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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49 views

A question about Fourier coefficients.

Is it true that the sequences $ (A_{n})_{n \in \Bbb{N}} = (0)_{n \in \Bbb{N}} $ and $ (B_{n})_{n \in \Bbb{N}} = \left( \dfrac{1}{\sqrt{n}} \right)_{n \in \Bbb{N}} $ are the Fourier coefficients of ...
7
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2answers
570 views

What are the limitations /shortcomings of Fourier Transform and Fourier Series?

I am fond of Fourier series & Fourier transform. But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about ...
1
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3answers
71 views

How to show that if all fourier coefficient of a function is zero, then the function is zero function?

Let $f$ be a continuous and integrable function with period $2\pi$. Consider its fourier coefficients with respect to the orthonormal system $\{ \frac {1}{\sqrt{2\pi} } e^{inx}\}$. If all the Fourier ...
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1answer
27 views

Convergence of Fourier series where function is continuous

I'm not able to understand how they worked out for x not equal to 2*pi*n the series converges to x mod 2*pi Any help would be much appreciated
4
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3answers
84 views

A uniform bound by an integrable function for a Fourier series' partial sums.

Consider \begin{equation} \sum\limits_{n=1}^\infty\frac{\cos(nx)}{n}=-\log|2\sin x/2|~~~ \big(x\in(0,2\pi)\big), \end{equation} and its $2\pi$-periodic extension $f$ (for a proof of the above ...
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0answers
15 views

Finding the Fourier series

Finding the Fourier series of $$y=\begin{Bmatrix} x & &1\geqslant x\geqslant 0 \\ 2-x & & 2\geqslant x\geqslant 1 \end{Bmatrix}$$ My attempt is: solution 1: ...
2
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1answer
49 views

Construction of a function $u$ such that $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ and $u \not\in W_0^{2,2}(\Omega)$

I'm wondering about an example of a function $u \in W^{2,2}(\Omega) \cap W_0^{1,2}(\Omega)$ such that $u \not\in W_0^{2,2}(\Omega)$. Clearly $W_0^{2,2}(\Omega) \subset W^{2,2}(\Omega) \cap ...
2
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3answers
53 views

Find the Fourier Transform of $2x/(1+x^2)$

I tried doing this the same way you would find the Fourier transform for $1/(1+x^2)$ but I guess I'm having some trouble dealing with the 2x on top and I could really use some help here.
1
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1answer
13 views

does frequency scaling property of Fourier transform not work for Fourier series?

So frequency/time scaling property of Fourier transform says that: fourier transform of $|c|f(ct)$ is $F(\omega/c)$. (I am using angular frequency $\omega = 2\pi f$ here) However, this doesn't seem ...
2
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0answers
42 views

Computing Fourier Series coefficients

Hello I have to calculate the Fourier series coefficients for the following function: $$f(t)=\sum_{n=-\infty}^{+\infty} \Pi(\dfrac{t-nT_o}{T_o/2})$$ where "$\Pi$" indicates the rectangular function. ...
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1answer
50 views

Fourier transform of indicator function

Given a set of complex numbers $\mathcal A$, is there a convenient solution for the Fourier transform of its indicator function $\chi_{\mathcal A}(z)$? More specifically, if $\mathcal A$ is a set of ...
2
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0answers
26 views

Drawing a continuous function with divergent Fourier series at $x=0$…

Does anyone know how the graph looks like for a continuous function with Fourier series diverging at $x=0$ ? The example due to Fejer (a variation of the du Bois-Reymond construction), is explicitly ...
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0answers
36 views

Complex Fourier series for $f(x)=\cos(3x)$

I want to find $c_n$ satisfying $$\sum_{n\in\mathbb Z}c_ne^{inx}=\cos(3x)$$ Noting that $\langle e^{inx},e^{-imx}\rangle=0$ for $n\neq m$ in $[0,2\pi]$, I have ...
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1answer
30 views

Boundary condition not compatible with initial condition…why?

Equation $$\frac{\partial^2 p}{\partial t^2} = \frac{\partial^2 p}{\partial x^2} + \frac{\partial^2 p}{\partial y^2}$$ Boundary conditions: $p = 0$ for $x = 0, \pi$ $p = 0$ for $y = 0, \pi$ ...
1
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1answer
38 views

Trigonometric Identities//Fourier Series

Basically I have to find the value of a constant $M$ from this equation: $$l(x)=0=\sum M\Big(\frac{n\pi}{L}\Big)\sin(n\pi x) $$ using the Fourier Series. However the usual Fourier Series formula is: ...
6
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2answers
66 views

Showing that $P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}\rightarrow 0$ uniformly on $[-\pi,-\delta]\cup[\delta,\pi]$ as $r\uparrow 1$

