Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

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Approximate identity for periodic integrable functions

I'm studying Fourier analysis now and learned the concept of approximate identity. If some sequence {h_n} satisfies the condition above, then holds for any integrable function f on (-π, π]. ...
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An and Bn values are same in fourier series [on hold]

hi friends i could't find the value of En(ie. An) using usual fourier series method. plz find the value of En.also i do not know to choose the f(x) for half range sine and cosine. please give the ...
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2answers
19 views

Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
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29 views

How Fourier decomposition is performed?

The Fourier decomposition explains a time series entirely as a weighted sum of sinusoidal functions and with the Fourier series,it is possible to do it. Suppose a sinusoidal periodic signal is ...
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97 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 ...
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22 views

randomly rough surface by ifft : real output from ifft

I'm trying to generate a randomly rough isotropic surface with predefined roughness amplitudes (standard deviation of heights). Suppose I have the absolute values of fourier components of the surface ...
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1answer
29 views

Partial Fourier series for $L^p$ functions, $p\ge 1$

Let $f$ be an $L^2$ function on the unit circle $f \in L^2(S^1, d \theta)$. This is equivalent to giving a Fourier series $\sum_{n \in \mathbb{Z}} a_n e^{i n \theta}$ with $\sum_{n \in \mathbb{Z}} | ...
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Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
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43 views

Integrate $ \int^{\pi}_{-\pi} (\pi^2-x^2)\sin nx \ dx$

Consider the function $f:(-\pi,\pi)\to\mathbb{R}$ be defined as $x \mapsto (\pi+x)(\pi-x)$ Compute the fourier series of $f$ So far, I've worked out $a_o$ by: \begin{equation} a_o = \frac{1}{\pi} ...
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1answer
39 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...
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1answer
14 views

fourier series notation question

Find the fourier series for the given function $$f(x)=-x \quad \text{for } -L\le x < L, f(x+2L)=f(x)$$ this is a question from my book, and im just wondering about one thing and that is what does ...
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29 views

question on Fourier Transformation

I have to find the Fourier Sine transform of $f(x)=1$ when $|x|<a$ and $f(x)=0$ when $|x|\ge a$ and hence show that $$\int_0^\infty {\sin(t)\over t} dt =\pi/2$$ and $$\int_0^\infty ...
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61 views
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28 views

Uniform convergence of real function

Let $f:[a,b] \to \mathbb{C}$ where $0 < a < b < 2 \pi$ be defined by $f(x) = \sum_{k=1}^{\infty} \frac{exp(ikx)}{k}$. Show that $f$ converges uniformly in $[a,b]$. The problem is that I ...
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29 views

Fourier series sketching

Whenever I am asked to draw fourier series, is it correct to first draw the function on the interval first (in this case 0<= x < pi), then extend the the graph to the desired interval ...
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413 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 ...
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1answer
360 views

Fourier and half range series for $\sin x$

Expand in Fourier series of $f(x) = \sin x$ for $0<x<l$. Deduce the result \[ \frac1{1 \cdot 3} - \frac{1}{3\cdot5} +\frac{1}{5\cdot 7} - \cdots = \pi-\frac{2}{4}. \] Obtain half range ...
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1answer
25 views

Partial sum of Fourier series of square wave

Let $f$ be a $2π$ -periodic square wave function so that $$f\, = -1 \quad -π \le x<0$$ $$f=1 \qquad 0 \le x< π$$ $S_{2n-1}(x)$ is the $(2n-1)st$ Fourier polynomial of $f$. Prove ...
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1answer
30 views

Fourier series on $e^x$ periodic $[-1,1]$

I got the Fourier series as $(e-e^{-1})(\frac 1 2+\sum \limits _{n=1} ^\infty \frac {(-1)^n(\cos(n\pi x)-n\pi \sin(n\pi x)} {1+n^2\pi^2})$. Although I've seen the answer online as being $\sum \limits ...
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1answer
382 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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34 views

Convergent series of a functional series

Is there any way to prove that the functional series $$g(t)=\sum\limits_{n=1}^{\infty} b_n\,n\,a\sin(n\,c\,t)$$ is uniform convergent, given $$b_n=\int_{-\pi}^{\pi} f(x)\,\sin(nx)dx$$ $f\in ...
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24 views

Fourier series of cotangent

I have found the Fourier series of $\cos(ax)$ and i get: $$\frac{ \sin(a \pi)}{a\pi}\left[ \left(\frac{1}{2a^2}\right)- \sum_1^\infty \frac{(-1)^n \cos(nx)}{n^2-a^2}\right]$$ How can I deduce the ...
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1answer
35 views

Problem with Fourier series

I'm trying to find the Fourier series of the function defined on the interval $(-2,2)$ $$ f(x)=\begin{cases} 0,& \,\,\, |x| <1 \\ 1, & \,\,\, 1<|x|<2 \end{cases} $$ This should be ...
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How to choose $f\in C_{c}^{\infty}(\mathbb R)$ so that $ \hat{g}\in \ell^{1}(\mathbb Z)$, where $g(x)=f(x+2\pi)$?

Suppose $K$ is compact proper subset of $[0, 2\pi]$ with the property $K\subset V \subset [0, 2\pi]$ where $V$ is open . My Question: Is it possible to choose $f\in C_{c}^{\infty}(\mathbb R)$ such ...
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Division of two series expansions

I have the two functions $u(x)$ and $v(x)$, both of which have known basis expansions $u(x) = \sum_n a_n f_n(x)$, $v(x) = \sum_n b_n f_n(x)$. I would like to calculate the function ...
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1answer
36 views

Find the Fourier Series of the following function.

