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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0
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2answers
34 views

Extend a function 2pi periodically and calculate fourier

I have the function $$f(x)= \begin{cases} \frac{\pi}{2}+x & x \in (-\pi,0] \\ \frac{\pi}{2}-x & x \in (0,\pi]\\ \end{cases} $$ I need to extend it $2\pi$ periodically and then ...
2
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1answer
23 views

Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin nx.$$...
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2answers
26 views

Fourier function expansion for extension over a $2\pi$ period

So I am currently looking at a fourier expansion for $$f(x)=\left\{\begin{array}{ccl}\sin x &\text{ if }& x\in[0,\pi]\\0 & \text{ if } & x\in[\pi,2\pi]\end{array}\right.$$ I am ...
1
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0answers
38 views

How to find particular solution of an ODE by fourier series expansion?

I encountered the question for the particular solution of, $$ k \frac{d^4y}{dx^4} = m x $$ where m and k are real numbers. I would solve this question with basic methods for ODEs but question ...
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1answer
14 views

Determine whether the fourier series converges

I have calculated the Fourier Series of $g\left(x\right)=x$ on $\left(-\pi,\pi\right]$ extended periodically to $\mathbb{R}$ to be $$g\left(x\right)=2\sum^{\infty}_{n=1}\dfrac{\left(-1\right)^{n+1}}{n}...
0
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1answer
16 views

Find a recurrence relationship for the following :

Find a recurrence relationhip for $a_{n}$: $a_{n}=\dfrac {2n+1}{2}\int^{1}_{-1}f\left( x\right) P_{n}\left( x\right) dx$ Where $f\left( x\right)= e^{-x}$ I have done it many times and keep ...
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0answers
26 views

When deriving the Fourier Series, how is $a_1$ calculated?

I am having difficulty understanding how the Fourier series is calculated. It starts like this; For any $f ∈ C_2π$ we would like to find coefficients $c_n(f)$ such that $$f(x) = \sum_{n=0}^{\infty} ...
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1answer
29 views

Find a recurrence relation and the Fourier-Legendre Series

Rodrique's Formula for the $n$th Legendre Polynomial is $$P_n\left(x\right)=\dfrac{1}{2^nn!}\dfrac{d^n}{dx^n}\left(\left(x^2-1\right)^n\right)$$ The Fourier-Legendre series of a function f is ...
0
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1answer
25 views

Determine the Fourier series considering the derivative of a function

Let $f\left(x\right)=x^2+1$ on the interval $\left[-\pi,\pi\right]$, which is extended periodically to $\mathbb{R}$. I have calculated the Fourier series of $f$ to be $$f\left(x\right)=\dfrac{\pi^2+...
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0answers
24 views

Find $a_0, a_1$ and $a_2$ by looking at a Fourier series

Given the Fourier series: $$F(x)=\sin{x}+\sum_{n=1}^\infty \frac{1}{5^n} \cos{nx}$$ How do I find $a_0, a_1, a_2$ when $$a_0=\frac{1}{\pi} \int_{-\pi}^\pi f(x) dx$$ and $$a_n=\frac{1}{\pi} \int_{-\...
0
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0answers
15 views

Converting Fourier Series into elementary expression

If a Fourier series corresponds to an elementary function, is there any algorithm that will produce the elementary expression of this function?
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0answers
32 views

Is $\frac{1}{n}\sin (\frac{n\pi}{2})-\frac{\pi}{2n}\cos (\frac{n\pi}{2})=\frac{(-1)^{n+1}}{(2n-1)^2}$, where $n \in \mathbb N$?