Let $0<r<1$ and consider the series $$s = \sum_{n=-\infty}^\infty r^{|n|}e^{inx}.$$ I have shown that the series converges uniformely to $$P_r(x)=\frac{1-r^2}{1-2r\cos x+r^2}$$ on all of ...
2
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1answer
28 views

Problem with a trigonometric function: $\arctan ( \sin x /(1-\cos x))$

I am studying Abel summability right now, and at a certain point I obtain the following identity: $$ \sum_{k=1}^{\infty}\frac{\sin kx}{k} r^k = \arctan \frac{r\sin x}{1-r\cos x} $$ By previous ...
1
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2answers
52 views

Divergence of Fourier series

Given $f(x)$ is the characteristic function of the interval $[a,b]\subset [-\pi,\pi]$ ($a\neq b$), so $f(x) = 1$ for $x\in [a,b]$ and $f(x)=0$ otherwise. From this definition, I obtained the Fourier ...
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1answer
18 views

Graphing Fourier transforms on a frequency versus intensity plot (how to deal with complex numbers)

I am trying to understand how Fourier transforms are used to make HNMR plots. HNMR basically consists of hitting some molecules with some radiation and listening to the radio signal that results. ...
0
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1answer
47 views

Riemann Lebesgue Lemma Clarification

If $f$ is continuous real-valued function, does the Riemann Lebesgue Lemma give us that $\int_{m}^k f(x) e^{-inx}\,dx \rightarrow 0\text{ as } n\rightarrow \infty$ for all $m\le k$? Specifically, is ...
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2answers
45 views

Solving the problem with Fourier

I want to solve the following problem: $$u_{xx}(x,y)+u_{yy}(x,y)=0, 0<x<\pi, y>0 \\ u(0,y)=u(\pi, y)=0, y>0 \\ u(x,0)=\sin x +\sin^3 x, 0<x<\pi$$ $u$ bounded I have done the ...
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0answers
11 views

Non periodic Fourier Series Point Convergence

If $f$ is a real-valued non-periodic continuous function that is differentiable at the point $x_0$, is it true that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
0
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1answer
31 views

Existence of Solutions to PDEs - How do I know I've got them all?

I'm taking a very computational course in partial differential equations. Because of this emphasis, I'm feeling very underwhelmed by the course, and have a lot of questions that really aren't ...
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2answers
38 views

Limit in combination with an infinite series

How would I go about showing the following limits that involve infinite series $$ \lim_{x \to 0^{+}} \sum_{n=1}^{\infty} \frac{(-1)^n}{n^{2k+1}} \sin (2\pi n(x - \frac{1}{2})) = 0 \text{ with } k \in ...
2
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2answers
27 views

Help establishing a bound on the Fourier coefficients of a bounded $2\pi$ periodic function that is discontinous at the end points?

This is from a practice midterm, and I'm having trouble with the first part. Suppose $f$ is a $2\pi$-periodic function that is continuous and differentiable on the interval $[-\pi, \pi]$, but has jump ...
4
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1answer
26 views

Fourier Series for a conformal map on unit disk

Given that a conformal map on the disk $\mathbb{D}$ will always have the form $f(z)=\lambda \displaystyle\frac{z-w}{1-\overline{w}z}$ for some $\lambda\in \partial \mathbb{D}$ and some $w\in ...
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0answers
18 views

Fourier Expansion of Hill's lunar problem

all! For my class I have to expand the following equation $y''(x)=4(\omega^2+q(x))y(x)$ in Fourier coefficients $y(x)=\frac{1}{2}y_0 + \sum^\infty_{n=1}y_n \cos(2nx)$ $q(x)=2\sum^\infty_{n=1}t_n ...
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0answers
19 views

How this result in archived in Fourier series

I was reading some notes about functions of symmetry in Fourier series and came across the following result for a function with symmetry of an odd quarter wave $$\begin{align} ...
3
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0answers
72 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
2
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0answers
18 views

half range fourier series, even and odd extension

Hello, I have some problems understanding what is above on the image. Firstly, he defines an "odd extension" of any function. I don't really understand what this means, how is it an "odd extension" ...
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1answer
20 views

Fourier series coefficient miscaluculation

In a nice introductory paper about Bernoulli numbers that I found, the following claim is made (p. 5, theorem 4.3) The Fourier series of $x$ is given by $b_n = \dots$ (not important, it is wrong in ...
2
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0answers
30 views

Invertibility of Fourier Transform implies a.e. convergence of Fourier Series?

