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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28
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Did Joseph Fourier ever make a pure mathematical mistake?

Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001: It is remarkable that the nineteenth century was a time of error for mathematics: not trivial oversights or amateur ...
0
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0answers
30 views

Is my formula for DFT correct?

I'm doing "Digital Image" online course. I tried to solve the following question $x(n_1,n_2)$ is defined as $x(n_1,n_2)=(−1)^{(n_1+n_2)}$ when $0≤n_1$, $n_2≤2$ and zero elsewhere. Denote by $X(k_1,k_2)...
0
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0answers
23 views

Numerical method for fourier transform other than FFT/DFT

FFT relies on uniform samples, which cause aliasing, so FFT can be inaccurate in a certain case. Suppose you can obtain samples of $f(t): \mathbb{C} \to \mathbb{C}$ at any point ($t$ can also be a ...
1
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0answers
39 views

Convergence of Fourier series in $L^2$ space

I'm learning about Fourier series, specifically $L^1$ and $L^2$ convergence, and need help with the following exercise: True or False (justify): $(1)$ The trigonometric series $2 + \sum_{k = ...
3
votes
3answers
73 views

Deriving formula from Fourier series: $\frac{\pi^2}{12} = \sum\limits_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$

The equation/formula $$ \frac{\pi^2}{12} = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^2}$$ is to be derived. I know that the Fourier expansion of $f(x)=x$ for $x \in (-\pi,\pi)$ is $$f(x)=x=\sum_{n=1}^{\...
0
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0answers
72 views

Problem: If $f\in L^1[0,1]$, show that $\lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}f(x+\frac{k}{n})=\int_0^1 f(s)ds$ in $L^1[0,1]$.

This is the solution from the Prof.: Define $f_n(x)=\frac{1}{n}\sum_{k=0}^{n-1}f(x+\frac{k}{n})$. Then, unless $j<n$, we get $\hat f_n(j)=\frac{1}{n}\sum_{k=0}^{n-1}e^{2\pi i j k/n} \hat f(j)=0$ ...
1
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1answer
80 views

$\sum\left|\sin\alpha n+\sin\beta n\right|/n=\infty$ when $\alpha\neq-\beta$

Is there an elementary way of showing that the series $$\sum\frac{\sin\alpha n+\sin\beta n}{n}$$ is not absolutely convergent? Assuming $\alpha,\beta$ are not $0$ or $\pi$, I can show that there ...
1
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1answer
32 views

Fourier series for $\min \{0, \cos x\}$

How can I find Fourier series and convergence of function $f(x)= \min \{0, \cos x \}$ ? Because it is an even function I am expanding it only for cosine. Doing that I get $a_{0} = 0$, after that I ...
4
votes
2answers
89 views

If $\int_0^{2\pi} q = 0$, then $\lim_{n \to \infty} \int_0^{2\pi}p(x)q(nx) = 0$

I'm learning about Fourier series and need help with the following exercise: Let the functions $p, q \in L^1([0, 2\pi])$ be bounded and $2\pi$-periodic. If $\int_0^{2\pi} q = 0$, show that $...
1
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1answer
25 views

Fourier series coefficients' relation to original function

I am doing some practice problems for fourier series and I don't fully understand the solution to the following problem. I understand part (c) and (e) but I dont understand part (b) without taking ...
0
votes
1answer
21 views

Orthonormal set of basis functions in $L^2([a,b])$

I am wondering about the basis functions for $$e_n(x)=\frac{e^{2\pi i n x/L}}{\sqrt(L)}$$ where $L = b - a$ on the domain of $L^2([a,b])$ when doing fourier series. Basically, we must scale it by $L = ...
1
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0answers
20 views

Finding the coefficients in fourier series and parseval's formula for $e^{x} = \sum_{n=-\infty}^{\infty}c_n e_n(x)$

I just wanted to check my answer for 2 practice problems that I am doing, which follows from one another, the questions are as follows: a) Find the coefficients in $c_n$ in the following fourier ...
1
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1answer
30 views

Fourier Series Notation

Can fourier series notation be written in different ways but get the same result? I ask because I've seen these two diferent notations. What is the difference between the two? If they produce the same ...
2
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1answer
37 views

A BBP-type series

The BBP-type series $$ \frac{\pi}{2} \, \left( \frac{\alpha^{2}}{5} \right)^{\frac{1}{4}} = \sum_{n=0}^{\infty} \left[ \frac{1}{10 n + 1} + \frac{\alpha}{10 n + 3} - \frac{\alpha}{10 n + 7} - \...
-1
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0answers
8 views

DFT: How to prove the relationship of DFT magnitude?