Is $\frac{1}{n}\sin (\frac{n\pi}{2})-\frac{\pi}{2n}\cos (\frac{n\pi}{2})=\frac{(-1)^{n+1}}{(2n-1)^2}$, where $n \in \mathbb N$? I am doing Fourier series, and my hand computed solution is the one on ...
0
votes
1answer
22 views

Fourier part series, missing one piece

$$F(x)=\left\{ \begin{array}{rl} ax,&0<x<\pi,\\ bx,&-\pi<x<0, \end{array} \right.$$ So, far i've got: $$a_0 = - \frac{b\pi}{2} + \frac{a\pi}{2}$$ $$bn = \frac{1}{\pi} \frac{(-1)^{...
0
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1answer
39 views

Fourier sine series of $\sin(x/2)$

$$f(x) =\sin \left(\frac{x}{2}\right)$$ on interval $0 < x < \pi$ Hello, I'm trying to do the sine series. I understand I have to do $b_n$ but somehow I always get $0$ as result, but it doesn'...
1
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1answer
43 views

Can an element of the closure of the span of an orthonormal sequence in a Hilbert space be represented by a Fourier series?

A problem I'm struggling with is this: If $(e_k)$ is an orthonormal sequence in a Hilbert space $H$, and we denote $M=\operatorname{span}(e_k)$, then for all $x\in \bar M$ we have that $x$ can be ...
3
votes
4answers
52 views

Is the sum of an infinite series of elements in the span of an orthonormal set also in that set?

If $(e_k)$ is an orthonormal sequence in some Hilbert space $H$ does it follow that, if for a set of scalars $\{\alpha_k\}$, the series $$\sum_{k=1}^{\infty}\alpha_ke_k$$ converges to an $x \in H$, ...
2
votes
1answer
46 views

Maximum value of $S_n(x)=\frac{4}{\pi} \sum_{k=1}^n \frac{\sin(2k-1)x}{2k-1}$

I'm doing the exercise $11.19$ from Apostol Real Analysis: Let $S_n(x)=\frac{4}{\pi} \sum_{k=1}^n \frac{\sin((2k-1)x)}{2k-1}$. Prove that $S_n(\frac{\pi}{2n}) \geq S_n(\frac{m \pi}{2n})$ for $m=1,...
1
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0answers
40 views

Uniform convergence fourier series |sin(x)|

As an exercise we have to calculate the fourier series of |sin(x)| (was no problem) and after that we are meant to show that this series converges uniformly towards |sin(x)|. After thinking about it ...
1
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2answers
40 views

Convergent Fourier series of continuous function

Let $f$ be a continuous function. It is known that its Fourier series is convergent almost everywhere to $f$ and it may fail to converge on some measure zero set. However I would like to know whether ...
1
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1answer
23 views

How does $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ become $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$?

How can you rewrite $\cos(x)\cdot\cos\left(\frac{3}{2}x\right)$ to $\frac{1}{2}\left(\cos\left(\frac{1}{2}x\right) + \cos\left(\frac{5}{2}x\right)\right)$? What rules have been used? I need it on ...
2
votes
2answers
64 views

Proving the convergence/divergence of $\sum_{n=1}^\infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+…+\frac{1}{\sqrt{n}})$ [closed]

Do the following series converges? Why? $$ \sum_{n=1}^ \infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}})$$
1
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1answer
66 views

Is $\sum_{k=1}^\infty \dfrac{1}{k^2}+\dfrac{1}{(k+1)^2}=\sum_{k=1}^\infty \dfrac{1}{k^2}$?

I think yes, since the first would be a kind of subsequence of the partial sums of $\dfrac{1}{k^2}$... To provide some context, the question arised while studying Fourier Series on Apostol, when was ...
0
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0answers
21 views

Evaluation of a series with absolute value

I want to estimate or evaluate the series $$S(\xi)=\sum_{n=1}^\infty\beta_n\left|\sin(\pi n \xi)\right|,~~ \xi\in(l_0,l_1)$$ with $\beta_n=\frac{\omega\sin\left(\pi^2 n^2 T\right)}{n^2\left(\omega^2-\...
2
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1answer
19 views

Find the Fourier series for $e^{-|x|}$ over $[-\pi,\pi]$

Calculate the Fourier series for $e^{-|x|}$ over $[-\pi,\pi]$. I know this function is even, there will no terms relate with $\sin$. To find $a_o$ and $a_k$, I need to calculate these two integrals ...
0
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0answers
14 views

How does one find the Fourier Series for a non-periodic function on an arbitrary interval $[-\frac{L}{2},\frac{L}{2}]$ using the complex exponential?

