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
43 views

Fourier series of: $[\log(\sin x)]^2$

What is the Fourier expansion of: $${ \left[ \log\left( \sin x \right) \right] }^{ 2 }$$ This is a well known Fourier series: $$-\log(\sin x ...
3
votes
1answer
70 views

Asymptotic expansion of $f(x)= \sum_{n=1}^\infty \frac{\sin nx}{\sqrt{n}}$ at the origin

The function $$f(x)= \sum_{n=1}^\infty \frac{\sin nx}{\sqrt{n}}$$ is odd, uniformly convergent on all intervals $[\epsilon,\pi]$ for $0 < \epsilon < \pi$. Hence $f$ is continuous on $(0,\pi]$. ...
0
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0answers
36 views

$1 / (2 \pi)$ factor in Fourier transform

I have been unable to see why the $1 / (2 \pi)$ appears in Fourier transform. Would you please justify it to me? Problem: Let $$ f(x) = \int_{-\infty}^{\infty} \mathrm{d} k \, e^{ikx} \tilde{f}(k) ...
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1answer
19 views

Why does this Fourier inner product equal this sum?

This is part of a derivation in a text that I am struggling to follow. It says that if we write $e_k(t) = e^{2 \pi i k t}$ then $$\langle \sum_{n=- \infty}^{\infty} \langle f, e_n \rangle ...
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0answers
13 views

Quesion about Parsevals formula for Fourier-Legendre Series

Question: A function $f(x)$ defined on $(-1,1)$ can be expanded as $$ f(x) \backsim \sum_{n=0}^{\infty} c_nP_n(x) $$ What do Parsevals formula look like for this expansion? My solution: Ok so I ...
5
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1answer
29 views

Deriving the coefficients during fourier analysis

I'm self-studying Fourier transforms, but I'm stuck on a basic point about integration during the derivation of an expression for the coefficients of the Fourier transform. For a function of period ...
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0answers
24 views

Complex coefficient in Fourier series

Why can $$\sum_{n=1}^N a_n \frac{e^{2 \pi i n t} + e^{-2 \pi i n t}}{2} + b_n \frac{e^{2 \pi i n t} - e^{-2 \pi i n t}}{2i} $$ be written as $$ \sum_{n=-N}^N c_n e^{2 \pi i n t}$$ for some setting of ...
3
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1answer
26 views

How to find the Fourier's coefficient $a_n$ of the Fourier's series of $\sin(x)$ on $(0,\pi]$, $0$on $(-\pi,0]$

Considering $g(x)$, periodical with a period of $2\pi$ defined by \begin{equation*} g(x)= \begin{cases} 0 & \text{for $x \in (-\pi;0]$} \\ \sin(x) & \text{for $x \in ...
0
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1answer
27 views

How to show that the Fourier's series of $f(x)=x$ uniformly converges?

How to show that the Fourier's series of $f(x)=x$ uniformly converges? After finding its coefficient, I got: $$\sum\limits_{n=1}^{+\infty}\frac{2(-1)^{n+1}}{n}\sin(nx)$$ I showed the pointwise ...
2
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0answers
33 views

Pointwise version of Fejer's theorem (convergence of Cesaro means)

Prove a pointwise version of Fejer's theorem: If $f\in \mathscr{R}$ and $f(x+),f(x-)$ exist for some $x$, then $$\lim \limits_{N\to \infty}\sigma_N(f;x)=\frac{f(x+)+f(x-)}{2},$$ where ...
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1answer
22 views

Why are the Fourier's coefficient on $0,2\pi$ and $-\pi,\pi$ the same?

I was given the Following Fourier's coefficient and I was happy with it: $$\left\{ \begin{array}{ll} a_n(f)=\frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos\left(nx\right)\,\mathrm{d}x\\ b_n(f) = ...
-2
votes
1answer
56 views

How these fractions becomes this?

