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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2
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2answers
38 views

Getting fourier coefficient by integrating over half the period?

In the book Schaum's Outlines of Analog and Digital Communications solved problem 1.2, the author calculates the fourier coeffecient $C_0$ for the rectangular pulse train: where $a$ is assumed to be ...
2
votes
1answer
37 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$
0
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1answer
42 views

Difficult integration

In my notes the lecturer takes the Fourier transform in $x, y$ and $t$ of $\phi(x,y,z,t)$ as: $$ \int_{-\infty}^{\infty}dt\, e^{i\omega ...
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0answers
27 views

How to represent a periodic function as the sum of sinc functions in fourier transform

Suppose function $f(t)$ is 1-periodic. This means that in fourier transform, $F(\omega)$ is sum of impulse signals (dirac delta function and its shifts) at the multiples of $1$. Now we can form $g(t)$ ...
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vote
2answers
93 views

Is Fourier transform still writing a function as a series of sines and cosines?

In the Fourier series we write a function as a series of sines and cosines. Fourier transform seems to me to be totally different, we are not finding a series but rather a function $\hat f(w)$. So ...
3
votes
1answer
27 views

Fourier series and evaluation of another series

I was given to expand in a Fourier series the function $f(x)=|x|, \; x \in [-\pi, \pi]$. The Fourier series is quite known and I had done the calculations and I ended up to the formula: ...
1
vote
1answer
23 views

What kind of information is available in a Fourier series expansion of an analytic function that is not (readily) available in a Taylor series?

What kind of information is available in a Fourier series expansion of a real analytic function that is not (readily) available in a power series? When would one know to work with one over the other?
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0answers
35 views

Help for solving limi of the Complex Fourier Series

I need help for this exercise. Let: $ f:\left[ -T /2, T/2 \right]\rightarrow \mathbb{R}. $ I need show that $$\lim_{N \to \infty} \int_{-T/2}^{T/2} \vert f(t)-f_{N}(t) \vert^{2} dt = 0 $$ ...
1
vote
2answers
129 views

Coefficient calculation on Fourier series !? [closed]

in a Fourier series for function $$f(x)=\begin{cases}-1&\text{for }-\pi<x<0\\\sin x&\text{for }0<x<\pi\end{cases}$$ with $f(x)=f(x+ 2 \pi)$, is $f(x)= \dfrac{a_0}{2}+ ...
1
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0answers
25 views

Non Riemann summable Fourier series but Abel summable

A Riemann summable Fourier series is also Abel summable. I am looking for an example of a non-trivial Fourier series that is Abel summable at a point but NOT Riemann summable at the same point. Such ...
0
votes
1answer
21 views

Rectangular Width Fourier Function

Working on #7, I've tried writing out the Fourier transformation and plugging it into the formula and multiplying it with Wf, but I'm getting mixed up about how I'm allowed to combine integrals and ...
1
vote
2answers
33 views

Complex Fourier coefficients for $e^{|x|}$

I'm new to Fourier expansions and transforms, and I'm not sure how to proceed with this question. I know a function f(x) can be expressed as an infinite sum of $c_ne^{in \pi x/L}$, and that $c_n = ...
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0answers
11 views

Trying to find the Fourier series of $f(x)$, where $f(x)$ is a piecewise function that includes $E\;sin(\omega\;t)$.

Here's the full function I'm trying to find the Fourier series to: $$f(x) = \left\{ \begin{array}{lr} 0 & : -\frac{\pi}{\omega}\leq t\lt 0 \\ E\;sin(\omega t) & : 0\leq ...
0
votes
0answers
16 views

Show that the convolution of the two time domain functions satisfy the relationship Y(q) = H(q) * U(q).