I was given three functions, and told to find the coefficients of their Fourier Series using $\tilde{f_k} = \frac{1}{\sqrt{L}}\int_{-\frac{L}{2}}^{\frac{L}{2}} f(x) e^{i2\pi kx/L}dx$ where $\tilde{...
0
votes
1answer
15 views

Possible to expand a constant function as a series of sines without phase?

Is it possible to expand a function such as $f(x) = C_0$, $C_0$ being an arbitrary positive real number, between $x = 0$ and $x = L$ in the form $$\sum_{n} C_n\sin\left(\frac{n\pi x}{L}\right)$$ ...
2
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0answers
15 views

Find the cosine series for the function defined by $f(x)=2$, $0 \le x \lt 1$ and $f(x)=0$, $1 \le x \lt 2$.

In class we only went over the series that are on an interval $x \in[-L,L]$ where $L$ is a positive real number. Here, we have $x \in [0,2]$, and I cannot do the transformation given in class where we ...
2
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1answer
27 views

Confusion regarding Gibbs phenomenon

I learned that the partial sum of the fourier series at a jump discontinuity always overshoots the value of the original function by about 9% and this percentage does not die out as we increase the ...
1
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2answers
53 views

Let $1 + \frac{1}{3^3} + \frac{1}{5^3} + \frac{1}{7^3} + \dots=s$, show that then $\sum_1^\infty\frac{1}{n^3}=\frac{8}{7}s$

Let $1 + \frac{1}{3^3} + \frac{1}{5^3} + \frac{1}{7^3} + \dots=s$, show that then $\sum_1^\infty\frac{1}{n^3}=\frac{8}{7}s$. This is the last part of a problem that I am working on. So far, we have ...
2
votes
1answer
45 views

Use the sine Fourier series for $x$ and $x^2$ to show $1-\frac{1}{3^3} + \frac{1}{5^3} - \frac{1}{7^3} + … = \frac{\pi ^3}{32}$

We were given that $x=2(\sin x - \frac{\sin 2x}{2}+ \frac{\sin 3x}{3}-\dots)$ and I computed that $x^2=\sum_{n=1}^{\infty}\frac{(-1)^n(4-2\pi ^2 n^2)-4}{\pi n^3}\sin(nx)$. How can I use these two to ...
0
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0answers
38 views

Proof verification : regarding pointwise and norm convergence of a fourier sequence of $L^2$ function

if $\{H_n\}$ fourier sequence of an $L^2$ function $f$ converges almost everywhere to a function $g$ then $f=g$ almost everywhere Where $H_n = n$th fourier series of $f$. could you verify the proof? ...
0
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1answer
24 views

Find the Fourier series for the function defined by:

$f(x)=\pi$, $- \pi \le x \le \pi/2$ $f(x)=0$, $\pi/2 \lt x \le \pi$ I got: $a_0=\frac{1}{\pi}\int_{-\pi}^{\pi/2}\pi dx=\frac{3\pi}{2}$ $a_n=\frac{1}{\pi}\int_{-\pi}^{\pi/2}\pi cos(nx)dx=\frac{1}{...
1
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0answers
13 views

reciprocal of a Fourier Cosine series

Assume that $a(t)$ is an even periodic function such that $1<a(t)<2$, and is continuously differentiable everywhere. Let its Fourier series expansion be $$a(t) = a_0+a_1 cos(t)+a_2 cos(2t)+a_3 ...
1
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1answer
40 views

Integrate the Fourier Legendre by parts :$\int_{-1}^{1}\left( x^{2}-1\right) ^{m}\cos \pi x\:dx$