This question comes from analysis the Fourier series. A writer published a paper in the journal of PAMI (IEEE Transactions on Pattern Analysis and Machine Intelligence), he just only finds this ...
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0answers
20 views

How to extend a function to be periodic and smooth?

Assume we have a function f(x) that is twice differentable on [0, L]. Let us define F(x) = f(x) on [0, L], F(x) = -f(-x) on [-L, 0], and F(x + 2L) = F(x) outside of [-L, L]. Thus, F(x) is ...
1
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1answer
113 views

How to justify interchange of sum and integral in fourier series?

$f$ is the $4$-periodic function $f(x) = 1$ if $x \in [0,2)$ and $f(x) = - 1$ if $x \in [2,4)$ The Fourier series of $f$ is $$F(t) = {4 \over \pi} \sum_{n=1}^{\infty} {\sin({\pi \over 2}(2n + 1)t) \...
0
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1answer
45 views

Find the Fourier series of $x^2$ over $[-\pi,\pi]$

I am finding difficulty solving this question. Find the Fourier series for the function $f(x)=x^2$ over $[-\pi,\pi]$.
0
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1answer
31 views

Writing an operator $T$ defined by $(T f)(t) = \int_{-\pi}^\pi h(t − s)f(s)ds$ as $\sum_{n \in \mathbb Z} \mu_n \langle f, \varphi_n\rangle \varphi_n$

Let $h$ be a continuous function with period $2\pi$. Define $T : L_2[−\pi, \pi] \to L_2[−\pi, \pi]$ by $(T f)(t) = \int \limits _{-\pi}^\pi h(t − s)f(s)ds$. Let $\{\varphi_n(t) =\frac{1}{\sqrt{2\pi}} ...
0
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1answer
25 views

Example of fourier series diverges at least two points

I know continuous function $f$ such that fourier series of $f$ diverges at one point. But, I don't know continuous function such that fourier series of $f$ diverges at least two point.
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1answer
30 views

The indefinite integral of an $L^1$ function has a convergent series of Fourier coefficients

Let $f\in L^1([0,2\pi])$ be a $2\pi$-periodic function with $\hat f(0)=0$ and $\hat f(\vert n\vert)=-\hat f(-\vert n\vert)\geq 0$. Define $F(t)=\int_0^t f(x)dx$. I know that F iscontinuous, $2\pi$-...
0
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0answers
53 views

Fourier transform of set of 2D points

I am searching for the Fourier transform $\mathcal{F}[\{P_z\}]$ of an infinite set of discrete points: $$ P_z = \left( \begin{array}{c} \sin(2 \pi\, \alpha \, z)\\ z \end{array} \right) $$ for $z \...
0
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1answer
46 views

Series expansion of {x}

Hello and sorry for my bad English. I am not mathematician, so sorry if this seems a silly question. I've seen this formula regarding the fractional part of a number in Wikipedia, and I would like to ...
0
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0answers
29 views

Fourier series of a complicated function

I am looking for a Fourier series of the following function: $g(x) = D\sin(C\arctan( Bx - E(Bx - \arctan(Bx)))) + A $ What mathematical software did you use to find the series? Thank you in ...
0
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1answer
34 views

Fourier Series of cos(ax)

I would like to ask some help on this problem.. 01) Expand the following function in fourier series. $f(x)=cosax,−π<x<π$ where 'a' is not an integer. Hence, Show that $\frac{1}{sint} = \frac{...
0
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0answers
21 views

Fourier basis of $L^2([-\pi,\pi])$

I have read that the Hilbert space $L^2([-\pi,\pi])$ has a Hilbert basis: $$\{e^{inx}|n\in\Bbb{Z}\}$$ This to me indicates that we can only represent a function $u(x)$ by a Fourier Series iff $u(x)\in ...
2
votes
2answers
47 views

Intuition for polynomial bases

In my linear algebra course I stumbled upon the following observations. We have some function $f: \Bbb{R} \to \Bbb{R}$, $f = f(x)$. $f(x)$ may be composed of elementary functions or not, but in ...
2
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1answer
66 views

Why the range of time period of exponential Fourier series is different from other two types of Fourier series?