The convolution of two time domain functions h(t) and u(t) is given by $$ y(t) = \int_{-\infty}^{\infty} h(t- \tau)u(\tau)d\tau $$ Show that the Fourier Transforms Y(q), H(q) and U(q) satisfy the ...
4
votes
1answer
63 views

Can we determine whether $f\in L^{p}$ or not ; if we know $\hat{f}$

Let $a_{n}:=\frac{1}{n}$ for all $n\in \mathbb Z\setminus \{0\}$ and $a_{0}= c$ where $c$ is some constant. Clearly, $a_{n}\in \ell^{2}(\mathbb Z)$, that is, $\sum_{n\in \mathbb Z} |a_{n}|^{2}< ...
1
vote
1answer
43 views

$\sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ does not converge as $\theta \rightarrow 0?$

We know that the series $H(\theta) := \sum \limits_{k=1}^{\infty} \frac{1}{k} \cos 2\pi k \theta$ is convergent for every $\theta \in (0,1)$ and for $\theta = 0$ the series tends to $+ \infty$. Is it ...
1
vote
1answer
39 views

Hard Integral [Heat Equation + Fourier Sine Series]

I encountered this integral while doing a heat equation problem in Advanced Calculus. How does the person evaluate the integral involving $$\int_0^\pi \sin x \cos (nx) \: dx $$ Can someone ...
-3
votes
1answer
24 views

Fourier transform of a scaled variable [duplicate]

If $f\hat(k)$ is the fourier transform of $f(x)$, what is the fourier transform of $f(x/c)$ where $c$ is a real number greater than $0$?
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0answers
12 views

Discrete Fourier Series Transformations

Let the DFT of f[n] be given by F[k]. Find the DFT G[k] of time series g[n] = f[n] * (-1)^n in terms of F[k]. I know that G[k] is related to F[k] by a shift in the frequency domain, but I'm not ...
1
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0answers
38 views

asymptotics of the Fourier transform of Daubechies wavelet

I want to evaluate the series \begin{equation} S(\alpha,\omega)=\sum_{k=-\infty}^{\infty}\frac{|\Psi(2k\pi-\omega)|^2}{|2k\pi-\omega|^\alpha} \end{equation} where $0\le\omega<2\pi$, ...
0
votes
0answers
23 views

Proving a fourier transform expression with green's formuls

Using Green's formula, show that: $${\cal F}\left[\frac{d^2f}{dx^2}\right]= -w^2F(w) + \frac{e^{iwx}}{2\pi}\left(\frac{df}{dx} - iwf\right) \\(evaluated\ from\ -\infty\ to\ \infty)$$ last part is ...
3
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0answers
72 views

About the closed form for $\lim_{y\to +\infty}\left(-\frac{2}{\pi}\log(1+y)+\int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx\right)$

Recently, when facing a baby Rudin's exercise, I proved that: $$ \int_{0}^{y}\frac{|\cos x\,|}{1+x}\,dx = \frac{2}{\pi}\log(1+y)+O(1) $$ holds by integration by parts. Now I wonder if ...
1
vote
2answers
54 views

Fourier inverse of a function to get dirac

I'm trying to get the dirac function from a fourier inverse tranform: $$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-iw(x-x_0)}dw$$ It is this last step I am stuck on to get the conclusion. Original ...
2
votes
0answers
45 views

Fourier Series of the batman equation

I want to represent the batman equation as a Fourier Series. (I got the equation here : Is this Batman equation for real?) But a part of it is an ellipse and when I tried to calculate an the integral ...
-1
votes
1answer
24 views

Complex Fourier Series of $t^3$

I am trying to find compute the complex Fourier series of the following function: $$f(t) = t^3$$ $$-\frac32 \le t \le \frac32$$ $$f(t) = f(t+3)$$ I am using the generic function for the complex ...
0
votes
1answer
51 views

The Fourier transform of exp(-x)*heaviside(x)

I'm trying to understand the Fourier transform of Y=exp^-x. Since the term tends to -infinity I have to multiply Y by the heaviside function to set everything below 0 to 0 so I can successfully ...
1
vote
1answer
12 views

Show behavior of Fourier Transform

If F(w) is the Fourier transform of f(x), show that F(aw) is the Fourier transform of (1/a)f(x/a). So if I apply a fourier transform to (1/a)f(x/a): $$ \frac{1}{2\pi}\int_{-\infty}^\infty ...
2
votes
0answers
25 views

I would like to find a generalization of the plane wave expansion to Hankel functions.