Today I was trying to solve an integral for a Fourier series. I looked at the solution and this was the solution: \begin{align*} C_n &= \frac{1}{2\pi} \int_0^{2\pi} x^2 e^{-inx} \,\text{d}x \\ ...
0
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1answer
21 views

Fourier Series of saw tooth function

I have a function $f(x) = \frac{x}{\pi} \in (-\pi , \pi]$ I googled but couldn't find a solution done using complex exponential and I tired to do it as follows. $$a_k = ...
0
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1answer
29 views

Dirichlet theorem and expansion of fourier series

Dirichlet's theorem says that any function $f(x)$ on the interval $[-a,+a]$ can be expanded as a Fourier series: $$f\left ( x \right )=\sum_{n=0}^{\infty}\left [ a_{n} \sin \left ( \frac{n\pi ...
1
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2answers
51 views

Proof of an infinite sum using Fourier Series

I was revising for my calculus exam and I came across a question that asked to find the Fourier Series of $f(x)=1+x$, on $-1<x<1$, which I did. Which I found to be: $$f(x) = ...
5
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3answers
113 views

Obtaining Fourier series of function without calculating the Fourier coefficients

In this question in one of the answers it's shown how to get from $$f\left ( x \right )=\sum_{n=1}^{\infty}\frac{\sin\left ( nx \right )}{10^{n}}$$ to $$f\left ( x \right )=\frac{10 \sin ...
3
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4answers
75 views

Application of Fourier Series and Stone Weierstrass Approximation Theorem

If $f \in C[0, \pi]$ and $\int_0^\pi f(x) \cos nx\, \text{d}x = 0$ , then $f = 0$ Define $ g(x) = \begin{cases} f(-x) & \text{if } -\pi \leq x < 0;\\ f(x) & ...
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1answer
24 views

Problem regarding Euler's formula and finding Fourier coefficient

I'm learning about Fourier series at the moment and there is an example in the literature that I have trouble following. (I'm sorry if I wrote a misleading title for the problem, I was not sure what ...
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1answer
30 views

Finding a limit involving Fourier series and Dirichlet's kernel

Find the limit $$\lim_{n\to\infty} \int_0^{2\pi} (x+\frac{\pi}{2})^2 \frac{\sin((n+\frac{1}{2})x + x\cos nx}{\sin\frac{x}{2}}\ dx$$ So we may define $f = (x+\frac{\pi}{2})^2$ and then look at the ...
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1answer
86 views

Representing the function $f\left ( x \right )=\frac{1}{e^{2}e^{\cos\left ( x \right )}-1}$ in terms of Fourier series

The function is periodic with main period of $2\pi$, and it is even. So only the coefficients of the cosine terms remain. Wolfram alpha gives the result for $a_{0}$ as follows: I guess it is only ...
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1answer
31 views

What is the Fourier series of $f(x)$

What is Fourier series of $$f(x) = \sum_{n=1}^\infty \frac{\cos nx}{2^n}$$ Now, it was claimed that since $f(x)$ converge uniformly and: $$f(x) = \sum_{n=1}^\infty \frac{e^{inx} + ...
3
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1answer
58 views

Fourier series of $\frac{1}{5+4 \cos x}$ using contour integration

The function $$f(x)=\frac{1}{5+4 \cos x}$$ is periodic with the main period being $T=2\pi$. The graph is easily obtained, but here is a graph from Desmos as it looks better: The function is even, ...
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0answers
33 views

Matlab FFT (dst) Diffusion equation with Fourier sine transform in Matlab

Good day, following the exposition presented on this site PDE with Fourier Transform I'm trying to solve a simple diffusion equation $u_t = u_{xx}$ with $u(x,0)=3\sin(2\pi x)$ and $u(0,t)=u(2,t)=0$. ...
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0answers
14 views

Why don't we use this expression as the Fourier transform series for the mod function?