I have been given the following question. "The function $f(x)$ is odd, has a period $2\pi$ and satisfies: $$f(x)=\begin{cases} 1 & 0\lt x \lt \pi \\ -1 & \pi \lt x \lt 2\pi \end{cases}$$ ...
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41 views

Fourier Expressions

In the Fourier series, what are all the ways we can express: $\displaystyle\sin\left(\frac{n\cdot\pi}2\right)$ $\displaystyle\cos(n\cdot\pi)$ I know we can express as $(-1)^{(n+1)}$, and as ...
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32 views

Fourier series of $f(x)=x^2+x$ ,$x\in(-\pi,\pi)$

Could you please help me solve this problem: I need use Fourier series of $f(x)=x^2+x$ ,$x\in(-\pi,\pi)$ to prove that $\sum_{n\ge1} \frac 1{n^2}= \frac{\pi^2}6$. I calculated the Fourier series: ...
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Mode of convergence for partial Fourier series in $B( L_p[-\pi; \pi ])$, $p \in [1; \infty]$

Which mode of convergence takes place, strong, weak, or in norm? If we have sequence of continuous linear operators in $L_p[-\pi; \pi]$: $(A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) ...
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1answer
21 views

Sine Fourier series and smooth function.

I was reading through my text on PDEs and came across a theorem (or perhaps Lemma) that states: "For any smooth function $g_1(y)$ with $g_1(0) = g_1(h) = 0$, it can be expressed as a Fourier sine ...
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1answer
24 views

Asymptotics of certain integrals

For $n\in\mathbb N$ put $\displaystyle a_n=\int_0^{2\pi}\int_0^{2\pi}\frac{\cos2n(x-y)}{\sqrt{|x-y|}}dxdy$ and $b_n=\displaystyle \int_0^{2\pi}\int_0^{2\pi}\frac{\sin2n(x-y)}{\sqrt{|x-y|}}dxdy$. Can ...
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What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

Let us take an example, a white ray (which is composed of bunch of frequency components) is passed through a prism, the ray gets split (decomposed) into its elementary vibgyor colours (i.e.different ...
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1answer
17 views

How to find the cosine series when solving a PDE with Dirichlet conditions?

Suppose I have to solve $\sum_{n=0}^{\infty} A_n \cos(\frac{(n+1/2)\pi x}{L}) = x $ from $0$ to $L$. If I we want to find $A_n$ my professor uses the formula for a cosine series: ...
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45 views

Evaluate two dimensional frequency domain for single point

I need to compute one specific value in the original domain from the 2D frequency domain data I have. I can't just use IFFT for a whole set for performance reasons. I know how to do this in 1D by ...
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71 views

How could I continue to show the inequality?

Let $g: [0, \pi]\rightarrow \mathbb{R}$ a $C^{\infty}$ function for which the following stands: $$g(0)=0 \ \ , \ \ g(\pi)=0$$ I have to show that $$\int_0^{\pi}g^2(x)dx \leq ...
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1answer
31 views

How can we find the sums ?

We have the function $$g: [0, 2\pi] \rightarrow \mathbb{R} \\ g(x)=\frac{(x-\pi)^2}{4}, x \in [0, 2\pi]$$ I found that the Fourier series of $g$ is the following: $$g \sim ...
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1answer
20 views

Orthognality of Trigonometric system [closed]

Considering the system $$\{1, \sin(x), \cos(x), \sin(2x), \cos(2x), \dots\}$$ Is the system orthogonal on any interval of length $\pi$?why? I have proved that it's not orthogonal on $[0,\pi]$.
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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 ...
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Using Fourier transform to compute Fourier series.

I have found an exercise on a signal processing book that asks to compute the Fourier series of a function by using its Fourier Transform, let: $$ x(t) = \sum_{n=-\infty}^{\infty} \Lambda \left( ...
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Fourier complex coefficients using derivative property

Consider the periodic function with period 2 given by $$ f(x) = 2x, 0 \leq x \leq 1 $$ $$f(x) = 2x -4, 0 < x \leq 2$$ If c_k denote the k-th complex fourier coefficient, we know, using the ...
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56 views

Question regarding Fourier Series

Things I understand (scroll down to see question in bold): Let $T$ be the function's period Let $w_0 = \frac{2π}{T}$ A function $x(t)$ can be written as the sum of its even and odd parts, that is ...
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inverse fourier transform of unit impulse function of omega

What is the inverse fourier transform of the unit impulse function of omega. Sorry I've not got the symbol in my phone. It Should looks like §(W).. Sorry for the special symbols.
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fourier transform of real and odd symmetric signal

What is the fourier transform of a real and odd symmetric signal.. Is it real and non negative or just real. Some of my friends say it's imaginary and some say it's complex.. What is the answer?
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83 views

Find $\sum_{n \ge 1} 1/n^2$ using the Fourier expansion of $f(x) = x$

The strategy I have been asked to take, is to show that Fourier coefficients of the function $f(x) = x$ on $[0, 1]$ are up to a constant equal to $1/n^2$.Then I should apply the norm ...
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Find the signal $m(t)$ by $C_k = j\delta[k-1]-j\delta[k+1]+\delta[k-3]+\delta[k+3] , w_0=4\pi$

I am trying to find the signal by the coefficients : $$C_k = j\delta[k-1]-j\delta[k+1]+\delta[k-3]+\delta[k+3] , w_0=4\pi$$ What I tried is to use some features of $cos$ but I dont know if its the ...
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Determine the type of signal $x(t)$ given by his coefficients [ Fourier Series ]

I want to determine if the signal $x(t)$ is even real $\frac{dx(t)}{dt}$ is even By the following coefficients $$C_k =\begin{cases} 2 & k=0 \\ j(\frac{1}{2})^{|k|} ...
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44 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 ...
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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
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73 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 ...