Trigonometric Fourier series is given as $$x(t)=\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T) \quad(1)$$ Polar FS is given as $$x(t)=...
1
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1answer
32 views

Three questions on Fourier Transforms

I just started Fourier Series and I have three questions on Fourier transforms. I am incredibly lost on the subject and I feel like i'm missing something more fundamental. This is going to be a long ...
1
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1answer
43 views

Finding Area to intergrate

I am having trouble figuring out where to integrate. The question asks to find the exponential Fourier series coefficients of a signal given: From the signal, I know that the period is $8$, thus $\...
1
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0answers
28 views

Showing $\int_0^1f\bar g =\sum\limits_{n\in\mathbf Z}\hat f_n\overline{\hat g_n}$

If $\int_0^1\lvert f\rvert^2=\sum\limits_{n\in\mathbf Z}\lvert\hat f_n\rvert^2$ then how can I derive $\int_0^1f\bar g =\sum\limits_{n\in\mathbf Z}\hat f_n\overline{\hat g_n}$ $\hat f_n$ is the ...
0
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0answers
11 views

What is/are the boundary condition for the steady-state temperature distribution in the half space , to use Fourier-Bessel series?

For, temperature distribution $u=0$ when r=a, we can write $J_n(Ka)=0$ so that we can use Fourier-Bessel series, and for half space, as $r$ goes to infinity $u$ will go to $0$, but then i can not use ...
0
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1answer
33 views

Parseval identity $\int_0^1|f(x)|^2dx=\sum\limits_{n\in\mathbf Z}|\hat f_n|^2$ weaker condition

Parseval identity $\int_0^1|f(x)|^2dx=\sum\limits_{n\in\mathbf Z}|\hat f_n|^2$ holds for square integrable $f$, what if the condition is dropped ? I have two questions, in both of which I have to ...
1
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1answer
37 views

Finding the Fourier series of $f(x)=\sin x \cos^2x$

Let us have $f(x)=\sin x \cos^2x$. We need to get the Fourier-series of this. Should we make $f(x)$ nicer using the known identities between $\sin, \cos$? I tried using, that $\cos^2x=\frac12 + \...
0
votes
1answer
32 views

The Fourier series of a continuously differentiable function converges to it pointwise

$S_N(f):=\sum\limits_{|n|\le N}\hat f_n\cdot e^{i2\pi nx}$ where $\hat f_n=\int_0^1f(x)e^{-i2\pi nx} dx$ and $f$ a $1$-periodic $C^1(\mathbb R,\mathbb C)$ function, then $S_N(f)$ converges pointwise ...
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0answers
34 views

General Form of Theta Functions from Functional Equations

From Elliptic Curves: Function Theory, Geometry and Arithmetic by McKean and Moll: Exercise 3.1.2. Discuss the general solution of the two identities (a) $f(x+2)=f(x)$ and (b) $f(x+2\omega)=e^{ax+b}f(...
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0answers
61 views

How to decompose a 2d shape into sin and cosin modes?

Assume that you have a circle with radius $r_0$, then you keep adding cosine modes as below: $r=r_0+a_1\cos(1\theta)+a_2\cos(2\theta)+a_3\cos(3\theta)+a_4\cos(4\theta)+~...$ if you plot this as ...
0
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1answer
58 views

If a fourier series converges to an elementary function, can I then find the closed form of this function?

Suppose that I am told that f(x) is some elementary function and that f(x) has the fourier series $\Sigma_{k=-\infty}^{\infty}c_ke^{ikx}$. By "elementary function" I mean: https://en.wikipedia.org/...
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1answer
28 views

Fourier expand with half integer

Let $f:[0,1] \to \mathbb{R}$ be a contious function. Then we have the Fourier series of $f$: $$f(x) \sim \sum_{n=0}^{\infty}a_n\sin(n\pi x)$$ with $\displaystyle a_n = 2 \int_0^1 f(z)\sin(n\pi z)dz.$ ...
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0answers
31 views

Suppose $\sum |A_n|^2 <\infty$, then $\sum r^{|n|} A_n e^{inx}$ converges uniformly?

Suppose $\sum_{-\infty}^\infty |A_n|^2$ converges. Show that for each $r\in (0,1)$, the series $\sum_{-\infty}^\infty r^{|n|} A_n e^{inx}$ converges uniformly in $x$. I know that the series $\sum_{-\...
0
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1answer
29 views

Fourier Series, finding coefficients

I am working on this problem: Let $f$ be a periodic function of period $12$ defined by $$f(x)=$$\begin{cases} 1 & \text{ if }-6<x\leq0\\ x & \text{ if }0<x\leq 6 \end{cases} Then the ...
4
votes
1answer
72 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 ...
1
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1answer
38 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 L^...
3
votes
1answer
48 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}] \\ (-x)^{\frac{1}{4}}...
1
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1answer
31 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 ...
0
votes
2answers
39 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 ...
1
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0answers
43 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: $S_{2N-1}[f](x)=\frac{4}{\pi}\displaystyle\sum_{k=0}^{N-1}{\frac{sen((...
1
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1answer
32 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; $EI\frac{d^4y}{dx^4}=w_0\frac{x}{L}$...
0
votes
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
votes
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 $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin nx.$$...