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
390 views

Convolution: $ f (-)*g = g(-)* f$ does this mean both $f$ and $g$ have to be even functions?

Assuming $f$ and $g$ are different functions, does $ f (-)*g = g(-)* f$ mean both $f$ and $g$ have to be even functions? In fact, this is equivalent to $f\star g = g \star f$ (i.e., cross-correlation ...
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22 views

Convergence of a Fourier series on the unit circle

I have a complex-valued function defined as $$\psi(z) = \sum_{j\in\Bbb Z} \psi_jz^j$$ We of course know that $\sum_j\lvert\psi_j\rvert < \infty$ implies $\psi(z)$ is well-defined (finite) on the ...
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38 views

Integration by parts thrice [on hold]

To find real fourier coefficients this resolves to a triple integration by parts. When I let $x=u$ and $dv=e^xcosxdx$ to find the $cos(nx)$ coefficients, This translates to $u∫dv-∫∫dvdu$, My ...
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0answers
9 views

Length of a line approximated by Fourier series

I was recently solving some simple exercises where you approximate a square wave with a Fourier series. As you add more and more terms to the Fourier series, the function becomes close in shape to ...
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0answers
52 views

Proof of $\sum\limits_{k=1}^{\infty} \frac{1}{k^4} = \frac{\pi^4}{90}$ using the Fourier series of $|x|$

I'm sure easier proofs exist, but I have to specifically use the method in the picture: This is what my attempt is: First, I did some manipulation to figure out that $$ \sum_{k=1}^\infty \frac ...
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1answer
19 views

Question Concerning Fourier Series

I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as: $$f(x) = c_0 \phi_0(x) ...
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1answer
28 views

$x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$

I am trying to establish the following $x\cos(x)=-\frac{1}{2} \sin(x) + 2\sum_{n=2}^{\infty} \frac{(-1)^n n \sin(nx)}{n^2-1}$ for $x\in (-\pi,\pi)$ The right sight looks the the Fourier expansion of ...
3
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1answer
663 views

heat equation solution

This is the question, I have solved it but I need someone to double check my solution. Question: Find the temperature $u(x, t)$ in a rod of length $L$ if the initial temperature is $f(x)$ throughout ...
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1answer
17 views

Closed form of integral $\int_a^b e^{-ix^2} dx$

Does any know how to find the closed form of integral $\int_a^b e^{-ix^2} dx$ for any real $a$ and $b$. It seems that I need to use the fresnel integrals.
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0answers
79 views

What is the name of that theorem?

Here is the statement : Let $f:\mathbb{R}\to \mathbb{C}$ a continuous map which is $\mathcal{C}^1$ by pieces and such that $f\in \mathcal{L}^1(\mathbb{R})$. Moreover, $\hat f \equiv 0$ in ...
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1answer
45 views

Find the solution of the Dirichlet problem in the half-plane y>0.

Find the solution of the Dirichlet problem in the half-plane $y>0$. $${u_y}_y +{u_x}_x=0, -\infty<x<\infty,y>0$$ $$u(x,0)=f(x),-\infty<x<\infty$$ $u$ and $u_x$ vanish as $$ \lvert ...
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0answers
15 views

Fourier Sequence Converges Uniformly Implies Almost Everywhere Pointwise Convergence

I'm trying to understand this problem: Let $f$ be Riemann integrable on $[0,2\pi]$ Suppose that the Fourier Series of $f$, $S_{n}^{f}(x)$, converges uniformly on the interval. I want to show that ...
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1answer
18 views

Fourier series on incomplete data [closed]

Given a periodic function that's only partly specified, e.g.: $$f(\theta)=\begin{cases}1 & \text{if } \cos(\theta)>a\\ -1 & \text{if } \cos(\theta)<-a\end{cases}$$ Obviously the ...
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2answers
85 views

How is the Fourier transform a generalization to the Fourier series?

I have taken a self-tought course on the subject of Fourier series and Fourier transform and I got the message that the latter is a generalization of the first. I know that the idea that the Fourier ...
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0answers
21 views

Fourier series for discontinuous function

I am a bit confused with the Fourier series. The first step should be to determine if my function is odd or even, then find the coefficients (with eventually the shortcut for odd or even function) and ...
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1answer
12 views

How would it looks the Fourier series of this signal?

