier Series: $$f\left(x\right)=\sum_{k=-\infty}^{\infty} c_k e^{ikx}$$ Define the inverse of $f\left(x\right)$ as, ...
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2answers
212 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
3
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0answers
100 views

Fourier transform of a logarithm

How can one go about computing the 2d (or 1d, in either variable) Fourier transform of the function $$\ln(w^2-k^2)?$$
2
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1answer
168 views

Parseval equation for a Fourier series

Consider $f(x):=\lvert x\rvert, x\in [-\pi,\pi]$. Then the Fourier series is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ Now my task is to write down the ...
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0answers
80 views

Need help on computing odd, even extensions of a function

OK I am going over d'Alembert solutions. And I came across the following example. $$ f(x) = \begin{cases} \frac{3}{10}x &0 \le x \le \frac{1}{3} \\ \frac{3(x-1)}{20} & \frac{1}{3} \le x \le ...
1
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1answer
104 views

Show absolute and uniform convergence of a Fourier series

Hello and good evening! The Fourier series of $f(x):=\lvert x\rvert$ on $[-\pi,\pi]$ is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ I have to examine if ...
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1answer
44 views

transformation of DFT matrix

$\mathbf{F}$ is a unitary DFT matrix where the $(m,n)$-th entry of $\mathbf{F}$ is given by $\frac{1}{\sqrt{M}}e^{-\imath2\pi(m-1)(n-1)/M}$. Note that $\imath=\sqrt{-1}$. Let $\mathbf{A}$ be a matrix ...
3
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1answer
311 views

Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
2
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1answer
112 views

Gibbs phenomenon in simple square wave

Given the square-wave function (later used to illustrate Gibbs phenomenon) $$f(x) = \left\{\begin{array}{c} \frac{h}{2} & 0 < x < \pi \\ -\frac{h}{2} & -\pi < x < 0 ...
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1answer
811 views

Is $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ continuous?

Considering the infinite series $\sum_{n=1}^{\infty}{\frac{\sin(nx)}n}$ , I can show that it is not convergent uniformly by Cauchy's criterion and that it is convergent for every $x$ by Dirichlet's ...
2
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1answer
123 views

Fourier series of $f(x)=1$ and $f(x)=x$

Would someone be kind enough to explain to me what would be the Fourier series of $f(x)=1$ and $f(x)=x$ on $[0,1]$? See, all the equations I can find are for intervals of the type $[-L,L]$. Now I ...
2
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1answer
53 views

Determining the Fourier series of a given function (Verification)

Determine the Fourier series for the function $$f(x)=\begin{cases} &0 \quad -\pi \leq x \leq 0\\ &e^{x} \quad 0 \leq x\leq\pi \end{cases}$$ Here is what I have come up with; I first ...
3
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1answer
50 views

Fourier coefficients converging

I'm thinking about this question, which has no answer yet despite being on a bounty and having 100+ views. Maybe it would be easier to start by asking this: Let $g\in C_0^{\infty}(\mathbb{R})$ ...
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1answer
167 views

Expanding a periodic function into a Fourier series

The solutions say that $$b_n=-\frac{4}{n\pi}$$ but I keep getting $$b_n=-\frac{2}{n\pi}$$ I checked with Wolfram Alpha and it seems that my integrals are right. Is this just a mistake in the ...
0
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1answer
137 views

Fourier Series (Even and Odd Functions)

Let $f \in E$ (where $E$ is a linear space of complex-valued piecewise continuous functions defined on the interval $[-\pi,\pi]$) and $$f(x) \sim ...
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1answer
63 views

Separation of Variables (Partial Differential Equation)

Does Separation of Variables work for the following PDE ? $$\nabla^2 W(x,y) \pm \alpha W(x,y) = \beta,$$ where $\alpha$ and $\beta$ are constants.
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1answer
39 views

$\int_a^b x^n \sin(mx)dx$ Identity

Are there nice memorable identity for the integrals $$\int_a^b x^n \sin(mx)dx$$ $$\int_a^b x^n \cos(mx)dx$$ where n can be an integer from $0$ to $n$. When I try to derive something by integration ...
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0answers
95 views

Searching for a counterexample in periodic functions.

