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.

learn more… | top users | synonyms

5
votes
1answer
61 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 ...
2
votes
1answer
31 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 ...
0
votes
1answer
23 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 ...
-4
votes
0answers
12 views

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 ...
0
votes
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 ...
3
votes
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}} | ...
0
votes
2answers
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} ...
1
vote
1answer
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
1answer
29 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 ...
1
vote
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 ...
0
votes
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 ...
2
votes
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 ...
0
votes
0answers
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. ...
0
votes
0answers
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 ...
3
votes
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 ...
1
vote
0answers
11 views

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 ...
2
votes
0answers
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 ...
0
votes
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: $$f(x)=\begin{cases} 1 & 0\lt x \lt \pi \\ -1 & \pi \lt x \lt 2\pi \end{cases}$$ ...
0
votes
0answers
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 ...
0
votes
0answers
18 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) ...
0
votes
1answer
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 ...
1
vote
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: ...
0
votes
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
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 ...
2
votes
2answers
94 views

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 ...
0
votes
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 $[0,\pi]$.
1
vote
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: ...
0
votes
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 ...
0
votes
0answers
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 ...
1
vote
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( ...
2
votes
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 ...
0
votes
0answers
10 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.
0
votes
0answers
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?
0
votes
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 ...
-4
votes
1answer
61 views

Find the fourier series for following signal $x(t)$ [closed]

Find the fourier series for following signal $x(t)$ :
0
votes
0answers
17 views

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 ...
1
vote
0answers
25 views

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|} ...
0
votes
2answers
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 ...
6
votes
2answers
417 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
vote
3answers
62 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 ...
0
votes
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
votes
3answers
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 ...
0
votes
0answers
13 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
votes
1answer
43 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 ...