This is a kind of digital signal I'd like to re-create. i.e. I'd like to get N samples that will describe this signal: even better if it satisfy the Nyquist theorem (thus, sample-rate is 2x ...
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0answers
13 views

Limit of Fourier-Stieltjes transform of a complex Borel measure

Let $\mu, \nu$ be complex Borel measures on $(\mathbb{T},\mathcal{B}_{\mathbb{T}})$. Suppose $$\lim_{|n| \to \infty} \int e^{-int}d\mu(t) = 0$$ and $|\nu|$ is absolutely continuous with respect to ...
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0answers
13 views

Complex exponential fourier series

Given $$ x(t) = \sum_{-\infty}^{\infty}\frac{1}{T_0}(t-nT_0)(-1)^n[u(t-nT_0)-u(t-(n+1)T_0)]$$ where $n\in \mathbb{Z}$, $T_0$ is the period, and $u(t)$ is the unit step function, sketch $x(t)$ and ...
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1answer
157 views

A trigonometric identity

If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind, $$\ln(2) + \sum _ {n=1} ^{\infty} ...
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1answer
24 views

Given any sequence $(a_n)_{n \in \mathbf{N}}$ is $\sum_{n \geq 0} a_n e^{2 \pi i n z}$ holomorphic on the upper half plane?

I've seen quite often that people consider some arbitrary sequence $(a_n)_{n \in \mathbf{N}}$ (say of real numbers), and form the sum $\sum_{n \geq 0} a_n e^{2 \pi i n z}$, $z \in \mathbf{H}$. Usually ...
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16 views

Poissions Equation (Laplace)

$$\begin{align} u''_{xx}&+u_{yy}= x, \quad 0<x<1, \quad 0<y<1,\\ \\ u(x,0)&=u(x,1) = 0, \\ u(0,y)&=u(1,y) = 0,\\ \end{align}$$ Having some problems with Poissons Equation. ...
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1answer
22 views

Fourier Series of a sum of two functions [closed]

Is the Fourier series of a sum of two functions $f,g$ the term by term sum of the Fourier Series?
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1answer
18 views

How to find the inverse Fourier transfmation of exp(-sk)/k.

I've tried this with the help of hint given by one of my friend.He told me to first find the Inverse fourier transformation of exp(-sk) which is $$ \frac{\sqrt2}{\sqrt pi}\frac{x}{x^2+ s^2}$$ .After ...
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2answers
111 views

Evaluating infinite series $\sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2}$

I have no idea to approach this problem. Mathematica gave the sum to be $$ \sum_{n=0}^{\infty} \frac{1}{a^{2}+(2n+1)^2} = \frac{\pi}{4a} \tanh(\frac{a \pi}{2}) $$ How can I analyze this?
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1answer
19 views

Identity for the sum of products of Sinc functions

The Sinc function is defined as follows: $$\mathrm{sinc}(t) = \begin{cases} \frac{\sin(\pi t)}{ \pi t} & \mathrm{if} \quad t \neq 0, \\ 1 & \mathrm{otherwise.} \end{cases}$$ I want to show the ...
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0answers
11 views

For fourier series g(x), prove that the fourier series for the integral G(x) can be found by term-by-term integration of g(x)

I want to prove that if I have a fourier series of the form $g(x) = a_0/2 + {\sum_i}^\infty a_icos(ix) + b_isin(ix) $, the fourier series of G(x) $-x*a_0/2$ can be found by simply integrating g(x) ...
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0answers
30 views

Why can we calculate the Fourier series of $x^2$ in any interval $[-l,+l]$?

We know that a function must satisfy Dirichlet's Conditions before it can be expanded in Fourier series. And Dirichlet's Conditions strictly require a function to be periodic in the interval in which ...
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2answers
18 views

Fourier series: can a function be odd and have a dc component?

Long story short: fourier series is taken in two subjects (for now). One doc says that the dc component is 0 if the function is odd. The other says that odd and even has no effect on the dc ...
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0answers
56 views

Find the Fourier series of the following function function

Would someone be able to help me solve this? The function $f:(0,\pi]\to\mathbb{R}$ is defined by $$f(x) = \begin{cases} x & 0 < x \le \frac\pi2 \\[5pt] 0 & \frac\pi2 < x \le \pi ...
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0answers
38 views

Is there a general rule to find period of multiplied functions?