Having difficulty integrating the Fourier Legendre series by parts : $$\alpha_{m}=\int_{-1}^{1}\left( x^{2}-1\right) ^{m}\cos \pi x\:dx$$ I understand we can use the general formula : $$uv-\int ...
0
votes
2answers
26 views

Reason for Fourier coefficients vanishing

I was computing the Fourier coefficient of the function: $$ F(t)=\left\{ \begin{array}{rl} F_0,&0<t<\pi,\\ -F_0,&\pi<t<2\pi, \end{array} \right. $$ with $F(t+2\pi)=F(t)$. Since ...
4
votes
1answer
68 views

$ \sum_{n = 1}^{\infty} \frac{1}{n^4}$?

Using Fourier series I have managed to show that $$ \frac{x^4}{12} = \frac{\pi^2 x^2}{6} + 4 \sum_{n = 1}^{\infty} \frac{(-1)^n}{n^4}(1-\cos(nx)) , x \in [-\pi,\pi]$$ From here apparently one need ...
0
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0answers
9 views

Fourier coefficients of a function with jump discontinuity

Suppose $f$ is a $2\pi$-periodic function with a single jump discontinuity at $\xi$. I have a result that says that the $k^{th}$ Fourier coefficient of $f$ is given by $$ \hat{f}_k = [f](\xi) \frac{ ...
-1
votes
1answer
42 views

Find the constant term in the Fourier series for $f$? [closed]

Let $f(x)$ be the function on [−3, 3] which is graphed below, Find the constant term in the Fourier series for $f$? $$\frac{a_{0}}{2}=\frac{1}{2L}\int^{L}_{-L}{f(...
0
votes
1answer
34 views

When the fourier series equal to the original function?

Let $f\in L^2([-1/2,1/2])$. Define $a_n=\int_{[-1/2,1/2]} f(x) e^{-2\pi i n x} dx$ for each $n\in\mathbb{Z}$. Define $S_N(x)=\sum_{n=-N}^N a_n e^{2\pi i n x}$ for each $N\in \mathbb{Z}^+$ and $x\in \...
0
votes
2answers
62 views

Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
0
votes
1answer
15 views

Solving solution given initial condition condition

Suppose we know that: $$u_t=ku_{xx},~~~~~~~~0<x<l,~~~t>0$$ and $$u(x,t)=\sum_{i=0}^\infty[C_n~cos(n\pi x/l) ~e^{-w_nkt}]$$ where $w_n=\frac{n\pi}{l} ~~~ for~~n=1,2,3,...$ What if the ...
0
votes
0answers
5 views

Fourier harmonics expansion of an invariant function

I have the complex function \begin{equation} \begin{gathered} A(t) = a(t) + b(t) i \\ A = R \, exp (i\theta) \end{gathered} \end{equation} under the invariant $A \rightarrow -A$, such that one ...
2
votes
3answers
65 views

Writing a matrix as a linear combination of basis matrices…

BACKGROUND: I have recently found (probably well known, but I had never seen this before) that a matrix can be written as a linear combination of the outer products of its eigenvectors where the ...
1
vote
1answer
50 views

Evaluate $\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$

Find a closed form expression for $$\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$$ I know that $\displaystyle\sum_{r=1}^{\infty} \dfrac{\sin(r \pi x)}{r} = \dfrac{\pi}{2} - \...
0
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0answers
16 views

Higher Order Fourier Transform

How do you extend Fourier transforms beyond the 1-d case? I just learned about the transform and I am curious if anyone has some good information I can study over the summer.
0
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0answers
27 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 ...
0
votes
0answers
11 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 ...
0
votes
0answers
24 views

What is support and spectrum of this nonnegative trigonometric function (or Finite Fourier Sum )?

This is a follow up of another question. The zeros of the following cosine sum shows the prime distribution, and the gap between the zeros can help to study the gap between prime numbers. $$ P(p_i,x)=\...
2
votes
0answers
58 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 1{k^...
0
votes
1answer
20 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.