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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Reference on Gibbs phenomenon

I need a reference that explains the following result (also known as the Gibbs phenomenon) Let $g$ be a $2\pi$-periodic function, $C^1$ per pieces (i.e., there exists a partition $x_1 < \cdots ...
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Let $f, \hat{f}\in L^{p}(\mathbb R) \cap C(\mathbb R)$ and locally in $FL^{1}$. Does it vanishes at infinity?

Let \begin{equation*} f, \hat{f}\in L^{p}(\mathbb R)\cap L^{\infty}(\mathbb R) \cap C(\mathbb R),~ 1<p< \infty \end{equation*} and $f$ belongs to locally in $ \mathcal{F}L^{1}(\mathbb ...
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Is there any mistake? Proof related to the Poisson summation formula.

I need to prove the following statement: Let $\varphi \in C(\Bbb R)$ with compact support. Then, $$ \Big \Vert \sum_{k\in \Bbb Z} \varphi(k) e^{ikx} \Big \Vert_{L_1(0,2\pi)} \leq C \Vert ...
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Fourier Analysis Help - $\mathcal L^2$

Let $\{e_k | e_k(x)= e^{ikx}/\sqrt{2\pi\,}\}$ be the orthonormal basis in $\mathcal L^2$ per. I first have to use this basis define two infinite dimensional orthogonal subspaces of $\mathcal L^2$ per. ...
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1answer
22 views

Finding the complex Fourier series… [on hold]

Here's is the problem: And this is my solution for part (i) of the problem. Is it correct? Can I simplify it more than I did? For part (ii), I don't know how to start. So please give me a ...
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1answer
10 views

Fourier Series Even Extension

For $f(x)=-x, 0\leq x \leq 1$, extend $f(x)$ into an even function into $-1 \leq x \leq 0$ and then regard $f(x)$ as a periodic function on $ -\infty < x < \infty $. Find the Fourier ...
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1answer
44 views

Proving that a trigonometric sum is in $L^2$

how can I use Perseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? thank you!
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27 views

Prove that the function $\xi\in R \mapsto {e^{i\cdot \xi\cdot λ}-1\over i\cdot \xi}-λ$ is $C^{\infty}$

Prove that the following function is $C^\infty$ in the point $\xi=0$: $$f:\Bbb R\to\Bbb C:\xi\mapsto {e^{i\cdot\xi\cdot λ}-1\over i\cdot\xi}-λ$$ Any ideas how to prove this? I am trying to think ...
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18 views

Solving differential equation with Fourier-series-inhomogenity

Let $\lambda$ be a real number , $(c_k)$ a complex sequence with $\mid c_k \mid \leq C(1+\mid k \mid)^{-2}$ for all k with a constant $C \geq 0 $. Find all periodic, two times differentiable ...
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2answers
24 views

Fourier series coefficient problem

I am having trouble calculating the $a_n$ coefficient for when $n=1$ for the following function. The function $f(x)$ is periodic with period 2 pi, and is defined on the interval $-\pi<x<\pi$ by ...
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24 views

Finite Fourier Transformation [on hold]

I have to solve the boundary value problem described by $\frac{\partial v}{\partial t} =\frac{\partial^2 v}{\partial t^2} $ using finite Fourier Transform where, $0 < x < 6 $, $t >0$, and ...
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1answer
15 views

is it possible to decompose nonperiodic sinusoidal signal?

Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one. Now, is it possible to decompose nonperiodic ...
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1answer
67 views

If $\gamma$ is irrational, then $\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)\to \int_T f(t)\,dt$

I need to show that $$ \lim_{n\to+\infty}\frac{1}{n}\sum_{k=1}^nf(2\pi k \gamma)=\int_T f(t)\,dt. $$ Here $\gamma$ is any irrational number on the real line and $f(t)$ is any continuous periodic ...
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1answer
38 views

On a property of Fourier coefficients

I need to prove the following: If $(\Phi_n)_{n\ge0}$ is an orthonormal system of integrable functions defined on some interval $[a,b]$, and $(c_n)_n$ is a sequence of reals such that $\sum c_n ...
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1answer
24 views

Approximate identity for periodic integrable functions

I'm studying Fourier analysis now and learned the concept of approximate identity. $$h_n\ge 0,\quad \int_{\mathbb{T}}h_n=1,\quad \lim_{n\to\infty}\int_{\mathbb{T}\setminus[-\delta,\delta]}h_n=0\quad ...
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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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1answer
23 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
30 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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2answers
44 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
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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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
30 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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2answers
20 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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1answer
32 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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3answers
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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1answer
28 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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18 views

Find the number of terms in non-arithmetic series? [closed]

For example part of my serie like this. ax + bx + cx =A As we see, I use x three times. ...
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25 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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26 views

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
37 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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20 views

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
35 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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1answer
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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1answer
22 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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1answer
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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2answers
106 views

What do $a_0$ ,$a_m$ and $b_m$ terms mean in the Fourier series formula?

We know that a Fourier series for signal $x(t)$ is given as $$\frac {a_0} 2 + \sum \limits _{m=1} ^\infty (a_m \cos \frac {2 \pi m t} T + b_m \sin \frac {2 \pi m t} T)$$ So my question is what ...
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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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1answer
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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14 views

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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2answers
44 views

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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1answer
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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13 views

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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15 views

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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2answers
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 ...