I am attempting to read Michael Lacey's proof (http://people.math.gatech.edu/~lacey/research/esi.pdf) of Carleson's Theorem about the almost everywhere pointwise convergence of Fourier Series of $L^2$ ...
2
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2answers
44 views

Fourier series of complex diff eq

Can I just use Euler's identity to construct the Fourier Series since it is complex? I was personally thinking I could, but I wanted to be doubly sure.
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0answers
16 views

Bibliographic reference for $\sum_{n\in\mathbb Z}(z-n)^{-k}$

I am currently writing a paper which requires the closed-form expression of \begin{equation} S_k(z)=\sum_{n\in\mathbb Z}\left(z-n\right)^{-k} \end{equation} I believe $S_2$ is extremely classical ...
2
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2answers
112 views

Fourier Decompositon problem

have a look at this video of Fourier Decomposition of an image (otherwise you can also refer the image, which shows few plots of different extracted waves from an image). We also know that a Fourier ...
1
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1answer
16 views

Determine the Fourier transform of $f(x) &

f(x)=1 if |x| < a or f(x) = 0 if |x| > a We use the formula $$ {1\over 2\pi} \int_{\infty}^\infty f(\bar x)e^{i\omega \bar x} $$ So is $f(\bar x)$ the same as $f(x)$ ?? In an answer they ...
0
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2answers
46 views

Fourier Series Convergence

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a continuous function that is differentiable at the point $x_0$. Prove that $S_n(f(x_0))$ converges to $f(x_0)$, where $S_n$ denotes the partial sums of the ...
0
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1answer
72 views

Calculating $a_n$ in $\sum_{n=1}^\infty a_n \sin(\frac{n \pi}{2})=T_0$

I'm looking to solve the following when $T_0$ is a constant: $$\sum_{n=1}^\infty a_n \sin\left(\frac{n \pi}{2}\right)=T_0$$ If it matters this was reached from the following: ...
1
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1answer
28 views

Fourier series phasor form and sin/cos form

can anyone give me a link on how to convert the forms (from phasor to sine/cos and vice versa)? I am new to this and I can't find the convertion table with a valid explaination.
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1answer
43 views

Parseval's identity does not hold for constucted basis

As part of an exercise, I was asked to show that given an orthonoraml basis $(\varphi_1,\varphi_2,\varphi_3,...)$ in $L_2[-\pi,\pi]$, we can construct an orthonormal basis $(\psi_1,\psi_2,\psi_3,...)$ ...
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0answers
23 views

how manipulating the coefficients vector effects on the result of DFT?

given: calculate: note that the given DFT is from n order and we want to compute DFT's from 2n order. edit: this is my try of B. i don't see where the given DFT is used and how to proceed: ...
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0answers
17 views

Complex exponential argument to a function

In many texts on signal processing, the following notation is used to describe the Fourier transform of a discrete time signal $x$: $$ \hat{X}\left(e^{j\omega}\right) = ...
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0answers
17 views

Fourier Series of Complex Valued Functions

Write the Fourier series of functions in the space of complex valued functions $L^{2}[0,1]$, which we view as periodic functions on $\mathbb{R}$. Specify the coefficients of the expansion and also ...
0
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1answer
34 views

Why do these equalities stand ?

In my notes there is the following theorem: Let $X_k : [a,b] \rightarrow \mathbb{R}$, $k=1, \dots , n$ an orthogonal system of functions and $X: [a,b] \rightarrow \mathbb{R}$, then $\forall c_1, ...
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0answers
38 views

Numeric Evaluation of Double Surface Integral over Greens Function with Singular Points

I'm currently using python to numerically evaluate the follow expression ...
0
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1answer
18 views

Complex Fourier Coefficients by Inspection?

This is the solution to a fourier series problem, of the function $sin(\omega_0t)$: I understand how the author has used Euler's formula to split this function into two exponential terms. However, ...
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1answer
33 views

Where does the imaginary unit dissapear in the Fourier transform of $f(t)= \exp(iat)$?

So I make the Fourier transform of$ f(t)= e^{iat} $on $[- \pi, \pi]$ for some real $a$ and i get: $$a_n=\frac{2a \sin(a \pi)(-1)^n}{\pi(a^2-n^2)}$$ $$b_n=\frac{2i(n\sin(a \pi) (-1)^n)}{\pi(a^2 - ...
0
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0answers
27 views

Exponential to Trigonometric function problem

Here is part of the solution to a fourier series problem involving a rectangular pulse train: I'm following along, and have integrated correctly. But I'm stuck at the second last step - I don't ...
2
votes
2answers
38 views

Getting fourier coefficient by integrating over half the period?

In the book Schaum's Outlines of Analog and Digital Communications solved problem 1.2, the author calculates the fourier coeffecient $C_0$ for the rectangular pulse train: where $a$ is assumed to be ...
2
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1answer
37 views

What is the sum over a shifted sinc function?

What is the sum of a shifted sinc function: $$g(y) \equiv \sum_{n=-\infty}^\infty \frac{\sin(\pi(n - y))}{\pi(n-y)} \, ?$$