The plane wave expansion is \begin{equation} e^{i\vec{k}\cdot \vec{x}}=\sum_{\ell=0}^{\infty}i^\ell(2\ell+1)j_{\ell}(kx)P_{\ell}(\cos(\theta)) \end{equation} where $j_\ell$ is the spherical Bessel ...
0
votes
1answer
16 views

Find the Fourier Transform of piecewise finction

$$f(x) = \begin{cases} 0 & |x|> a \\ 1 & |x|< a \end{cases}$$ I have most of the solution, I'm just faltering on obtaining the sin(ax) part of the solution, I'm missing an exponential ...
1
vote
3answers
29 views

How do I find the solution to this summation after computing the following power series?

I have found that the Fourier cosine series from $({-\pi},{\pi})$ of the function $f(x)=\cosh(x)$ is $$ \frac{2\sinh({\pi})}{\pi}\left[\frac{1}{2}+ \sum_{n\: =\: 1}^{\infty}\:\ ...
4
votes
2answers
63 views

Integral using Parseval's Theorem

How would I integrate $$\int_{-\infty}^{+\infty} \frac{\sin^{2}(x)}{x^{2}}\,dx$$ using Fourier Transform methods, i.e. using Parseval's Theorem ? How would I then use that to calculate: ...
0
votes
1answer
75 views

Solve differential equation using fourier series

I am trying to solve this problem in my analysis book in a chapter on Fourier series: Solve the differential equation $$(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}) u(x,y) = ...
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0answers
24 views

Chladni patterns

So I was watching this video on Chladni figures (https://www.youtube.com/watch?v=wvJAgrUBF4w) and thought that it would be nice to replicate a few of these, especially the more complicated, high ...
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0answers
27 views

Table of Fourier series

I found that there are very good references on Fourier integral transform but none on Fourier series. Do you happen to know one?
1
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1answer
17 views

ODE with finite Fourier expansion periodic coefficients

Regard the ordinary differential equation $$ \dot a(t) = z(t) a(t) $$ where $a(t)$ and $z(t)$ are matrix valued such that $z$ is periodic ($z(t+2\pi)=z(t)$). Then it is well-known (Floquet theory), ...
0
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0answers
16 views

Compute the Fourier series of a piecewise function.

Consider the function: $ f(\theta) = \begin{cases} 0 & \text{if } |\theta| >\delta \\ 1-|\theta|/\delta & \text{if } |\theta| \leq \delta \end{cases} $ I need to show ...
0
votes
0answers
36 views

Show that the Fourier series is $\frac{8}{\pi} \sum_{k \;odd \ge 1} \frac{sin(k \theta)}{k^3} $

Consider the odd function $f(\theta)=\theta (\pi - \theta)$, then I need to show that: $f(\theta)=\frac{8}{\pi} \sum_{k \;odd \ge 1} \frac{sin(k \theta)}{k^3}$ then I computed the Fourier ...
2
votes
2answers
37 views

Show that $\widehat{f}(n)$ is zero for odd $n$

The following problem is from Stein´s Introduction to Fourier analysis: Suppose that $f(\theta + \pi)=f(\theta)$ for all $\theta \in \mathbb{R}$ Show that $\widehat{f}(n)$ is zero for odd $n$. My ...
1
vote
1answer
32 views

Writing a Fourier series of a $2\pi$-periodic function.