We know that $g(x)$ and $f(x)$ are both periodic and trigonometric functions and we also know its period interval. How can we find the period of the function $f(x)g(x)$?
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1answer
19 views

Suppose $f(x,y)$ has double Fourier series, find Fourier series of $\Delta f$

Suppose $f(x,y)$ has double Fourier series $\sum a_{n1n2} e^{in_1 x} e^{in_2 y}$. Then I have $$\Delta f(x,y) = \frac{\partial}{\partial x^2} f + \frac{\partial}{\partial y^2}f$$ ...
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1answer
19 views

Combination of even and odd functions

Can someone help me how to show that any function $f(x)$ defined on a symmetrically placed interval can be written as a sum of an even and a odd function? What is the special role played by ...
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0answers
20 views

Given Fourier coefficients of a function , find the function

Given these Fourier coefficients: $$ X[k]=\begin{cases} 1 & \text{, k even}\\ 2 & \text{, k odd}\\ \end{cases} $$ I want to find the analytical expression for the function. What i tried was ...
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23 views

Need some help with wave equation [closed]

Sorry for the structure, maybe someone help me fix it? Solve the waveproblem : $u(t,x)$ $u''_{tt} + 2u'_t = u''_{xx}$ when $0 < x < \pi$ and $t > 0$. $u(0,x) = 0$ when $0 \leq x \leq ...
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28 views

Dirac function expansion

In my book it is said that Dirac function $\delta(\tau)$ can be expanded as: $$ \delta(\tau)=(\beta \hbar)^{-1}\sum_{n \in even} e^{-i\omega_n\tau} $$ where $\omega_n=\frac{n\pi}{\beta\hbar}$, and ...
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25 views

Geometrical interpretation of complex exponential integral

Coefficients of Fourier series of a function $f$ are computed by multiplying $f(x)$ by the exponential term $e^{-inx}$, then by integrating $f(x)e^{-inx}$ from $-\pi$ to $\pi$ and dividing by $2\pi$ ...
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42 views

Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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0answers
19 views

Uniform convergence of Fourier series given certain conditions

If $f$ is a continuous, $a$-periodic and piecewise differentiable function on $[0,a]$ with piecewise continuous derivative on $[0,a]$, then $(f_N)$ converges uniformly to $f$ over $\Bbb R$. ...
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1answer
62 views

Minimal modulus for the finite field NTT

I need your support. Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity. I am using it to compute the convolution of two vectors of ...
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1answer
11 views

Coefficients of Fourier Series of (Cos(t))^3

I have to do the problem through the Sine/Cosine formulation of Fourier Series, so I'm talking about those coefficients. The interval is [-π, π]. I did the problem and checked it via Wolfram ...
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1answer
34 views

Fourier series coefficients in PDEs

I have a problem that involves solving a PDE using separation of variables. For context, here is the question: $u(x,t)$ is the displacement of a string at position $x$ and time $t$, which is ...
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0answers
25 views

How do I calculate the Fourier Transform of this signal?

The Context: Find $X(ω)$ which is the frequency domain representations of $x(t)$. $$ x(t) = \sum_{k=-\infty}^\infty \delta(t-4k) $$ This my professor's solution: As we can see, the ...
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19 views

Shifting the Fourier Series?

If $f(x)$ is some periodic function, I know how to express the shift $f(x-a)$ in the complex formulation of the Fourier series. However, I was wondering how such shifting affects the coefficients ...
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0answers
34 views

Use the Fourier Series of $f(x)=x^2+1$ to find the sum of the series

I have found the Fourier Series of $f\left(x\right)=x^{2}+1$ on the interval $\left[-\pi, \pi\right]$ extended periodically to $\mathbb{R}$ to be $$ ...
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1answer
30 views

How to prove that $f(x) = x(1-x)$ converges to a Fourier series?

The solution to an exercise I've done approximates $ f(x) = x(1-x)$ as a Fourier series, but does not mention how I can prove that $f(x)$ is indeed equal to the solution series. What I've done is : ...
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0answers
27 views

Fourier series in spherical coordinates?

I'm reading an article and he just state: let $f\left(\theta,\varphi\right)$ be of this form $$f\left(\theta,\varphi\right)={\sum}g_{m}\left(\theta\right)e^{im\varphi},$$ I'm on the unitary ...
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1answer
50 views

How to evaluate this series using fourier series?

With the help of Hermite's Integral,I got $$\sum_{n=1}^{\infty }\frac{1}{n}\int_{2\pi n}^{\infty }\frac{\sin x}{x}\mathrm{d}x=\pi-\frac{\pi}{2}\ln(2\pi)$$ I'd like to know can we solve this one using ...
0
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1answer
32 views

Determing an inverse Fourier transform

The inverse Fourier transform is defined as: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$ I can't get an inverse Fourier Transform to ...
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19 views

Why we need fourier series to analysis wave spectrum?

I am interested to study water wave spectrum to analysis irregular wave height. To analysis wave spectrum we need fourier series but i have a question to mine why we need to study fourier series.
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1answer
14 views

inverse fourier transform of w*e^w

I have the function \begin{align} F^{-1}\{{λe^{-|λ|}}\} \end{align} How can we find the inverse Fourier transform? The correct answer is: \begin{align} \frac{-2ix}{π(1+x^2)^2} \end{align} Can ...