According to Wikipedia page on floor and ceiling functions, $x\mod k$ has a Fourier series expansion of $$\frac{k}{2}-\frac{k}{\pi}\sum_{n=1}^\infty\frac{\sin{\left(\frac{2\pi nx}{k}\right)}}{n}$$ The ...
1
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1answer
36 views

Problem with the Fourier cosine coefficients of $1/(60 + 2 \cos t +0.3 \cos 2t) $

In Mathematical Astronomy Morsels, Meeus defines: $$ A = 60 + 2 \cos t +0.3 \cos 2t $$ And gives the following expression for calculating the inverse of $A$ ...
2
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0answers
57 views

Solving Fourier Integration of $ f(x)=\begin{cases} \sin(x)&0\leq x\leq\pi\\ 0 & \text{remaining} \end{cases}$

I tried to compute Fourier integration of below function : $$ f(x)=\begin{cases} \sin(x)&0\leq x\leq\pi\\ 0 & \text{remaining} \end{cases} $$ my solution ended up as follows but I need a ...
3
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0answers
31 views

Lebesgue measurable with two periods

I am trying to prove that a Lebesgue Measurable function with two periods $a $ and $b$ such that $b/a $ is irrational must ne constant almost everywhere.... I really dont know what to do, it says that ...
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1answer
54 views

Solving Fourier series of $ f(x)=\begin{cases} x+1 ;-1<x<0\\ 1-x;0<x<1 \end{cases} $

Please take a look at below Fourier series : $ f(x)=\begin{cases} x+1 &-1<x<0\\ 1-x & 0<x<1 \end{cases} $ I tried to solve it as follows : $ a_n=\displaystyle ...
4
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3answers
106 views

Develop $x\sin(x)$ into a specific series

I have to find a series that in $0\leq x\leq \pi$: $$x\sin(x) = \sum_{n=0}^{\infty} a_n\sin(2nx) $$ It seems to be impossible because on $x=\pi/2$ we get $ \pi/2 = 0$. However I tried to do it (as ...
0
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1answer
34 views

Why this equation “Fourier series” is important?

I am a student majoring in electrical engineering. There is three equations about Fourier series. \begin{align} x(t)&=\sum_{n=-\infty}^{\infty}X_n e^{j2\pi nf_0t} &&&& (1)\\ ...
1
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1answer
18 views

Why $X^*_n = X_{-n}$ in Fourier Series?

I am studying about Fourier Series. \begin{align} x(t)&=\sum_{n=-\infty}^{\infty}X_ne^{j2\pi nf_0t}\\ X_n&=\frac1{T_0}\int_{T_0}x(t)e^{-j2\pi nf_0t}dt \end{align} I understand the process ...
0
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1answer
56 views

What's exactly the output of Fourier Transform?

I'm new to Fourier Transform. I need to get a bit of understanding on it for my CompSci dissertation. I've looked at several tutorials online. Most of them explain the Fourier Series very well. ...
3
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2answers
151 views

Convergence of Fourier series at $x=0$

Let $f$, $2\pi$-periodic and intergrable function defined as follows: $$f(x) = \begin{cases} 1+\sin\frac{\pi^2}{x} & x\in[-\pi,\pi),x\ne 0 \\ 1 & x=0 \end{cases} $$ Does the Fourier ...
2
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2answers
78 views

Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$

Let $G(x,y) = \sum_{n=1}^\infty \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$ I'm trying to compute this sum by understanding it as an integral kernel. This question comes from Dym and Mckean ...
0
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1answer
28 views

is there any good way to figure out number of fourier series frequencies of some signal?

Suppose you have $f(t)$, but you do not know the exact function and can only measure $f(t)$ at certain time. Assume $f(t)$ is complex-valued with $t$ being "time." One wishes to find out the number ...
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0answers
24 views

What is the analogue of $f(x)=e^{-x^2}$ on the torus? What about its Fourier transform?