This problem was taken from Stein's Introduction to Fourier analysis, and it goes like this: Let $f$ be a $2\pi$-periodic Riemman integrable function defined on $\mathbb{R}$. Show that the Fourier ...
2
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2answers
62 views

A difficult trigonometric integral involving absolute value

$$ \int_{0}^{2\pi}\lvert\sin(x)\rvert\cos(nx)\,dx= -\frac{4\cos^2\bigl(\frac{\pi n}{2}\bigr)\cos(\pi n)}{n^2-1} $$ I'm not actually trying to solve this myself. The answer appears in my lecture notes ...
2
votes
1answer
80 views

A trigonometric integral identity from Krylov's “Approximate Calculation of Integrals”

In the theory of Fourier series the following expansion is known $$ \operatorname{sign}\left(\sin\left((n + 1) x\right)\right) = \frac{4}{\pi} \sum_{k = 0}^\infty \frac{\sin\left((2k + 1) (n + 1) ...
0
votes
1answer
29 views

Do I have to transform the solution into $u(x, y)$?

Find the solution of the problem $$u_{xx}(x,y)+u_{yy}(x,y)=0, x^2+y^2>1 \\u=1+3\sin^3 \theta , 0 \leq \theta <2\pi$$ $u$ is bounded. I have done the following: $$u(x,y)=v(\rho, \theta) \\ ...
0
votes
1answer
32 views

Can we expect $\|fg\|_{\mathcal{F}L^{1}} \leq C \|f\|_{L^{2}(\mathbb R)} \|g\|_{\mathcal{F}L^{1}}$?

Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is, $$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$ with the ...
3
votes
1answer
39 views

$\|fg\|_{A (\mathbb T)} \leq C \|f\|_{L^{2}} \|g\|_{A (\mathbb T)}$?

Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space, $$A(\mathbb T):= \{f\in ...
0
votes
1answer
134 views

Is there a closed-form of $\sum_{n=1}^{\infty} \frac{\sin(n)}{n^4}$

Is there a closed-form summation result for Fourier series: $$\sum_{n=1}^{\infty}\frac{\sin(n)}{n^4}?\tag{1}$$ I tried using available result of the following (odd) function : ...
2
votes
1answer
38 views

On the weak closedness of a closed ball with fixed $L^2$-norm in a periodic Sobolev space

Preliminaries: Let $\mathrm{L}_P^2$ denote the Hilbert space of $P$-periodic, locally square-integrable functions $f\colon \mathbb{R} \to \mathbb{C}$ with Fourier series representation $$f(x) \sim ...
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vote
0answers
23 views

Properties of Fourier coefficients of real valued functions

Let $\hat{f}(n)$ be the Fourier coefficients of $f:[0,2\pi]\to \mathbb{C}$ defined as $$\hat{f}(n)=\int_{0}^{2\pi}f(x)e^{-{\rm{i}}nx}\,\mathrm{d}x$$ Note $f$ is Riemann-integrable on $[0,2\pi]$. We ...
0
votes
0answers
9 views

Finding Fourier coefficients of (discrete ) $cos(\frac{6*n*\pi}{N})$

What is the Fourier coefficients of (discrete ) $cos(6*n*pi/N)$? The answer says $0.5[delta(k-3)+delta(k+3)]$ (delta is Dirac delta function)...my attempt was to use a formula $1/N(sum from 0 to ...
0
votes
0answers
8 views

Solutions of $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$

Is there a solution for the equation $\sum_{n=1}^N a_n n\sin{(n x+\theta_n)}=\sum_{n=1}^N a_n n^2\cos{(n x+\theta_n)}=0$ in terms of the variable $x$, for some choice of coefficients $a_n$ and ...
1
vote
1answer
34 views

periodicity of an exponential sum

I wish to rigorously prove that the function $f(x), x \in \mathbb{R}$ is not periodic. A function is defined to be periodic with period $M$ if $f(x+M)=f(x), \forall x \in \mathbb{R}$. Here $f(x) ...