I am looking for a function $f\in L_{\infty}[-\pi,\pi]$ which is $2\pi$ periodic and $$||f||_{\infty}\leq 1$$ and $$||\sum_{k=-N}^{N} \hat{f}(k) e^{ikx}||_{\infty} >1$$ Does such a function exist? ...
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1answer
254 views

Computing Fourier transform for $L^2$ function

For a function $f\in L^1(\mathbb{R})$, its Fourier transform is defined as $$\hat{f}(y)=\int_{-\infty}^\infty f(x)e^{-ixy}dx$$ For a function $f\in L^2(\mathbb{R})$, its Fourier transform is ...
3
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1answer
109 views

Computing Fourier sum for infinitely differentiable functions

Let $f\in C^{\infty}(\mathbb{R})$ be a periodic function of period $2L$. I want to show that $$f(x)=\sum_{n=-\infty}^\infty \left(\dfrac{1}{2L}\int_{-L}^Lf(y)e^{-in\pi y/L}dy\right)e^{i\pi nx/L}$$ ...
3
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1answer
375 views

Change of variables to derive Fourier series

Let $f\in C^{\infty}(\mathbb{R})$ be a periodic function of period $2L$. Define $$a_n=\dfrac{1}{2L}\int_{-L}^Lf(x)e^{-in\pi x/L}dx$$ Show by change of variables that $$f(x)=\sum_{n=-\infty}^\infty ...
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0answers
214 views

Matlab FFT conj Ifft

i have 2 questions 1) How can i get something to work 2) Why does it do what it does. 1) In matlab i have an image. On the image i apply the function (-1)^(x+y) in order to center the DFT (fft). In ...
2
votes
2answers
171 views

Inner Product vs. Integrals with Fourier Series, When to include 1/2pi?

I am confused about when to include a prefactor of $\frac{1}{2\pi}$ when dealing with integrals of functions that are expressed as fourier series. This is what I understand (please correct me if I'm ...
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1answer
56 views

Smooth lower envelope of a function

I want to determine if this problem has a unique solution. Given a continuous function $f(x)$ in a bounded interval or the real line, say [-1/2,1/2],  the problem is to find a function $L(x)$ that ...
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0answers
82 views

Representing real function as integral over trigonometric functions

Since one can clearly express any function g(x) as integral from 0 to infinity of A(k)cos(kx)dk + integral from 0 to infinity of B(k)sin(kx)dk, how would G(k) relate to A(k) and B(k)? In other words, ...
3
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1answer
44 views

Compare mixed derivatives to laplacian

Suppose $u,f$ periodic and smooth in $Q=[0,1]^n$ such that $\Delta u=f$. Show that for each $i,j$, $$\int_Q \left| \frac{\partial^2 u}{\partial x_i \, \partial x_j} \right|^2 \leq C \int_Q |f|^2.$$ ...
3
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2answers
58 views

$C$ such that $\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^4|a_{ij}|^2$

More generally, can we find $C_n>0$ such that $$\sum_{k\in \mathbb{Z}^n} k_i^2k_j^2|a_{ij}|^2 \leq C \sum_{k\in \mathbb{Z}^n} \|k\|^2|a_{ij}|^4$$ for all $\{a_k\}_{k\in \mathbb{Z}^n} \in ...
1
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1answer
33 views

Fourier series on $\mathbb{R}^n$

In setting up a Fourier series on $\mathbb{R}^n$, we use that for $l, m \in \mathbb{Z}^n$ $$\int_{[0,2\pi]^n}e^{i\langle x, l-m\rangle }=\begin{cases} 0 &\text{ if }l\neq m \\ (2\pi)^n & ...
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2answers
47 views

Why include negative $n$ in a Fourier expression?

If we make the Fourier series $$\sum_{n=-\infty}^\infty a_n e^{in\theta},$$ what is the point of explicitly including the negative terms? It seems just using evenness and oddness of cosine and sine ...
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1answer
77 views

Finding the complex Fourier Series

I want to solve the following: Given the $2\pi$ periodic function $f$: $$ f(x) = \begin{cases} 2\pi & for \;\; 0 < x < K \\ 0 & for \;\; K < x<2\pi \end{cases} $$ Where K is ...
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2answers
42 views

Finding real part of fourier series

I have encountered the following problem in one of my textbooks but I'm not really getting anywhere: Let $f$ be complex-valued and piecewise continuous on the interval $[-\pi,\pi]$. Find the complex ...
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0answers
67 views

Recovery of Bandlimited Signals

Let $\Omega > 0$ and denote by $\mathcal{B}_\Omega$ the subspace of $L^2(\Bbb R)$ consisting of signals that are bandlimited to $(-\Omega, \Omega)$. Denote $\mathcal{L}_{\Omega} : L^2(\Bbb R) ...
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0answers
170 views

Q: Calculating Fourier Coefficients and Inverse Fourier Transform

Let $\Omega >0$ and $x \in \mathcal{B}_{\Omega/2}$ is continuous. Define $\hat{y}(\omega) = \sum_{n \in \Bbb Z} \hat{x}(\omega - n\Omega)$. If $\hat{y}$ is expressed as \begin{equation} ...
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0answers
83 views

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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2answers
195 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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1answer
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 ...
0
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1answer
268 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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0answers
39 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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0answers
67 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 ...
1
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2answers
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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0answers
43 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 ...