Let $f:\mathbb R \to \mathbb R$ such that $f(x)=e^{-x^2} \ (x\in \mathbb R).$ We know that $f, \hat{f} \in L^{1}(\mathbb R).$ My Question is: What is the natural analogue function of ...
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0answers
42 views

is there something wrong with the Fourier transform coefficients?

define the Fourier transform and its inverse $$\hat{u}(y)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n}e^{-ixy}u(x)\,dx$$ $$\check{u}(y)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb R^n}e^{ixy}u(x)\,dx$$ then ...
5
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0answers
85 views

Does this inequality hold? Proof / Counterexample

Does the following inequality $ \int_0 ^\infty x^2 |\frac{d}{dx}f(x)|^2 dx - \int_0 ^\infty x |f(x)|^2 dx + 2\pi (\int_0 ^\infty x^2 |f(x)|^2 dx) (\int_0 ^\infty x |f(x)|^2 dx) > -\frac{1}{8\pi} ...
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0answers
39 views

Change from Fourier Space to Real Space

I have a function in 3D fourier Space $$g(\textbf {k})=\frac{\hat{k}_i}{\hat{k_j}}f(\textbf {k}),$$ where $\hat{\alpha}$ is a fixed vector and $i$ and $j$ are the components of the relevant vector, ...
0
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1answer
32 views

Fourier series for $\cos( \frac x2)$

I am trying to get the Fourier series for $\cos( \frac x2)$ from $[- \pi, \pi]$. I know the general equation for a Fourier series. Since this is an even function, I know that the coefficients for ...
1
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1answer
90 views

Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$ using Fourier series

Consider the function $f(x) = \frac{x}{2}$, defined over the interval $[0, 2\pi]$. Show that $\frac{\pi}{4} = 1 − \frac13 +\frac15 −\frac17 + \cdots$.
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1answer
33 views

$u(x, t)$ of \begin{equation} \begin{cases} u_t = k u_{xx} + u \\ u(x, 0) = f(x) \end{cases} \end{equation}

How do I use a Fourier transform to find a formula for the solution $u(x, t)$ of \begin{equation} \begin{cases} u_t = k u_{xx} + u \\ u(x, 0) = f(x) \end{cases} \end{equation} ...
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0answers
43 views

Leibniz formula using Fourier Series

I have to show the Leibniz formula i.e $$\pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...$$ and I have to do so using $f(x) = x/2$ on the interval $[0,2\pi]$ for this function being $2\pi$ periodic. It is clear ...
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2answers
38 views

Why does $\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$

Why does $$\int_{-L}^{L} \sum_{n=1}^{\infty}a_n\cos \frac{n\pi x}{L}=\sum_{n=1}^{\infty}a_n\int_{-L}^{L}\cos \frac{n\pi x}{L}$$ This is used in a derivation of the Fourier coefficients. I see why ...
2
votes
1answer
17 views

Unitary Operator on Hilberspace to show that Fourierbasis is a maximal Orthogonal Set

I have looked at the proof Proving that the Fourier Basis is complete for C(R/$2*\pi$ , C) with $L^2$ norm but am having trouble understanding the argumentation about the Hilberspace. I think the ...
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0answers
50 views

An exercise from stein's fourier analysis

I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows: Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in ...
0
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1answer
52 views

Solving a simple trigonometric equation for coefficients

Is it possible to solve the equation $$ a \sin x + b \cos x + c \cos^3 x = d \cos x $$ where $c\neq0$ using some coefficients $a$, $b$, $c$, and $d$? I can't see how to make the frequency of ...
1
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0answers
22 views

Determining coefficients to a nonhomogenous differential equation.

Consider the following ODE: $$\frac{d^2 x}{dt^2} + kx = f(t)$$ where $$f(t) = \frac{1}{2} +\sum_{n = 1}^{\infty}{ \frac{-4}{n \pi} \sin\left(\frac{n \pi t}{2}\right)}$$ was derived using fourier ...
1
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1answer
43 views

Algebraic way to see why only $n=3$ is a valid coefficient

I'm a bit of a sucker for brute force calculations. Say I want to calculate a coefficient with Fourier theory, in my case \begin{align*} a_n = \int_0^1 \sin (3\pi x) \cos (n\pi x) dx. \end{align*} ...
0
votes
2answers
30 views

Find the Fourier series for the function $f(x) = x^4$

How do I start solving this question, what are the steps? a) Find the Fourier series for the function $f(x) = x^4$ on the interval $[−π, π]$. b) Hence prove that ...