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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32 views

If the series $\sum_{k=1}^{\infty} a_k$ is Cesàro summable and $n a_n \to 0$ as $n \to \infty$, then the series converges

I'm learning about Fourier series, specifically Cesàro summable sequences and series, and need help with the following problem: Show that if the series $\sum_{k=1}^{\infty} a_k$ is Cesàro ...
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1answer
25 views

If $f: \mathbb R \to \mathbb R$ is continuous and $2\pi$-periodic, then $f \in L^2[-\pi, \pi]$

I'm learning about Fourier series, specifically $L^2$ convergence, and need help with the following problem: Let $f: \mathbb R \to \mathbb R$ be continuous and $2\pi$-periodic. Show that $f \in ...
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1answer
24 views

Fourier series of piecewise-defined function and convergence

I'm learning about Fourier series and need help with the following problem: Consider the function $$g(x) = \begin{cases} x^{\frac{1}{3}}, & x \in [0, \frac{\pi}{2}] \\ ...
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1answer
24 views

Find widest subset on which Fourier series can be integrated and derived term by term

As part of one problem I need to find the widest subset of $\mathbb{R}$ on which the obtained Fourier series can be integrated and derived term by term. I found that it has something to do with ...
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2answers
35 views

Proving, that $\text{Arg}(-i\sin(x))=\pi/2\text{sgn}(x)$ on $(-\pi,\pi)$

Alright. I thought, that $\text{Arg}(-i\sin(x))=3\pi/2$, however, the Wolfram Alpha tells a different story. I am sure that it must be kind of true, because $\text{Arg}(\sin(x))$ is the result of sum ...
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0answers
25 views

Absolute maximum

I´m trying to find the absolute maximum of $(2N-1)$ partial sum of the Fourier´s series of signum function on $[0,\pi]$, I have: ...
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1answer
29 views

How to solve this Nonhomogeneous ODE problem of beam deflection and find particular solution

Problem states that the load on the beam having length L and fixed on both end is; $\omega(x)=w_0\frac{x}{L}$ Function of the deflection of the beam is given as; ...
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2answers
33 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 ...
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1answer
22 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 ...
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2answers
24 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 ...
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0answers
33 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 ...
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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 ...
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1answer
23 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 ...
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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} ...
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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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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 ...
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1answer
18 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} ...
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1answer
36 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 ...
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1answer
40 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 ...
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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 $x \in H$, ...
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1answer
43 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 ...
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0answers
35 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 ...
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2answers
36 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 ...
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1answer
22 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 ...
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2answers
63 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? $$ \sum_{n=1}^ \infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}})$$
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1answer
65 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 ...
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0answers
20 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 ...
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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 ...
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0answers
13 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 ...
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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 $$\sum_{n} C_n\sin\left(\frac{n\pi x}{L}\right)$$ ...
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14 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 ...
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1answer
25 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 ...
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2answers
49 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 ...
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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 ...
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0answers
36 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? ...
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1answer
23 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 ...
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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 ...
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1answer
31 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 ...
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2answers
25 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 ...
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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 ...
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0answers
8 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{ ...
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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 $f$? ...
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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 ...
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2answers
60 views

Coefficients of a cosine series

Let $u$ have the cosine series representation $$u = \sum_{k_1=0}^{\infty} \sum_{k_2=0}^{\infty} a_\underline{k} \cos\left(\frac{2\pi k_1 x }{L_1}\right) \cos\left(\frac{2\pi k_2 y }{L_2}\right) $$ ...
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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 ...
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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
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3answers
56 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 ...